Posted October 8, 2003

Last modified January 27, 2004

Colloquium Questions or comments?

Anton Deitmar, Mathematical Sciences Department, University of Exeter

Class number asymptotics in degree 3

Visit supported by Visiting Experts Program in Mathematics, Louisiana Board of Regents. LEQSF(2002-04)-ENH-TR-13

Posted September 2, 2003

Last modified October 4, 2021

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3:40 pm – 4:30 pm Lockett 243
Yuri Antipov, Mathematics Department, LSU

Functional-difference equations and applications

Posted August 26, 2003

Last modified October 4, 2021

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3:40 pm – 4:30 pm 243 Lockett Hall
Amha Lisan, Mathematics Department, LSU

Transitive flows and associated congruences and groups

Refreshments will be served in the lounge one half hour before the talk.

Posted September 4, 2003

Last modified October 4, 2021

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3:40 pm – 4:30 pm 243 Lockett Hall
Kalyan B. Sinha, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore

Non-commutative Analysis

Refreshments will be served in the lounge one half hour before the talk.

Visit supported by Visiting Experts Program in Mathematics, Louisiana Board of Regents.
LEQSF(2002-04)-ENH-TR-13

Posted September 10, 2003

Last modified January 27, 2004

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3:40 pm – 4:30 pm 243 Lockett Hall
Marco Schlichting, Universität Essen, Germany

Hermitian K-theory and Algebraic Bott Periodicity

Refreshments will be served in the lounge one half hour before the talk.

Visit supported by Visiting Experts Program in Mathematics, Louisiana Board of Regents.
LEQSF(2002-04)-ENH-TR-13

Posted August 26, 2003

Last modified October 4, 2021

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3:40 pm – 4:30 pm 243 Lockett Hall
Michael M. Tom, Mathematics Department, LSU

Kadomtsev-Petviashvili and RLW-KP models

Refreshments will be served in the lounge one half hour before the talk.

Posted October 14, 2003

Last modified October 4, 2021

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3:40 pm – 4:30 pm 243 Lockett Hall
Horst Beyer, Max Planck Institute for Gravitational Physics, Golm, Germany, and Dept. of Mathematics, LSU

On the Completeness of the Resonance Modes of the Pöschl–Teller Potential

Refreshments will be served in the lounge one half hour before the talk.

Posted September 24, 2003

Last modified October 4, 2021

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3:40 pm – 4:40 pm 243 Lockett
Yuan Wang, Florida Atlantic University

Input-to-State Stability of Nonlinear Control Systems

Refreshments will be served in the lounge one half hour before the talk.

Visit supported by Visiting Experts Program in Mathematics, Louisiana Board of Regents.
LEQSF(2002-04)-ENH-TR-13

Posted December 14, 2003

Last modified January 27, 2022

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2:30 pm – 3:30 pm 285 Lockett
David Levin, University of Utah

Modern Topics in Random Walks

I will survey some of my work relating to random walks: dynamical random walks and reconstruction of sceneries visited by a random walk. Dynamical random walks are easily constructed "coin tossing" analogues of infinite dimensional diffusions. We discuss the existence of times where atypical random walk behavior is seen, and give connections to the Ornstein-Uhlenbeck process on Wiener space. (Joint work with Khoshnevisan and Mendez.) Hidden Markov chains are widely applicable probabilistic models: a noisy function of an underlying stochastic process is seen, while the process itself is unobserved. We describe such models where an unknown scenery is explored by a hidden random walk, and discuss when reconstruction of this underlying scenery is possible. (Joint work with Pemantle and Peres.)

Refreshments at 2:00 in the Keisler Lounge.

Posted December 17, 2003

Last modified March 3, 2021

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2:30 pm – 3:30 pm
Patricia Hersh, University of Michigan

A GL_n(q) analogue of the partition lattice and discrete Morse theory for posets

Coffee at 2:00 in the Keisler Lounge

Posted January 8, 2004

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2:40 pm – 3:30 pm Lockett 285
Diane Maclagan, Stanford University

Toric Hilbert schemes

Abstract: Toric Hilbert schemes have broad connections to other areas of mathematics, including optimization, geometric combinatorics, algebraic geometry, and representations of finite groups and quivers. They parameterize all ideals in a a polynomial ring with the simplest possible multigraded Hilbert function. I will introduce these objects, and discuss some of the applications.

Posted January 15, 2004

Last modified March 3, 2021

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3:30 pm – 4:30 pm 285 Lockett
Scott Baldridge, Louisiana State University

Elementary Mathematics for Teachers: A mathematician's perspective

This talk describes a mathematics course, designed by mathematicians, for prospective elementary teachers. I will describe three unique features of the course: the extensive use of the Primary Mathematics books from Singapore, the idea of a “teaching sequence”, and the use of “teacher's solutions” in class and in homework. The course is based on a new textbook I wrote with T. Parker: Elementary Mathematics for Teachers. The goal of the textbook and the course is to present the mathematics clearly and correctly while keeping the focus on material that elementary school teachers will be addressing in their classrooms.

Refreshments in the Lounge at 3:00

Posted January 15, 2004

Last modified January 16, 2004

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3:30 pm – 4:30 pm 285 Lockett
Scott Baldridge, Louisiana State University

Seiberg-Witten invariants of 4--manifolds with circle actions, with applications to symplectic topology.

Ever since the introduction of Donaldson invariants in the early 1980's, efforts to calculate diffeomorphism invariants of 4-manifolds centered upon large classes of smooth manifolds that have some additional structure. One such class of manifolds thought to have promise was 4-manifolds with effective circle actions, but the extra structure given by such manifolds turned out to be insufficient for calculating Donaldson invariants. However, it is possible to calculate their Seiberg-Witten invariants. In this talk I will give an overview of the Seiberg-Witten invariants and describe formulas for calculating the Seiberg-Witten invariant of 4-manifolds with circle actions. I will also discuss some results on the topology of symplectic 4-manifolds which follow from those calculations.

Refreshments in the Lounge at 3:00

Posted January 15, 2004

Last modified January 26, 2004

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2:30 pm – 3:30 pm Lockett 237
Malabika Pramanik, University of Wisconsin–Madison

Averaging and maximal operators for curves in R^3

We consider the L^p regularity of an averaging operator over a curve in R^3 with nonvanishing curvature and torsion. We also prove related local smoothing estimates, which lead to L^p boundedness of a certain maximal function associated to these averages. The common
thread underlying the proof of these results is a deep theorem of T. Wolff on cone multipliers. This is joint work with Andreas Seeger of University of Wisconsin, Madison.

Refreshments in Lounge at 2:00

Posted January 15, 2004

Last modified January 27, 2004

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3:30 pm – 4:30 pm Lockett 285
Susan Wilson, Michigan State University, College of Education

Reforming Mathematics Education; Lessons from California

Refreshments in the Lounge at 3:00

Posted January 29, 2004

Last modified October 5, 2023

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3:40 pm – 4:30 pm Lockett 285
Pramod Achar, University of Chicago

Equivariant K-theory of the unipotent variety

The equivariant K-theory of the unipotent variety in a complex algebraic group has two natural bases, one indexed by the set $\Lambda^+$ of dominant weights, the other by the set $\Omega$ of irreducible representations of centralizers of unipotent elements. Lusztig's work on cells in affine Weyl groups led him to conjecture that the change-of-basis matrix relating these two bases is upper-triangular, and that in particular there is a natural bijection between $\Lambda^+$ and $\Omega$. This question has been treated in the work of Bezrukavnikov, Ostrik, Xi, and others. I will discuss an approach to the problem that, in the case of $GL(n)$, results in an explicit combinatorial algorithm for computing the bijection. I will also discuss connections to the Springer correspondence, duality, and other topics.

Posted January 29, 2004

Last modified February 11, 2004

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2:40 pm – 3:30 pm Lockett 243
Tatyana Foth, University of Michigan

Quantization, Kahler manifolds, and automorphic forms

Abstract:

I shall talk about results and problems that appear in the interplay between three subjects:

1. quantization (which can be regarded as an attempt to construct a finite-dimensional representation of the Lie algebra of smooth functions on a compact symplectic manifold with the Poisson bracket);

2. varying Kahler structure on a compact Kahler manifold with the symplectic form being kept fixed;

3. holomorphic automorphic forms on a bounded symmetric domain in C^n (for example, on the open unit ball in C^n with the Bergman metric).

Refreshments served in Keisler Lounge at 2pm.

Posted February 4, 2004

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3:40 pm – 4:30 pm Lockett 285
Irina Mitrea, Cornell University

On the Spectral Radius Conjecture in Two Dimensions

Posted February 16, 2004

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3:40 pm – 4:30 pm Lockett 285
Ruhai Zhou, University of North Carolina at Chapel Hill

Analysis and computations of nematic polymers

Posted February 2, 2004

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3:40 pm – 4:30 pm Wednesday, March 10, 2004 Lockett 285
Serge Lang, Yale University
Member, National Academy of Sciences; Recipient, Frank Nelson Cole Prize in Algebra

The Heat Kernel and Theta Inversion Formulas

Visit supported by Visiting Experts Program in Mathematics, Louisiana

Board of Regents LEQSF(2002-04)-ENH-TR-13

Posted January 30, 2004

Last modified March 1, 2004

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3:40 pm – 4:30 pm Lockett 285(Originally scheduled for Thursday, February 26, 2004)

Carl Mueller, University of Rochester

Properties of the Random String

Visit supported by Visiting Experts Program in Mathematics, Louisiana

Board of Regents LEQSF(2002-04)-ENH-TR-13

Posted January 29, 2004

Last modified October 4, 2021

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3:40 pm – 4:30 pm Lockett 285
Stig Larsson, University of Gothenburg, Chalmers University

The finite element method for a linear stochastic parabolic partial differential equation driven by additive noise

Visit supported by Visiting Experts Program in Mathematics, Louisiana

Board of Regents LEQSF(2002-04)-ENH-TR-13

Posted March 4, 2004

Last modified March 20, 2004

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3:40 pm – 4:30 pm Lockett 285
Mark Meerschaert, University of Nevada, Reno

The Fractal Calculus Project

Posted March 4, 2004

Last modified February 14, 2023

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3:40 pm – 4:30 pm Lockett 285
Loukas Grafakos , University of Missouri, Columbia

Calderón's program, the bilinear Hilbert transforms, and the Carleson-Hunt theorem

Posted January 27, 2004

Last modified October 4, 2021

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3:40 pm – 4:30 pm Lockett 285
John Willis, Cambridge University
Fellow, Royal Society of London (FRS)

Bounds for the Effective Constitutive Relation of a Nonlinear Composite

For a nonlinear composite, a bound on its effective energy density does not induce a corresponding bound on its constitutive relation, because differentiating a bound on a function does not automatically bound its derivative. In this work, a method introduced by G.W. Milton and S.K. Serkov for bounding directly the constitutive relation is refined by employing a linear comparison material, in a similar way that Talbot and Willis introduced such a material to obtain bounds of “Hashin–Shtrikman” type for the effective energy of a nonlinear composite. The original Milton–Serkov approach produces bounds with a close relationship to the classical energy bounds, of Voigt and Reuss type. The bounds produced in the present implementation are closely related to bounds of Hashin–Shtrikman type for the composite. It is demonstrated by means of examples that the approximate constitutive relation that is obtained by differentiating the energy bound can be on the boundary of the bounding set, obtained here, for the exact constitutive relation, but a simple counterexample is presented to show that this is not always the case.

(This talk reports on joint work with D R S Talbot.)

Visit supported by Visiting Experts Program in Mathematics, Louisiana
Board of Regents LEQSF(2002-04)-ENH-TR-13

Posted March 2, 2004

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Simon Gindikin, Rutgers University

Complex geometry and complex analysis on real symmetric spaces

Posted March 4, 2004

Last modified March 2, 2021

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3:40 pm – 4:30 pm Lockett 285
Fritz Gesztesy, University of Missouri, Columbia

On the spectrum of Schrödinger operators with quasi-periodic algebro-geometric KdV potential

Refreshments will be served in Keisler lounge at 3pm.

Posted March 3, 2004

Last modified March 20, 2004

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3:40 pm – 4:30 pm Lockett 285
Alberto Setti, Università dell'Insubria, Como

Maximum principle on Riemannian manifolds: an overview

Refreshments will be served in Keisler lounge at 3pm.

Visit supported by Visiting Experts Program in Mathematics, Louisiana Board of Regents.

LEQSF(2002-04)-ENH-TR-13

Posted February 11, 2004

Last modified October 4, 2021

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3:30 pm – 4:30 pm Lockett Hall 285
Thomas Kerler, Ohio State University

Topological Quantum Field Theories

TQFT can be thought of as measures on topological spaces that behave nicely functorially under the gluing of spaces. We will motivate the formalism and give an elementary construction of TQFTs starting from nothing more than the basic Seifert–van Kampen Theorem. From there we will expand on more general TQFT properties, formalisms and constructions, sketch some problems of finiteness and quantization, and present a few typical applications of TQFTs.

Refreshments will be served in the lounge one half hour before the talk.

Visit supported by Visiting Experts Program in Mathematics, Louisiana Board of Regents. LEQSF(2002-04)-ENH-TR-13

Posted March 4, 2004

Last modified March 28, 2004

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3:40 pm – 4:30 pm Lockett 285
Salah-Eldin Mohammed, Southern Illinois University, Carbondale

The Stable Manifold Theorem for Stochastic Partial Differential Equations

Posted March 5, 2004

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3:40 pm – 4:30 pm Lockett 285
David Kirshner, Department of Curriculum and Instruction, LSU

What Ails Elementary Algebra Education: Historical, Psychological, and Philosophical Perspectives

Refreshments in Keisler Lounge at 3pm

Posted March 23, 2004

Last modified March 3, 2021

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3:30 pm – 4:30 pm
Ronald Stanke, Baylor University

Differential Operators, SL(2,R) Invariance and Special Functions

Refreshments in the Lounge one half hour before talk. Visit supported by Visiting Experts Program in Mathematics, Louisiana Board of Regents, LEQSF(2002-04)-ENH-TR-13.

Posted March 28, 2004

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3:40 pm – 4:30 pm Lockett 281
Boris Rubin, The Hebrew University of Jerusalem

Selected problems of integral geometry and small denominators on the sphere

Posted April 7, 2004

Last modified April 25, 2004

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3:30 pm – 4:30 pm 285 Lockett Hall
Paul Saylor, University of Illinois

What Does Radar Have to Do with Solving Sets of Linear Equations?

Posted March 31, 2004

Last modified April 26, 2004

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3:40 pm – 4:30 pm Lockett 281
Thierry Lévy, École normale supérieure and CNRS

Two-dimensional Yang-Mills theory is almost a topological field theory

Posted March 16, 2004

Last modified April 13, 2004

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3:30 pm – 4:30 pm 285 Lockett
Paulo Lima-Filho, Texas A&M

Applications of operads and ternary trees to polynomial map

Using ternary trees we build operads and use them to define a family of ideals in the (non-commutative) algebra generated by pointed ternary trees. These constructions have several applications to iterations of polynomial maps and conjectures in algebraic geometry. This is an essentially self-contained talk, accessible to a general audience and to graduate students.
Refreshments will be served in the lounge one half hour before the talk.

Visit supported by Visiting Experts Program in Mathematics, Louisiana Board of Regents.

LEQSF(2002-04)-ENH-TR-13

Posted April 3, 2004

Colloquium Questions or comments?

3:30 pm – 4:30 pm 285 Lockett Hall
Milton C. Lopes Filho, Penn State University and UNICAMP, Brazil

Open problems on mathematical hydrodynamics

Abstract: Mathematical hydrodynamics is primarily concerned with the behavior of solutions of the incompressible Euler and Navier-Stokes equations. These nonlinear systems of PDEs have a rich mathematical structure that keeps hydrodynamics a topic of current interest in mathematical research. One illustration of the cogency of this topic is the choice of the singularity problem for the Navier-Stokes equations as one of the seven Millenium Prize Problems. Problems in the field of mathematical hydrodynamics often reduce to proving that solutions of the incompressible flow equations behave as actual fluids are known to behave. In this talk we will examine a few instances where the known behavior of real fluids leads to open problems on the behavior of solutions of the incompressible flow equations, exploring the power, and the limitations, of modern analytic techniques used in the treatment of these problems.

Refreshments will be served in the lounge one half hour before the talk. Visit supported by Visiting Experts Program in Mathematics, Louisiana Board of Regents. LEQSF(2002-04)-ENH-TR-13

Posted May 13, 2004

Colloquium Questions or comments?

3:30 pm – 4:30 pm Lockett 285
Cun-Quan Zhang, West Virginia University

Some Results about Integer Flows

Refreshments in Keisler Lounge at 3pm

Posted September 17, 2004

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3:30 pm – 4:30 pm Lockett 277
Daniel Sage, Mathematics Department, LSU

Group and quantum group actions on algebras and composite materials

Posted September 23, 2004

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3:30 pm – 4:30 pm Lockett 277
Ricardo Estrada, Mathematics Department, LSU

On the regularization of generalized functions

Posted September 30, 2004

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3:30 pm – 4:30 pm Lockett 239
Mark Davidson, Mathematics Department, LSU

Generating functions associated to Highest Weight Representations

Posted October 20, 2004

Last modified March 2, 2021

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3:40 pm – 4:30 pm Lockett 277
Fernando Rodriguez-Villegas, University of Texas at Austin

The Many Aspects of Mahler's Measure

Visit supported by Visiting Experts Program in Mathematics, Louisiana Board of Regents LEQSF(2002-04)-ENH-TR-13. Refreshments will be served in the lounge at 3pm.

Posted November 29, 2004

Last modified March 3, 2021

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3:40 pm – 4:30 pm 277 Lockett
Christopher Leininger, Columbia University
Candidate for Assistant Professor Position in Topology

Teichmüller disks in geometry and topology

Refreshments in Keisler Lounge at 3:00 PM

Posted December 13, 2004

Last modified January 5, 2005

Colloquium Questions or comments?

3:40 pm Lockett 285
Selim Esedoglu, UCLA
Candidate for Assistant Professor Position in Applied Math

Threshold dynamics for the piecewise constant Mumford-Shah

Posted January 5, 2005

Last modified January 12, 2005

Colloquium Questions or comments?

3:40 pm Lockett 285
Burak Aksoylu, University of Texas at Austin, Institute for Computational Eng. and Science
Candidate for Assistant Professor Position in Scientific Computation

Local refinement and single/multi level preconditioning with applications in biophysics, computer graphics, and geosciences

Posted December 13, 2004

Last modified January 14, 2005

Colloquium Questions or comments?

3:40 pm Lockett 285(Originally scheduled for Thursday, January 20, 2005, 3:40 pm)

Valeriy Slastikov, Carnegie Mellon University
Candidate for Assistant Professor Position in Applied Math

Geometrically Constrained Walls

Posted January 24, 2005

Last modified January 27, 2005

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1:40 pm Lockett 138
Tara Brendle, Mathematics Department, Cornell University
Candidate for Assistant Professor Position in Topology

Mapping class groups and complexes of curves

Posted January 24, 2005

Last modified January 26, 2005

Colloquium Questions or comments?

3:40 pm Lockett 285
Brendan Owens, Mathematics Department, Cornell University
Candidate for Assistant Professor Position in Topology

Four-manifolds with prescribed boundary and applications to knot theory

Posted January 21, 2005

Last modified January 26, 2005

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3:40 pm Lockett 285
Masako (Marta) Asaeda, Mathematics Department, University of Iowa
Candidate for Assistant Professor Position in Topology

Generalizations of Khovanov homology

Posted January 26, 2005

Last modified January 31, 2005

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3:40 pm Lockett 285
Hongyu He, Mathematics Department, LSU

Tensor Products of Oscillator Representations

Posted January 26, 2005

Last modified February 3, 2005

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3:40 pm Lockett 285
Ai-Ko Liu, U. C. Berkeley, Mathematics
Candidate for Assistant Professor in Geometric Analysis

Cosmic String, Family Seiberg-Witten theory and Harvey-Moore Conjecture

Posted January 31, 2005

Last modified February 1, 2005

Colloquium Questions or comments?

3:40 pm Lockett 285
Justin Sawon, Department of Mathematics, SUNY at Stony Brook
Candidate for Assistant Professor in Topology

Derived equivalence of algebraic varieties

Posted January 31, 2005

Last modified February 1, 2005

Colloquium Questions or comments?

3:40 pm Lockett 285
Yann Rollin, MIT, Department of Mathematics
Candidate for Assistant Professor in Geometric Analysis

Construction of Kaehler surfaces with constant scalar curvature

Posted February 16, 2005

Last modified March 3, 2021

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3:30 pm – 4:30 pm Lockett 285
Leticia Barchini, Oklahoma State University at Stillwater

Positivity of Zeta distributions and small representations

We study positivity of zeta distributions associated to noneuclidean Jordan algebras. The values of the complex parameter s for which the distributions are positive is determined. A theory analogous to the classical theory of Riesz distributions and Wallach set is developed. We calculate the distributions when they are positive. For each value of s for which the zeta distribution is positive we build a Hilbert space. These Hilbert spaces are representations spaces for the conformal groups of the Jordan algebras involved. In this way we build an explicit family of small (non holomorphic) representations.

Posted February 21, 2005

Colloquium Questions or comments?

2:00 pm – 2:50 pm Lockett 277
Stephen(Steve) Bryson, NASA Ames Research Center
Candidate for Assistant Professor Position in Scientific Computation

Central Methods for Balance Laws

Posted February 23, 2005

Last modified February 25, 2005

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3:40 pm Lockett 285
Yaniv Almog, Department of Mathematics, Technion I.I.T.
Candidate for Assistant Professor Position in Applied Math

Abrikosov lattices in finite domain

Posted February 23, 2005

Last modified March 4, 2005

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3:40 pm Lockett 285
Petr Plechak, Mathematics Institute, University of Warwick
Candidate for Associate Professor Position in Scientific Computation

Approximation and coarse-graining of stochastic lattice systems

Posted March 1, 2005

Last modified March 3, 2021

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3:40 pm – 4:30 pm Lockett 285
Brian Hall, University of Notre Dame

The range of the heat operator

I will consider the heat operator both on Euclidean space and on certain symmetric manifolds such as spheres and hyperbolic spaces. I will begin by reviewing the heat equation itself, which describes how temperature distributions evolve in time. Then I will consider the following question: What class of functions does one obtain by taking an arbitrary initial temperature distribution and then running the heat equation for some fixed time t? The heat equation is very smoothing: the diffusion of heat smoothes out any rough edges in the initial temperature distribution. Thus the functions obtained must be very nice ones and I will characterize them in terms of their analyticity properties. My talk will follow a recent reprint, available at www.arxiv.org/abs/math.DG/0409308.

Posted February 22, 2005

Last modified March 28, 2005

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3:30 pm – 4:30 pm Lockett 285
Alexander Figotin, University of California at Irvine

Nonlinear dispersive media

We study the basic properties of the Maxwell equations for nonlinear inhomogeneous media. Assuming the classical nonlinear optics representation for the nonlinear polarization as a power series, we show that the solution exists and is unique in an appropriate space if the excitation current is not too large. The solution to the nonlinear Maxwell equations is represented as a power series in terms of the solution of the corresponding linear Maxwell equations. This representation holds at least for the time period inversely proportional to the appropriate norm of the solution to the linear Maxwell equation. We derive recursive formulas for the terms of the power series for the solution including an explicit formula for the first significant term attributed to the nonlinearity. Coffee will be served in the Keisler Lounge at 3:00pm

Posted March 28, 2005

Last modified April 7, 2005

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3:40 pm – 4:30 pm Lockett 285
Xiao-Song Lin, University of California Riverside

A folding problem of polygonal arcs in 3-space

Posted March 1, 2005

Last modified October 4, 2021

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3:40 pm – 4:30 pm Lockett 241
Dror Bar-Natan, University of Toronto

Local Khovanov Homology

Board of Regents LEQSF(2002-04)-ENH-TR-13

Posted April 15, 2005

Colloquium Questions or comments?

11:00 am Coates 202
Martin Olbrich, Universität Göttingen

Automorphic distributions and dynamical zeta functions

Posted March 15, 2005

Last modified January 27, 2022

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11:00 am – 12:00 pm 104 Hill Memorial Library(Originally scheduled for Tuesday, April 12, 2005)

Roger Howe, Yale University

More than Mathematics for Teaching

There has been substantial agreement among professionals concerned with mathematics education that the mathematical skills of the teaching corps needs to be substantially upgraded. This is an urgent project which will require huge effort. At the same time we work on this, however, we should not lose sight of the fact that there are certain jobs, in particular, mathematics supervisor, which require substantially higher levels of expertise than classroom teaching. Furthermore, the system as a whole needs means to improve its understanding of both mathematics teaching practice and curriculum. This talk will discuss these issues, and some possible means for addressing them.

Posted September 2, 2005

Last modified October 4, 2021

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3:40 pm – 4:30 pm Lockett 235
Jacob Rubinstein, Indiana University

The weighted least action principle

Posted September 28, 2005

Last modified October 4, 2021

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3:40 pm – 4:30 pm Lockett 235(Originally scheduled for Thursday, October 20, 2005, 3:00 pm)

Richard Anderson, Louisiana State University (Emeritus)
Emeritus Boyd Professor

My Three Lives in Mathematics

Refreshments will be served in the James E. Keisler Lounge one half hour before the talk.

Posted November 30, 2005

Last modified January 27, 2022

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Changyou Wang, University of Kentucky
Candidate for an Associate/Full Professor Position in Partial Differential Equations

Calculus of variations in L-infinity and Aronsson's equation

In this talk, I will discuss the basic issues for L-infinity variational problems, where one considers minimization problem of the supernorm functional:
$$F(u,\Omega)=\esssup H(x,u(x),\nabla u(x)), u\in W^{1,\infty}(\Omega).$$
We will survey some recent developments on: (1) the existence of absolute
minimizers (AMs), (2) the PDE characterization of AMs (i.e., Aronsson's
equation or AE), (3) the relationship between AM and AE, and (4) regularity
and uniqueness of AE.

We will also discuss its connection with image interpolation, random
game theory.

Posted January 13, 2006

Last modified January 16, 2006

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 284
David Damanik, California Institute of Technology
Candidate for Full Professor Position

Structures of intermediate complexity and quantum dynamics

Abstract: We discuss the spreading properties of quantum particles in structures of intermediate complexity. Examples of interest include quasicrystals. We carry out a complete analysis for the Fibonacci Hamiltonian, which is the most prominent object in the mathematics and physics literature on quasicrystals.

Posted February 3, 2006

Colloquium Questions or comments?

3:00 pm – 4:00 pm Lockett 284
Dana Scott, Carnegie Mellon University

Parametric Sets and Virtual Classes

Posted February 15, 2006

Last modified March 3, 2021

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3:30 pm – 4:30 pm Lockett 284
Dorin Dutkay, Rutgers University

Wavelets and self-similarity

In the past twenty years the theory of wavelets has proved to be extremely successful, with important applications to image compression and signal processing. The theory involves the construction of orthonormal bases in euclidean spaces generated by translations and dilations. A key feature of these constructions is the property of self-similarity. We exploit this property and, using operator algebra methods, we offer a wider perspective on the subject. We show how techniques from the theory of wavelets can be used in many other contexts such as fractals, dynamical systems, or endomorphisms of von Neumann algebras. Thus, we can construct rich multiresolution structures with scaling functions and wavelets on fractals, solenoids, super-wavelets for Hilbert spaces containing L^2(R), or harmonic bases on fractal measures.

Posted February 17, 2006

Last modified January 27, 2022

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11:40 am – 12:30 pm Johnston 338
Susanne Brenner, Department of Mathematics, University of South Carolina

Fast Solvers for $C^0$ Interior Penalty Methods

In this talk we will discuss fast solvers for $C^0$ interior penalty methods for fourth order elliptic boundary value problems. We will give a brief introduction to $C^0$ interior penalty methods, and present convergence results for the V-cycle, W-cycle and F-cycle multigrid algorithms, and also condition number estimates for two-level additive Schwarz preconditioners. Numerical results will also be reported.

Posted February 15, 2006

Last modified March 3, 2021

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1:40 pm – 2:30 pm Lockett 16
Dave Larson, Texas A&M

Wavelet Sets and Interpolation

A wavelet is a special case of a vector in a separable Hilbert space that generates a basis under the action of a collection, or "system", of unitary operators defined in terms of translation and dilation operations. We will begin by describing an operator-interpolation approach to wavelet theory using the local commutant of a unitary system that was developed by the speaker and his collaborators a few years ago. This is really an abstract application of the theory of operator algebras, mainly von Neumann algebras, to wavelet theory. The concrete applications of operator-interpolation to wavelet theory include results obtained and partially published results, and some brand new results, that are due to this speaker and his former and current students, and other collaborators.

Posted February 16, 2006

Last modified January 27, 2022

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3:40 pm – 4:30 pm Lockett 284
Chris Hruska, University of Chicago
Candidate for an Assistant Professor Position in Topology

Nonpositively curved spaces with isolated flats

In the 1980s Gromov popularized the study of finitely generated groups using geometric techniques. He introduced and popularized notions of negative and nonpositive curvature in group theory, which have been highly influential in shaping the field of geometric group theory over the last two decades. The theory of negatively curved groups is extremely rich and exhibits many strong stability properties. On the other hand, the theory of nonpositively curved groups is much more delicate and less understood. In my thesis, I introduced the class of nonpositively curved groups with “isolated flats”. These groups occur naturally throughout group theory and low-dimensional topology and can be considered as the simplest nontrivial generalization of a negatively curved group. They have many properties in common with negatively curved groups. In particular, in joint work with Bruce Kleiner, I have shown that such a group is relatively hyperbolic with respect to virtually abelian subgroups.

Posted February 16, 2006

Last modified January 27, 2022

Colloquium Questions or comments?

1:40 pm – 2:30 pm Lockett 16
(Jennifer) Suzanne Hruska, Indiana University
Candidate for an Assistant Professor Position in Analysis

The Dynamics of Polynomial Skew Products of C^2

Our goal is to develop and use rigorous computer investigations to study the dynamics of polynomial skew products of $C^2$; i.e., maps of the form $f(z,w) = (p(z), q(z,w))$, where $p$ and $q$ are polynomials of the same degree $d \ge 2$. The skew products we are most interested in studying are those maps which are Axiom A. Such maps have the “simplest” chaotic dynamics, and stability under small perturbation, thus are amenable to computer investigation. In this talk, we will describe a new class of skew products with interesting dynamics, and sketch how we have proven using rigorous computer techniques that sample maps from this class are Axiom A. This leads us to conjecture that all (or nearly all) maps in this class are Axiom A.

Posted February 17, 2006

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 284
Li-yeng Sung, University of South Carolina
Candidate for Possible Senior Professor Position

Fokas Transforms

Just as initial value problems for evolution partial differential equations in one spatial variable can be solved by means of the Fourier transform on the full-line, initial-boundary value problems on the half-line can be solved using Fokas transforms. In this talk we will present unified derivations of these transforms and discuss their applications to linear and nonlinear partial differential equations.

Posted February 23, 2006

Colloquium Questions or comments?

1:40 pm – 2:30 pm Lockett 277
Olga Plamenevskaya, Massachusetts Institute of Technology
Candidate for Assistant Professor Position in Topology

Heegaard Floer theory, knots, and contact structures

Abstract: Heegaard Floer theory is one of the most significant recent developments in low-dimensional topology. Reminiscent of gauge theory, it provides powerful invariants for 3-manifolds. Although defined via holomorphic disks, these 3-manifold invariants have an unexpected connection to combinatorial knot invariants developed by Khovanov. I will outline the construction of Heegaard Floer and Khovanov theories, as well as their relation (due to Ozsvath and Szabo). Then, I will expand these results to the world of contact topology, providing a new invariant for transversal knots, and bringing the correspondence between the two theories to a new level.

Posted February 23, 2006

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 284
Alexander Retakh, University of Texas at Arlington
Candidate for Assistant Professor Position in Algebra

Structure Theory and Representations of Conformal Algebras

The last several decades saw a great deal of interaction between representation theory and modern mathematical physics. The search for rigorous algebraic formalism in areas such as string theory and conformal field theory led Kac and others to the concept of a conformal algebra. Apart from their physical applications, conformal algebras also turned out to be extremely useful in the study of infinite-dimensional Lie algebras. I will define conformal algebras, explain their relation to vertex algebras and superconformal algebras of string theory, the connection to Hamiltonian formalism in calculus of variations, and describe recent progress and conjectures in the field.

Posted February 27, 2006

Last modified March 6, 2006

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 284
Chris Larsen, WPI and California Institute of Technology

Epsilon Stability - A new tool for studying local minimizers

Posted March 17, 2006

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 284
Leonid V. Berlyand, Department of Mathematics, Pennsylvania State University

The discrete network approximation and asymptotic fictitious fluid approach in modeling of highly packed composites

Posted March 21, 2006

Last modified October 4, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 284
Achim Jung, Department of Computer Science, University of Birmingham, England

Semantic domains, or the curious inapplicability of mathematics in computer science

In the mid-60s, Christopher Strachey and others began a
program of describing the meaning of computer programs in a mathematical
style. The approach is known as denotational semantics. From the
beginning, Strachey was aware that set theory is not a good basis for such
an endeavor, but it was not until Dana Scott developed his domain
theory several years later that there was any mathematical basis at all.

In this talk, I will try to explain why sets—without further structure—do not reflect well the realities of computing, and I will try to motivate
why the domains of Scott, which carry an order and a topology, do a better job.
There are several further computational phenomena which required Scott to
restrict the concept of domain even further, but once this is done, a
fairly pleasing and flexible semantic universe is obtained.

In the spirit of this lecture, I will not dwell too much on the successes
that domain theory has had in modeling computation, but rather present
those phenomena which have resisted being incorporated into the model. One
issue that is still not completely understood is the treatment of exact
real numbers. On the one hand, real numbers seem ideally suited for a
topological model but recent work by Escardó, Hofmann, and Streicher
suggests that there is an inherent conflict between efficiency of the
programming language and faithfulness to the mathematical concept.

Posted March 9, 2006

Colloquium Questions or comments?

3:40 pm – 4:30 pm Johnston 338
Qian-Yong Chen, University of Minnesota
Candidate for Assistant Professor Position

A new basis for spectral methods

Abstract: The spectral methods have been very successful in many applications, such as weather prediction, seismic imaging and etc. The main reason for their success is the exponential accuracy: For smooth problems on simple domains, spectral methods can achieve 10 digits accuracy, compared to 2 ~ 3 digits for finite difference or finite element methods with similar computational cost. However, there are still two issues with the Legendre/Chebyshev polynomials based spectral methods for non-periodic problems: the time-step size and the number of points needed to resolve a wave. In this talk, I address this two issues by using a new basis, the prolate spheroidal wave functions (PSWFs), for spectral methods. The relevant approximation theory will be covered. The advantage of the new basis over Legendre/Chebyshev polynomials will be showed for marginally resolved broadband solutions.

Posted March 24, 2006

Last modified January 27, 2022

Colloquium Questions or comments?

9:00 am – 10:00 am 338 Johnston Hall/CCT
Christopher King, Northeastern University

Mathematical problems in quantum information theory

Attempts to extend Shannon's noisy coding theorem to a quantum setting have led to interesting mathematical questions concerning products of completely positive maps on matrix algebras. Specifically for trace-preserving maps it is conjectured that the output state with minimal entropy is always a product state—this is equivalent to the statement that minimal output entropy is additive for such maps. I will review the background to this problem, and describe its relation to some inequalities that arose in different contexts, including strong subadditivity and Hanner's inequality. I will then show how some new matrix inequalities can be used to prove special cases of this result.

Posted April 10, 2006

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 284
Atle Hahn, University of Bonn and LSU

Towards a path integral derivation of 3-manifold quantum invariants

Abstract: The study of the heuristic Chern-Simons path integral by E. Witten inspired (at least) two general approaches to quantum topology. Firstly, the perturbative approach based on the CS path integral in the Landau gauge and, secondly, the quantum group approach by Reshetikhin and Turaev. While for the first approach the relation to the CS path integral is obvious for the second approach it is not. In particular, it is not clear if and how one can derive the relevant R-matrices or quantum 6j-symbols directly from the CS path integral. In my talk, I will sketch a strategy that should lead to a clarification of this issue in the special case where the base manifold is of product form. This strategy is based on the torus gauge fixing procedure introduced by Blau and Thompson for the study of the partition function of CS models. I will show that the formulas of Blau and Thompson can be generalized to Wilson lines and that the evaluation of the expectation values of these Wilson lines leads to the same state sum expressions in terms of which shadow invariant of Turaev is defined. Finally, I will sketch how one can obtain a rigorous realization of the path integral expressions appearing in this treatment.

Posted April 21, 2006

Last modified April 26, 2006

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 284
Tomasz Przebinda, University of Oklahoma

Invariant Eigen-Distributions and Howe's Correspondence

Abstract: The notions of a group reduction, a character and an invariant eigen distribution play a crucial role in Harmonic Analysis on a Real Reductive Group. The classical groups may be organized in pairs. This leads to a correspondence of representations, which is compatible with Capelli identities. We shall explain a recent microlocal construction of invariant eigen distributions which is also compatible with Capelli identities. The hope is that this construction explains the behavior of the characters under Howe's correspondence.

Posted April 19, 2006

Last modified February 2, 2022

Colloquium Questions or comments?

3:40 pm 284 Lockett
Franco Rampazzo, Dipartimento di Matematica Pura ed Applicata, Università degli Studi di Padova

Commutators of Flows of Non-Smooth Vector Fields

My talk will concern the problem of finding a nonsmooth analogue of the
notion of Lie bracket—the *commutator*—of two vector fields. The results I am presenting are contained in joint works with H. Sussmann. In particular, by
means of a notion of set-valued Lie bracket introduced in [1], in [2] we have
extended some classical results valid for smooth vector fields to the case when
the vector fields are just Lipschitz. For instance, we have proved that the flows
of two Lipschitz vector fields *commute* for small times if and only if their set-valued Lie bracket vanishes everywhere (or, equivalently, if their classical Lie bracket vanishes almost everywhere). We have also extended the asymptotic formula that gives an estimate of the lack of commutativity of two vector fields in terms of their Lie bracket, and we have proved a *simultaneous flow box theorem* for commuting families of Lipschitz vector fields. Finally, I shall mention the question of higher order Lie brackets. In particular, the extension of the concept of set-valued Lie bracket to an order greater than two requires some care, and it cannot be merely treated by inductively applying the notion of degree-two bracket. However, we are proposing a concept of higher degree bracket which, in particular, allows us to generalize the classical Chow’s Theorem.

[1] Rampazzo, F., and H. J. Sussmann, “Set-valued differentials and a nonsmooth version of Chow’s theorem,” in *Proceedings of the 40th IEEE Conference on Decision and Control (Orlando, FL, December 2001)*, Vol. 3, pp. 2613-2618.

[2] Rampazzo, F., and H. J. Sussmann, “Commutators of flows of nonsmooth
vector fields,” submitted for publication.

Professor Rampazzo's visit is sponsored by the Louisiana Board of Regents Grant
"Enhancing Control Theory at LSU".

Posted September 1, 2006

Last modified September 12, 2006

Colloquium Questions or comments?

3:40 pm – 4:30 pm Locket 285
Kalyan B. Sinha, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore

Hilbert Tensor Algebras and Stochastic Differential Equations

Posted September 13, 2006

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Michael Malisoff, LSU
Roy P. Daniels Professor

Lyapunov Functions, Stabilization, and Engineering Applications

Posted October 5, 2006

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Habib Ouerdiane, University of Tunis

Infinite Dimensional Complex Analysis and Application to Probability

There will be coffee and cookies in the lounge at 3:00.

Posted October 4, 2006

Last modified January 27, 2022

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Sarada Rajeev, Department of Physics and Astronomy, University of Rochester

The Stochastic Geometry of Two Dimensional Turbulence

Fluid motion is notoriously hard to predict due to the growth of small instabilities (the “Butterfly Effect”). Arnold showed that these instabilities can be explained geometrically as due to the negative curvature of the group of diffeomorphisms: ideal fluid flow is the geodesic motion on this group. A detailed understanding of fluid flow is still out of reach in the three dimensional case. I will describe an approach to two dimensional turbulence based on a surprising connection to geodesic motion on unitary groups. Including dissipation and fluctuation leads to a stochastic differential equation for which a steady state solution can be obtained. The eventual goal (not yet realized) is to explain how stable structures such as hurricanes arise from such an unstable chaotic system.

Visit supported by National Science Foundation Grant DMS 0601141

Posted September 26, 2006

Last modified October 4, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Darren Crowdy, Imperial College London and MIT

Vortex motion in complex domains: new theoretical perspectives

There will be coffee and cookies in the lounge at 3:00.

Posted October 5, 2006

Last modified November 1, 2006

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Paul Kirk, Indiana University

The geography of 4-manifolds with specified fundamental group

There will be coffee and cookies in the lounge at 3:00.

Posted October 4, 2006

Last modified January 27, 2022

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Thierry Lévy, École normale supérieure and CNRS

Combinatorial aspects of the heat kernel measure on the unitary group

Computing asymptotic quantities related to the heat kernel measure on the unitary group U(N) as N tends to infinity is one of the basic questions of large N Yang-Mills theory, and a natural problem in the framework of large random matrices. P. Biane (1995) and F. Xu (1997) have independently computed limiting distributions and proved asymptotic freeness results. Recently, A. Sengupta has reformulated Xu’s computation of the limiting distribution in a very clear and attractive way. In this talk, I will explain how physical ideas related to “string theories” developed in the context of two-dimensional Yang-Mills theory by D. Gross and W. Taylor (1993) shed some light on the approach of Xu and Sengupta, in particular on its combinatorial aspects. More concretely, I will explain how elementary computations related to the Schur-Weyl duality allow one to relate the Brownian motion on the unitary group and the most natural random walk on the symmetric group. Then I will derive and discuss convergent series expansions for expectations of products of traces of unitary matrices under the heat kernel measure. This discussion will involve paths in the Cayley graph of the symmetric group and the lattice of non-crossing partitions.

Refreshments will be served in the lounge at 3pm

Visit supported by National Science Foundation Grant DMS 0601141

Posted October 20, 2006

Last modified October 4, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Larry Gerstein, University of California at Santa Barbara

Quadratic forms: classification and other problems

There will be coffee and cookies in the lounge at 3:00.

Posted November 2, 2006

Last modified January 27, 2022

Colloquium Questions or comments?

3:40 pm – 4:30 pm 109 Nicholson
John Baez, Univeristy of California at Riverside

Higher Gauge Theory

Gauge theory describes the parallel transport of point particles using the formalism of connections on bundles. In both string theory and loop quantum gravity, point particles are replaced by 1-dimensional extended objects: paths or loops in space. This suggests that we seek some sort of “higher gauge theory” that describes parallel transport as we move a path through space, tracing out a surface. To find the right mathematical language for this, we must “categorify” concepts from topology and geometry, replacing smooth manifolds by smooth categories, Lie groups by Lie 2-groups, bundles by 2-bundles, and so on. Some interesting examples of these concepts show up in the mathematics of topological quantum field theory, string theory and 11-dimensional supergravity.

This is a joint Mathematics and Physics & Astronomy Event.

Posted October 5, 2006

Last modified November 20, 2006

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 235
Samuel M. Rankin, Director, American Mathematical Society Washington Office

Activities of the American Mathematical Society's Washington Office

There will be coffee and cookies in the lounge at 3:00.

Posted September 29, 2006

Last modified October 4, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Phuc Nguyen, Purdue University

Nonlinear equations with power source terms and measure data

There will be coffee and cookies in the lounge at 3:00.

Posted January 14, 2007

Last modified January 16, 2007

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Carlos A. Berenstein, University of Maryland

Internet Tomography

Abstract: The problem to be discussed is how to detect as early as possible an attack on a network by saturation. There will be coffee and cookies in the lounge at 3:00.

Posted January 15, 2007

Last modified October 4, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Jan Dijkstra, Vrije Universiteit Amsterdam

On sets with convex shadows

There will be coffee and cookies in the lounge at 3:00.

Posted January 18, 2007

Last modified March 3, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Kariane Calta, Cornell University
Candidate for Assistant Professor Position

Billiards, translation surfaces and associated dynamical systems

In this talk I will discuss some of the basic notions relevant to the study of translation surfaces and provide several interesting examples of such surfaces, including those that arise from billiard tables. I will focus on the dynamics of the geodesic flow on an individual surface and the related dynamics of the action of SL(2,R) on the moduli space of translation surfaces. I will also discuss recent advances in this field, including some of my own results and their relationship to the work of a number of other authors.

Posted January 26, 2007

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Ioan Bejenaru, University of California, Los Angeles
Candidate for Assistant Professor Position

Local and global solutions for Schroedinger Maps

Abstract: We introduce the Schroedinger Maps which can be thought as free solutions of the geometric Schroedinger equation. More exactly, while the classical Schroedinger equation is written for functions taking values in $\\mathhb{C}$ (complex plane), the range of a Schroedinger Map is a manifold (with a special structure). We explain the importance of these Maps and what are the fundamental aspects one would like to understand about them. Then we focus on the particular case when the target manifold is $\\mathbb{S}^2$ (the two dimensional sphere) and review the most recent results along with our contribution to the field.

Posted January 29, 2007

Last modified March 3, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Mahta Khosravi, Institute for Advanced Study, Princeton, NJ
Candidate for Assistant Professor Position

Spectral Asymptotics on Heisenberg Manifolds

Let $R(t)$ be the error term in Weyl's law for $(2n+1)$-dimensional Heisenberg manifolds. Based on the Petridis-Toth conjecture $R(t)=O_\delta(t^{n-1/4+\delta})$. We discuss new pointwise and moment results that provide evidence for this conjecture in three dimensions and a proof for it in higher dimensions. The methods used also allow a proof of a new fifth moment result in the case of the Dirichlet Divisor problem.

Posted January 22, 2007

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Milen Yakimov, University of California, Santa Barbara

Poisson structures on flag varieties

Abstract: The geometry of Poisson structures originating from Lie theory found numerous applications in representation theory, ring theory, and dynamical systems. The linear Poisson structure on the dual of a Lie algebra provides the setting for the orbit method of Kirillov, Kostant, and Dixmier for the study of the unitary duals of Lie groups and the spectra of universal enveloping algebras. In this talk we will describe in detail the geometry of a class of Poisson structures on complex flag varieties and some of their relations to combinatorics (Schubert cells and their Deodhar partitions, cluster algebras, total positivity, the Springer and the Lusztig partitions of wonderful compactifications), ring theory (spectra of algebras of quantum matrices and other quantized algebras), integrable systems (Kogan-Zelevinsky systems). In the special case of hermitian symmetric spaces of compact type, these Poisson structures further elucidate the works of Wolf, Richardson, R\\\"ohrhle, and Steinberg on the structure of the orbits of certain Levi factors.

Posted January 29, 2007

Last modified February 7, 2007

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Christian Haesemeyer, University of Illinois at Urbana-Champaign
Candidate for Assistant Professor Position

On the algebraic K-theory of singularities

Abstract: Algebraic K-theory is a highly complicated invariant of algebraic varieties and rings, encoding arithmetic, geometric and algebraic information. In this talk, I will try to explain these different notions and give some idea as to how to isolate geometric from algebraic information in the case of singularities.

Posted February 14, 2007

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Milen Yakimov, University of California, Santa Barbara

Poisson structures on flag varieties

Abstract: The geometry of Poisson structures originating from Lie theory found numerous applications in representation theory, ring theory, and dynamical systems. The linear Poisson structure on the dual of a Lie algebra provides the setting for the orbit method of Kirillov, Kostant, and Dixmier for the study of the unitary duals of Lie groups and the spectra of universal enveloping algebras. In this talk we will describe in detail the geometry of a class of Poisson structures on complex flag varieties and some of their relations to combinatorics (Schubert cells and their Deodhar partitions, cluster algebras, total positivity, the Springer and the Lusztig partitions of wonderful compactifications), ring theory (spectra of algebras of quantum matrices and other quantized algebras), integrable systems (Kogan-Zelevinsky systems). In the special case of hermitian symmetric spaces of compact type, these Poisson structures further elucidate the works of Wolf, Richardson, R\\\"ohrhle, and Steinberg on the structure of the orbits of certain Levi factors.

Posted January 5, 2007

Last modified January 27, 2022

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Max Karoubi, University of Paris 7

Twisted K-theory, old and new

Twisted K-theory has its origins in the author's PhD thesis [K1] and in the paper of P. Donovan and the author about 37 years ago. The objective of the lecture is to revisit the
subject in the light of new developments inspired by Mathematical Physics. See for instance E. Witten ArXiv hep-th/9810188, J. Rosenberg and M.F. Atiyah-G. Segal ArXiv math/0407054. The unifying theme is the notion of K-theory of graded Banach algebras, already present in [K1], from which most of the new theorems in twisted K-theory are derived. Some explicit computations are also given in the equivariant case, related to previous known results. (See https://webusers.imj-prg.fr/~max.karoubi/ or ArXiv mathKT/0701789 for more details.)

There will be coffee and cookies in the lounge at 3:00.

Posted February 14, 2007

Last modified February 15, 2007

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Michael Otto, University of Arizona

The moment map in symplectic geometry and elsewhere

Abstract: The moment map is a central object of study in symplectic
geometry. It also appears (in disguise) in several other branches of
mathematics, such as linear algebra, classical mechanics,
representation theory of Lie groups etc. This talk is intended to
give an overview over some of its most interesting properties, most
notably several convexity results and formulas of Duistermaat-Heckman
type.

There will be coffee and cookies in the lounge at 3:00.

Posted April 11, 2007

Last modified January 27, 2022

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Kevin Knudson, Mississippi State University

Algorithms in discrete Morse theory

Discrete Morse theory was developed by Robin Forman to provide a combinatorial analogue, for simplicial complexes, of classical smooth Morse theory on manifolds. Constructing efficient discrete Morse functions is a nontrivial task. In this talk, I will present an algorithm that begins with a function h defined on the vertices of a complex K and extends it to a discrete Morse function on the entire complex so that the resulting discrete gradient field mirrors the large scale behavior of h. This has applications to the analysis of point cloud data sets and several examples will be given. No prior knowledge of Morse theory (discrete or smooth) will be assumed.

There will be coffee and cookies in the lounge at 3:00.

Posted March 30, 2007

Last modified January 27, 2022

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
William Velez, The University of Arizona

Increasing the number of mathematics majors

In the late 1980's I began my efforts to increase the success rate of minorities in first semester calculus. The interventions that I devised were very time consuming and as the number of minority students increased, I could not manage that kind of effort. I developed my Calculus Minority Advising Program in an effort to meet with scores of minority students each semester. This program consists of a twenty-minute meeting with each student at the beginning of each semester. These meetings with students eventually transformed my own attitude about the importance of mathematics in their undergraduate curriculum.

I took over the position of Associate Head for Undergraduate Affairs in the department four years ago. I set a very modest goal for myself: to double the number of mathematics majors. With almost 500 mathematics majors I have reached that goal. I think the next doubling is going to be much harder to achieve. My work with minority students provided me with the tools to accept this new challenge of working with all students.

This talk will describe my own efforts to encourage ALL of our students that a mathematics major, or adding mathematics as a second major, is a great career choice.

Posted April 23, 2007

Last modified October 4, 2021

Colloquium Questions or comments?

3:40 pm Lockett 285
Raul Quiroga, Centro de Investigacion en Matematicas (CIMAT)

Actions of Noncompact Simple Lie Groups on Pseudo Riemannian Manifolds

Let $G$ be a noncompact simple Lie group acting on a compact manifold $M$, and suppose that the $G$-action has a dense orbit and preserves a pseudo-Riemannian metric. A general conjecture of R. Zimmer states that, for this geometric/dynamical setup, $M$ is essentially a coset space of a semisimple Lie group $H$ containing $G$. In this talk we will discuss some geometric conditions on the $G$-action on $M$ that ensure the conclusion of Zimmer's conjecture.

Posted September 19, 2007

Last modified October 1, 2007

Colloquium Questions or comments?

10:00 am Johnston Hall 338
Doug Arnold, Institute for Mathematics and its Applications, Minneapolis
Director

The Institute for Mathematics and its Applications

Abstract: The Institute for Mathematics and its Applications (IMA) in Minneapolis is a leading research center, founded by the National Science Foundation in 1982. The primary mission of the IMA is to increase the impact of mathematics by fostering interdisciplinary research linking mathematics with important scientific and technological problems from other disciplines, industry, and society. Through a variety of programs, it provides opportunities for scientists, mathematicians, and engineers from academia and government labs and industry to make contact and interact with each other, to learn about new developments, and to stimulate the study of interesting and relevant problems and their solution. In this informal presentation the director of the IMA will discuss IMA operations, upcoming programs, and outcomes, in order to promote participation and gather input from LSU researchers.

Posted September 28, 2007

Last modified October 8, 2007

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
John Etnyre, Georgia Institute of Technology

Invariants of embeddings via contact geometry

Abstract:
I will describe a method to define, hopefully new, invariants of any
embedded submanifold of Euclidean space. To define this invariant we
will need to take an excursion into the realm of contact geometry and
a recent generalization of Floer homology called contact homology.
More specifically, after recalling various notions from contact
geometry, I will show how to associate a Lagrangian submanifold to
any embedded submanifold of Euclidean space. The invariant of the
embedding will be the contact homology of this Lagrangian. Though the
definition of this invariant is somewhat complicated it is possible
to compute it for knots in Euclidean 3-space. Lenny Ng has
combinatorially studied this invariant for such knots and has shown
that it does not seem to be determined by previously known invariants
but non the less has some connections with the classical Alexander
polynomial of a knot. I will concentrate on the more geometric
aspects of the invariant and ongoing work of Tobias Ekholm, Lenny Ng,
Michael Sullivan and myself aimed at a better understanding of the
invariant (in particular, showing that it is well defined in some
generality).

There will be coffee and cookies in the lounge at 3:00.

Posted September 14, 2007

Last modified March 3, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285(Originally scheduled for Thursday, September 20, 2007)

Alex Iosevich , University of Missouri–Columbia

Some examples of interaction between harmonic analysis, geometric measure theory, combinatorics and number theory

Many problems stated in analytic terms often turn out to be, in essence, problem in combinatorics or number theory. The opposite phenomenon, where number theoretic or combinatorial problems are fundamentally analytic in nature is equally ubiquitous. We shall discuss this phenomenon from a systematic point of view and will outline mechanisms that allow one to transfer techniques and ideas from area to another.

There will be coffee and cookies in the lounge at 3:00.

Posted September 14, 2007

Last modified November 5, 2007

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Clint Dawson, University of Texas at Austin

Simulation of Coupled Ground Water/Surface Water Flow

Abstract: There is strong evidence that the supply and quality of
water are influenced by interactions between water stored at the
surface and water stored in the subsurface. There have been a few
efforts at modeling this interaction, but a number of outstanding
questions still remain. In this talk we will address the mathematical
modeling and numerical simulation of coupled ground water/surface
water flow. Mathematical modeling issues include determining the
appropriate models of flow within each subdomain, and determining the
interface or boundary conditions to couple the models. Given a model,
the next question is how to solve it. Here we will discuss an approach
based on the discontinuous Galerkin (DG) method. A priori error
analysis for a DG formulation for a shallow water model coupled with
saturated ground water flow will be presented. Numerical results will
also be discussed for several practical scenarios. We will also
discuss recent analysis and results for a simplified surface water
flow model, the diffusive wave approximation.

There will be coffee and cookies in the lounge at 3:00.

Posted November 12, 2007

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Michael Lacey, Georgia Institute of Technology

Pointwise convergence of Fourier series

Abstract: Lennart Carleson\'s celebrated theorem on the pointwise
convergence of Fourier series was one of three results cited by the
Abel Prize committee, in making their award to him. This result states
that any square integrable function on the unit circle is the limit,
almost everywhere, of the Fourier partial sums. We will recall the
theorem, explain why it is worthy of an Abel prize, and give a brief
description of a proof. The theorem is related, even required, for a
range of related questions, a much more recent development
investigated by the speaker and Christoph Thiele, among many
others. We close with a very recent new result of Victor Lie on the
Quadratic Carleson Theorem.

There will be coffee and cookies in the lounge at 3:00.

Posted January 23, 2008

Last modified October 4, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Jan Dijkstra, Vrije Universiteit Amsterdam

Homeomorphism groups of manifolds and Erdős space

There will be coffee and cookies in the lounge at 3:00.

Posted January 22, 2008

Last modified October 4, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Dan Barbasch, Cornell University

Unipotent representations and unitarity

There will be coffee and cookies in the lounge at 3:00.

Posted January 29, 2008

Last modified October 4, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Joseph Wolf, University of California, Berkeley

Limits of nilpotent commutative spaces

There will be coffee and cookies in the lounge at 3:00.

Posted March 17, 2008

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Toshiyuki Kobayashi, Harvard and University of Tokyo

Existence Problem of Compact Locally Symmetric Spaces.

Abstract: The local to global study of geometries was a major trend of 20th

century geometry, with remarkable developments achieved particularly in

Riemannian geometry. In contrast, in areas such as Lorentz geometry,

familiar to us as the space-time of relativity theory, and more generally

in pseudo-Riemannian geometry, as well as in various other kinds of

geometry (symplectic, complex geometry, ...), surprising little is known

about global properties of the geometry even if we impose a locally

homogeneous structure.

I will give a survey on the recent developments regarding the question

about how the local geometric structure affects the global nature of

non-Riemannian manifolds, with emphasis on the existence problem of compact

forms, rigidity and deformation.

There will be coffee and cookies in the lounge at 3:00.

Posted February 17, 2008

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Ron Goldman, Department of Computer Science, Rice University

Three Problems in Search of a Graduate Student

Abstract: Three unsolved problems that originated from research in

Computer Graphics and Geometric Modeling will be presented. The first

problem involves understanding the notion oscillation for Bezier

surfaces, the freeform surfaces most common in Computer Graphics and

Geometric Modeling. The second problem is related to Bezier curves and

univariate Bernstein polynomials, and concerns the combinatorics of

symmetrizing multiaffine functions. The third problem pertains to

fractals and asks if there is an algorithm to determine whether two

arbitrary sets of contractive affine transformations generate the same

fractal.

There will be coffee and cookies in the lounge at 3:00.

Posted January 22, 2008

Last modified April 4, 2008

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Oleg Viro, SUNY Stony Brook

Compliments to bad spaces

Abstract:
Could Mathematics be made any better by a different choice
of basic definitions? Definitions in some mathematical theories
exclude any mentioning of objects, which are believed to be nasty.
We will consider few examples of such "political correctness".
Speaking on differential manifolds, we usually pretend that they have no
singular siblings. This causes lots of inconveniences.
Another example is finite topological spaces. Most of mathematicians
believe that all finite topological spaces
are either trivial or nasty. Topology appears to be the only
mathematical field that feels ashamed of its finite objects.

There will be coffee and cookies in the lounge at 3:00.

Posted September 16, 2008

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 235
Ravi Rau, Department of Physics and Astronomy, LSU

Possible links between physics problems in quantum computing and fibre bundles, projective geometry and coding/design theory

Abstract:

Quantum Information, the field that embraces quantum computing,

cryptography and teleportation, involves as a central object an

entangled pair of spin-1/2 (or two-level) systems. I have been

interested in developing geometrical pictures for manipulating the

fifteen-operator su(4) algebra that describes such systems. For a

single spin, its su(2) algebra's fibre bundle of a two-sphere

(called Bloch sphere by physicists) and a u(1) phase plays a major

role throughout the field of magnetic resonance. I will present

analogous geometrical descriptions of fibre bundles for su(4) and

its sub-algebras (and also higher su(N)). One of these sub-algebras,

su(2) X su(2) X u(1), also "maps" onto octonions and the Fano Plane.

Other sub-algebras and the full su(4) can be similarly related to

Desargues's and other diagrams of projective geometry. These relate

to the subjects of coding and design theory, and Hadamard matrices.

I am looking for help from mathematical experts in each of these

areas to see how these connections may be exploited for application

in quantum information.

There will be coffee and cookies in the lounge at 3:00.

Posted September 19, 2008

Last modified October 17, 2008

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 235
Ricardo Cortez, Tulane University

Regularized Stokeslets and other elements with applications to biological flows

Biological flows, such as those surrounding swimming microorganisms or beating cilia, are often modeled using the Stokes equations due to the small length scales. The organism surfaces can be viewed as flexible interfaces imparting force on the fluid. I will present the Method of Regularized Stokeslets and other elements that are used to compute Stokes flows interacting with immersed flexible bodies or moving through obstacles. The method treats the flexible bodies as sources of force or torque in the equations and the resulting velocity is the superposition of flows due to all the elements. Exact flows are derived for forces that are smooth but supported in small spheres, rather than point forces. I will present the idea of the method, some of the known results and several examples from biological applications.

There will be coffee and cookies in the lounge at 3:00.

Posted August 28, 2008

Last modified March 3, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 235
Mihai Putinar, University of California at Santa Barbara

Poincaré's variational problem in potential theory

The simultaneous diagonalization of two quadratic forms naturally attached to a domain in the Euclidean space has guided Poincaré in his study of the Dirichlet problem. Put in modern setting, due to a pioneering work of Mark G. Krein, Poincaré's variational principle offers a deep understanding of modern aspects of function theory (quasiconformal mappings, Beurling-Schiffer transform) and provides the theoretical background of some recent studies of an inverse problem in electrostatics. Based on joint work with D. Khavinson and Harold S. Shapiro.

There will be coffee and cookies in the lounge at 3:00.

Posted October 27, 2008

Last modified November 3, 2008

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 235
Jinhyun Park, Purdue University

What can an algebraist do for Euclidean geometry?

Abstract: The 3rd problem of Hilbert, one of the firstly solved

Hilbert problems, studied the scissors congruence for the

3-dimensional Euclidean space: two polyhedra are said to be in

scissors congruence if one can cut the first along straight lines and

reassemble the components to get the second. Are all polyhedra of a

fixed volume then scissors congruent? The answer was negatively given

by Max Dehn, a student of Hilbert in around 1900. Though the problem

was solved, it opened many new doors during the next 100 years. We

will describe this problem from scratch, and mention how it is related

to some number theoretic questions and how it mysteriously gives new

problems in algebraic geometry.

There will be coffee and cookies in the lounge at 3:00.

Posted November 11, 2008

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 235
Ivo Babuska, Institute for Computational Engineering and
Sciences, University of Texas, Austin

Computational Science, Mathematics and Where Are We Going

Abstract: A major goal of Computational Science is to predict physical

and other phenomena. The problem is how confident can we be that the

computed results describe reality well enough so that they can be the

basis for crucial decisions. ( Do we have enough courage to sign the

blueprints based on the computation?) The notions of Verification and

Validation and their mathematical contains will be explained. These

notions are the basis for the confidence that the computed results

could be used for the decisions. A few examples of engineering

accidents and their reasons will be presented. Brief comments of the

repercussions for the educations at the universities will be

made. References to the basic literature will be given.

There will be coffee and cookies in the lounge at 3:00.

Posted December 8, 2008

Last modified December 12, 2008

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285 (provisionally)
Burak Aksoylu, Department of Mathematics and CCT

Rigorously Justified Solvers for Rough Coefficients

Abstract:

Roughness of PDE coefficients causes loss of robustness of

preconditioners. The main goal is to recover robustness and obtain

rigorous structural understanding of the involved process. A

qualitative understanding of the PDE operators and their dependence on

the coefficients is essential for designing preconditioners. This

process draws heavily upon effective utilization of theoretical tools.

According to experience, the performance of a preconditioner depends

essentially on the degree to which the preconditioned operator

approximates the properties of the underlying infinite dimensional

operator. For this reason, controlling the infinite dimensional

problem provides a basis for the construction of preconditioners. We

use tools from operator theory for this. On the other hand, another

basis is the control of the finite dimensional discretized

problem. For that, we use singular perturbation analysis (SPA). After

obtaining a preconditioner through SPA, a fundamental need is to

explain the effectiveness of the preconditioner and to justify that

rigorously. With the insights provided by operator theory and SPA, we

are in control of the effectiveness and computational feasibility

simultaneously.

Based on ideas developed for porous media flow, we present a new

preconditioning strategy which is computationally comparable to

algebraic multigrid, but with rigorous justification. We will also

demonstrate how SPA gives valuable insight to the asymptotic behavior

of the solution of the underlying PDE, hence, provides feedback for

preconditioner construction.

There will be coffee and cookies in the lounge at 3:00.

Posted December 8, 2008

Last modified January 27, 2022

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285 (provisionally)
Michael Shapiro, Escuela Superior de Fisica y Matematicas del Instituto Politecnico Nacional, Mexico

On some basic ideas of hypercomplex analysis

“Hypercomplex analysis” is a generic name for those generalizations of one-dimensional complex analysis which involve hypercomplex numbers. Quaternionic analysis is the oldest and the most known version of it. In the talk, it will be discussed, first of all, in which sense quaternionic analysis is a “proper” or a “closest” version in low dimensions which includes as particular cases, or sub-theories, such classic theories as vector analysis and holomorphic mappings in two complex variables, as well as some systems of partial differential equations. This allows one, by developing quaternionic analysis, to obtain new results for the above classic theories and to refine known ones; some applications of this approach to harmonic analysis, operator theory, mathematical physics, will be mentioned. Some comments on Clifford analysis and its applications will be also made.

There will be coffee and cookies in the lounge at 3:00.

Posted December 8, 2008

Last modified October 4, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285 (provisionally)
Maria Elena Luna-Elizarraras, Escuela Superior de Fisica y Matematicas del Instituto Politecnico Nacional, Mexico

On functional analysis with quaternionic scalars

There will be coffee and cookies in the lounge at 3:00.

Posted February 2, 2009

Last modified October 4, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Stephen Sawin, Fairfield University

Supersymmetry, Quantum Mechanics and the Gauss-Bonnet-Chern Theorem

In joint work with Dana Fine at UMass Dartmouth, we will give a rigorous construction of the path integral describing the time-evolution operator for imaginary time quantum mechanics, with and without $N=1$ supersymmetry. The path integral is constructed as a limit of finite-dimensional approximating integrals, with concrete uniform estimates on the convergence. Consequences of this include an alternative construction of the Laplace and Laplace–Beltrami heat kernels. We will use this construction to give a rigorous version of Witten–Alvarez-Gaumé–Friedan and Windey's path integral "proof" of the Gauss–Bonnet–Chern Theorem, and explain how we expect a minor variation to make rigorous their proof of the general Index Theorem.

There will be coffee and cookies in the lounge at 3:00.

Posted February 19, 2009

Last modified March 3, 2009

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Charles Livingston, Indiana University

Four-dimensional aspects of classical knot theory

Abstract: Classical knot theory studies knots in 3-space; that is,
embeddings of the unit circle, S^1, into R^3. Higher dimensional knot
theory generalizes this, for instance by considering embeddings of the
2-sphere, S^2, into R^4. In this talk I will discuss an aspect of knot
theory between the low and high-dimensional realms: the study of knots in
3-space in terms of the surfaces they bound in upper 4-space, H^4, the set
of points (x,y,z,w) in R^4 with w > 0. Three of the goals of the
presentation will be to: (1) give some intuitive insight into how knots can
bound such surfaces; (2) describe a few of the central topics in geometric
topology that motivate looking at knots in this way; and (3) summarize some
recent advances in the area.

There will be coffee and cookies in the lounge at 3:00.

Posted February 10, 2009

Last modified February 19, 2009

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Stratos Prassidis, Canisius College

Detecting Linear Groups

Abstract: Linear groups are subgroups of general linear groups. Deciding if

a group is linear or not is an old problem in group theory. Linear groups

became important in topology after the Isomorphism Conjecture was proved for

discrete linear groups. We present criteria that guarantee that a group is

linear and some applications. At the end, we will show a hands-on proof that

the holomorph of the free group on two generators is linear.

There will be coffee and cookies in the lounge at 3:00.

Posted February 2, 2009

Last modified October 4, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Jiazu Zhou, Southwest University, China

Geometric measures and geometric inequalities

There will be coffee and cookies in the lounge at 3:00.

Posted March 26, 2009

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Gregor Masbaum, University Paris 7

Trees , Pfaffians and Complexity (or How not to win a million dollars)

There will be coffee and cookies in the lounge at 3:00.

Posted April 12, 2009

Last modified October 4, 2021

Colloquium Questions or comments?

2:40 pm – 3:30 pm Lockett 285(Originally scheduled for Monday, April 20, 2009)

Ivan Dimitrov, Queen's University (Canada)

A geometric realization of extreme components of the tensor product of modules over algebraic groups

In this talk I will explain how the celebrated theorem of Borel–Weil–Bott provides a natural realization of some extreme components of the tensor product of two irreducible modules of simple algebraic groups. I will also discuss a number of connections of our construction with problems coming from Representation Theory, Combinatorics, and Geometry, including questions about the Littlewood–Richardson cone related to Horn's conjecture, settled by Knutson and Tao in the late 1990s.

The talk is based on a joint work with Mike Roth.

There will be coffee and cookies in the lounge at 2:00.

Posted August 11, 2009

Last modified October 5, 2023

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Pramod Achar, Mathematics Department, LSU

Positivity, sheaves, and representation theory

The celebrated “Kazhdan-Lusztig polynomials” of an algebraic group have an elementary combinatorial definition, but the proof that all their coefficients are nonnegative requires very deep results from algebraic geometry—the Weil conjectures, proved by Deligne in the 1970s. The link between combinatorics and algebraic geometry is furnished by sheaf theory, especially the so-called "perverse sheaves." I will explain how "positivity" results come out of the interaction of these topics, and I will also discuss more recent developments in which perverse sheaves are replaced by vector bundles and coherent sheaves.

Posted August 11, 2009

Last modified August 14, 2009

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Habib Ouerdiane, University of Tunis El Manar

Generalized fractional evolution equations

Abstract:

In this talk we study the generalized Riemann-Liouville (resp. Caputo)

time fractional evolution equation in infinite dimensions. We show that

the explicit solution is given as the convolution between the initial

condition and a generalized function related to the Mittag-Leffler

function. The fundamental solution corresponding to the Riemann-Liouville

time fractional evolution equation does not admit a probabilistic

representation while for the Caputo time fractional evolution equation

it is related to the inverse stable subordinators.

There will be coffee and cookies in the lounge at 3:00.

Posted August 11, 2009

Last modified March 3, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Fernanda Cipriano, University of Lisbon

Statistical solutions for the 2D Euler equation

In the study of Euler and Navier-Stokes equations we can consider two
different approaches. The most classical one consist in the study of the
equations with specific initial and boundary conditions.
Another approach, the so-called stochastic approach, consist in the
construction of suitable probability measures and study its evolution
in time according to the corresponding dynamic. The framework of
stochastic analysis can be used to construct solutions. In our
presentation, we follow the second point of view to present some
results on the 2D Euler equation with periodic boundary conditions.
We construct surface type measures on the level sets of the renormalized
energy and establish the existence of weak solutions living on such level
sets. We also prove the existence of weak solutions for the forward and
backward transport equations associated with the 2D Euler equation. Such
solutions can be interpreted, respectively, as a statistical Lagrangian
and statistical Eulerian description of the motion of the fluid.

There will be coffee and cookies in the lounge at 3:00.

Posted August 11, 2009

Last modified September 11, 2009

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Milen Yakimov, LSU

Poisson geometry of flag varieties, ring theory and combinatorics

Abstract:

The geometry of Poisson Lie groups and Poisson homogeneous spaces was

actively studied after Drinfeld's celebrated 1986 ICM talk in which he

describe its importance for the representation theory of quantum groups.

In this talk we will go over various aspects of the geometry of Poisson

structures on flag varieties for complex simple Lie groups

(results with K. Brown and K. Goodearl). We will apply them to ring theory

to resolve several problems for the De Concini-Kac-Procesi algebras from

about 10-15 years ago: determining the torus invariant prime ideals of those

algebras, their inclusions, finding effective generating sets for the

ideals. We will also describe applications to combinatorics: 1. obtaining a

simple proof of the recent result of Knutson, Lam and Speyer for cyclicity

of the Lusztig stratification of Grassmannians, 2. combinatorial formulas

for Hecke algebras (with F. Brenti), 3. Deodhar's stratification of open

Richardson varieties (with B. Webster).

There will be coffee and cookies in the lounge at 3:00.

Posted July 27, 2009

Last modified January 27, 2022

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Stephen Smith, University of Illinois Chicago Circle

Revisiting the classification of the finite simple groups (an outline)

Most mathematicians are aware that the classification of the finite simple groups involved hundreds of researchers and thousands of journal pages.

In 1983, Daniel Gorenstein published the first volume of a general outline of this massive work—covering the “non-characteristic 2 type” case (correspondingly roughly to simple matrix groups over fields of odd order).

But he could not publish the projected second and final volume, on the characteristic 2 type case—due to the non-publication of Mason's expected work on “quasithin groups”. That gap was not filled until the 2004 publication by Aschbacher and Smith of a more general quasithin treatment. This finally left the way open for the second volume of the overall outline of the CFSG---a first draft of this outline has now been completed by Aschbacher, Lyons, Solomon, and Smith.

The talk will be an elementary exposition of some of the ideas in this overall outline; including mention of certain of the new ideas and approaches which have arisen since the 1980s.

There will be coffee and cookies in the lounge at 3:00.

Posted August 11, 2009

Last modified October 4, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Dmitry Ryabogin, Kent State University

On the local version of Mahler Conjecture

There will be coffee and cookies in the lounge at 3:00.

Posted August 11, 2009

Last modified October 4, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Irina Mitrea, Worcester Polytechnic Institute

Boundary Value Problems for Higher Order Elliptic Operators

There will be coffee and cookies in the lounge at 3:00.

Posted January 12, 2010

Last modified January 14, 2010

Colloquium Questions or comments?

2:00 pm – 3:00 pm Lockett 16
Jiajun Wang, California Institute of Technology
Candidate for Assistant Professor Position in Topology

On combinatorial Floer homology

Posted January 7, 2010

Colloquium Questions or comments?

3:40 pm – 4:30 pm 338 Johnston Hall
Jason Howell, Carnegie Mellon University
Candidate for Assistant Professor Position in Math and CCT

Dual-Mixed Finite Element Methods For Fluids

Jason Howell, Carnegie Mellon University Postdoctoral Associate In The Center For Nonlinear Analysis And The Department Of Mathematical Sciences Bio Jason Howell is a Postdoctoral Associate in the Center for Nonlinear Analysis and the Department of Mathematical Sciences at Carnegie Mellon University. He earned a PhD in Mathematical Sciences at Clemson University in 2007 with a specialty in numerical analysis and computational mathematics. During 2004-2006, he had three appointments as a summer scholar with the Center for Applied Scientific Computing at Lawrence Livermore National Laboratory. He also holds an MS in Mathematical Sciences from Clemson and a BS in Mathematics from the College of Charleston. His research interests lie at the intersection of analysis, computation, and applications, and he currently works on projects in finite element methods for fluid and solid mechanics, numerical methods for non-Newtonian fluids, and numerical methods for fluid/fluid and fluid/structure interaction problems. Abstract Accurate and efficient numerical methods to approximate fluid flows are important to researchers in many fields, including mechanical, materials, and biomedical engineering. In many applications within these fields, it is of paramount importance to accurately predict fluid stresses. However, in most existing numerical schemes for fluids, the primary unknown of interest is the fluid velocity. This motivates the development of dual-mixed finite element methods for fluids, in which the stress is a primary unknown of interest, and the study of inf-sup conditions for single and twofold saddle point problems is an important component of the construction of these methods. This study has led to results that give equivalent sets of inf-sup conditions for twofold saddle point problems, yielding new tools for proofs of well-posedness and finite element compatibility. These tools, together with a macroelement technique, show compatibility of a new dual-mixed method for fluids employing Arnold-Winther symmetric tensor finite elements for stress.

Posted January 7, 2010

Last modified October 4, 2021

Colloquium Questions or comments?

2:00 pm – 3:00 pm 338 Johnston Hall
Shawn Walker, New York University
Candidate for Assistant Professor Position in Math and CCT

Shape Optimization Of Peristaltic Pumping

Shawn W. Walker, New York University

Research Scientist, Courant Institute Of Mathematical Sciences

Bio: Shawn W. Walker is a postdoctoral researcher and instructor at New York University's Courant Institute of Mathematical Sciences. He earned his PhD in aerospace engineering and an MSc in applied mathematics and scientific computing from the University of Maryland in 2007 and also holds an MSc in engineering and applied science from Yale University and a BSc in electrical engineering from Virginia Polytechnic Institute & State University. His research interests include finite element methods and PDEs, free boundary problems, shape optimization, and fluid-structure interaction and control. http://www.cims.nyu.edu/~walker/

Abstract: Transport is a fundamental aspect of biology and peristaltic pumping is a fundamental mechanism to accomplish this; it is also important in many industrial processes. We present a variational method for optimizing peristaltic pumping in a two dimensional periodic channel with moving walls to pump fluid. No a priori assumption is made on the wall motion, except that the shape is static in a moving wave frame. Thus, we pose an infinite dimensional optimization problem and solve it with finite elements. Sensitivities of the cost and constraints are computed variationally via shape differential calculus and $L^2$-type projections are used to compute quantities such as curvature and boundary stresses. Our Optimization method falls under the category of sequential quadratic programming (SQP) methods. As a result, we find optimized shapes that are not obvious and have not been previously reported in the peristaltic pumping literature. Specifically, we see highly asymmetric wave shapes that are far from being sine waves. Many examples are shown for a range of fluxes and Reynolds numbers up to Re=500 which illustrate the capabilities of our method.

Posted December 24, 2009

Last modified October 4, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Jan Dijkstra, Vrije Universiteit Amsterdam

Topological Kadec norms with applications

There will be coffee and cookies in the lounge at 3:00.

Posted January 12, 2010

Last modified January 15, 2010

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Andrew Putman, MIT
Candidate for Assistant Professor Position in Topology

The Picard Group of the Moduli Space of Curves with Level Structures

Posted January 14, 2010

Last modified October 4, 2021

Colloquium Questions or comments?

2:10 pm Lockett 10
Richard Kent, Brown University
Candidate for Assistant Professor Position in Topology

Analytic functions from hyperbolic manifolds

There will be coffee and cookies in the lounge at 3:00.

Posted January 7, 2010

Last modified October 4, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Michael Farber, University of Durham

Stochastic algebraic topology and robotics

I will describe solutions to several problems of mixed probabilistic-topological nature which are inspired by applications in topological robotics. These problems deal with systems depending on a large number of random parameters, $n\to \infty$. Our results predict the values of various topological characteristics of configuration spaces of such systems.

There will be coffee and cookies in the lounge at 3:00.

Posted January 12, 2010

Last modified January 14, 2010

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Mark Colarusso, University of Notre Dame
Candidate for Assistant Professor Position in Lie Theory

THE GELFAND-ZEITLIN INTEGRABLE SYSTEM ON g[(n, C)

Posted August 11, 2009

Last modified January 25, 2010

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285(Originally scheduled for Tuesday, September 15, 2009)

Frank Sottile, Texas A&M

Orbitopes

Abstract:

An orbitope is the convex hull of an orbit of a compact

group acting linearly on a vector space. Instances of these

highly symmetric convex bodies have appeared in many areas

of mathematics and its applications, including protein

reconstruction, symplectic geometry, and calibrations in

differential geometry.

In this talk, I will discuss Orbitopes from the perspectives

of classical convexity, algebraic geometry, and optimization

with an emphasis on ten motivating problems and concrete examples.

This is joint work with Raman Sanyal and Bernd Sturmfels.

There will be coffee and cookies in the lounge at 3:00.

Posted January 14, 2010

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Richard Oberlin, UCLA
Candidate for Assistant Professor Position in Analysis

A variation-norm Carleson Theorem

Posted January 13, 2010

Last modified January 25, 2010

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Karl Schwede, University of Michigan
Candidate for Assistant Professor Position in algebraic geometry

Singularities of polynomials in characteristic 0 and characteristic p

Posted January 7, 2010

Last modified January 28, 2010

Colloquium Questions or comments?

3:40 pm – 4:30 pm 338 Johnston Hall
Xuemin Tu
Candidate for Assistant Professor Position in Math and CCT

Inexact Balancing Domain Decomposition By Constraints Algorithms

Abstract and Bio available at www.cct.lsu.edu

Posted January 15, 2010

Last modified January 28, 2010

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Juhi Jang, New York University
Candidate for Assistant Professor Position in PDEs/applied math

Vacuum in Gas and Fluid dynamics

Posted January 7, 2010

Last modified January 28, 2010

Colloquium Questions or comments?

3:40 pm – 4:30 pm 338 Johnston Hall
Yingda Cheng
Candidate for Assistant Professor Position with Math and CCT

Discontinuous Galerkin Finite Element Methods And Applications To Boltzmann-Poisson Models In Semiconductor Device Simulation

Abstract and Bio at www.cct.lsu.edu

Posted January 15, 2010

Last modified February 4, 2010

Colloquium Questions or comments?

2:00 pm – 3:00 pm Lockett 112
Leonardo Mihalcea, Baylor University
Candidate for Assistant Professor Position in Algebraic Geometry

Quantum K-Theory of Grassmannians and the Geometry of Spaces of Curves

Posted February 1, 2010

Last modified February 4, 2010

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
John Oprea, Cleveland State University

Gottlieb Groups, LS Category and Geometry

Abstract: Gottlieb groups are special subgroups of the homotopy groups which

arise in many homotopical contexts. LS category is a numerical homotopy

invariant that was originally invented to give a bound on the number of

critical points of smooth functions. Strangely enough, these two things are

related, and --- what's more --- they are related via analogues of geometric

theorems. This talk will recall basic notions of algebraic topology,

introduce Gottlieb groups, LS category and their relationships and see how

geometry fits into the mix.

There will be coffee and cookies in the lounge at 3:00.

Posted January 7, 2010

Last modified February 12, 2010

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Bojko Bakalov, North Carolina State University

Singularities, root systems, and W algebras

Abstract:

Gromov-Witten invariants are naturally organized in a generating function,

which is a formal power series in infinitely many variables. In many cases

this function is a highest-weight vector for a certain

infinite-dimensional algebra and at the same time is a solution of an

integrable hierarchy of partial differential equations. Similar generating

functions can be introduced for the Frobenius structures coming from

singularities of hypersurfaces. We will start by reviewing the marvelous

relations among singularities, root systems and reflection groups. The

generating function of a simple singularity was shown recently to be a

solution of the Kac-Wakimoto hierarchy. Our main result is that it is also

a highest weight vector for the corresponding W algebra.

There will be coffee and cookies in the lounge at 3:00.

Posted February 9, 2010

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Scott Armstrong, Department of Mathematics, Louisiana State University

The infinity Laplace equation, tug-of-war games, and minimizing Lipschitz extensions

Abstract: Given a nice bounded domain, and a Lipschitz function defined on

its boundary, consider the problem of finding an extension of this

function to the closure of the domain which has minimal Lipschitz

constant. This is the archetypal problem of the calculus of variations

in $L^\\infty$. There can be many such minimal Lipschitz extensions,

but there is there is a unique minimizer once we properly \"localize\"

this Lipschitz minimizing property. The uniquely specified function is

a solution of the infinity Laplace equation: the Euler-Lagrange

equation for our optimization problem. This PDE is a highly degenerate

nonlinear elliptic equation which does not have smooth solutions. In

this talk we will discuss what we know about the infinity Laplace

equation, what the important open questions are, and some recent

developments. We will even play a two-player random-turn game called

\"tug-of-war\". One advantage of our topic is that it is completely

accessible to graduate students and even perhaps some undergraduates.

There will be coffee and cookies in the lounge at 3:00.

Posted March 3, 2010

Last modified October 4, 2021

Colloquium Questions or comments?

2:40 pm – 3:30 pm Prescott 205
Gisele Goldstein, University of Memphis

Derivation and Interpretation of Dynamic Boundary Conditions for the Heat and Wave Equations

There will be coffee and cookies in Prescott 205 at 2:00.

Posted March 3, 2010

Last modified October 4, 2021

Colloquium Questions or comments?

2:10 pm – 3:00 pm Lockett 10
Jerome Goldstein, University of Memphis

Instantaneous blowup and related nonexistence issues

There will be coffee and cookies in the lounge at 3:00.

Posted January 27, 2010

Last modified October 4, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Charles Weibel, Rutgers University

The Lichtenbaum and Bloch-Kato Conjectures are now Theorems

The Lichtenbaum Conjecture related algebraic K-theory of the integers to the values of the Riemann Zeta function. The Bloch-Kato Conjecture said that the étale cohomology of a field with (odd) finite coefficients) should have a presentation with units as generators and simple quadratic relations (the ring with this presentation is now called the "Milnor K-theory"). For Z/2 coefficients, this is a famous theorem of Voevodsky.

This talk will be a non-technical overview of the ingredients that go in to the proof, and why this conjecture matters to non-specialists. Most of the ingredients are due to Rost and Voevodsky.

Here is a fun consequence of all this. We now know the first 20,000 groups K_n(Z) of the integers, except when 4 divides n. The assertion that these groups are zero when 4 divides n (n>0) is equivalent to Vandiver's Conjecture (in number theory), and if it holds then we have fixed Kummer's 1849 "proof" of Fermat's Last Theorem. If any of them are nonzero, then the smallest prime dividing the order of this group is at least 16,000,000.

There will be coffee and cookies in the lounge at 3:00.

Posted January 7, 2010

Last modified March 17, 2010

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Boris Rozovsky, Brown University

Quantization of Stochastic Navier-Stokes Equation

Abstract: We consider a stochastic Navier-Stokes equation driven by a

space-time Wiener process. This equation is quantized by transformation of

the nonlinear term to the Wick product form. An interesting feature of this

type of perturbation is that it preserves the mean dynamics: the expectation

of the solution of the quantized Navier-Stokes equation solves the

underlying deterministic Navier-Stokes equation. From

the stand point of a statistician it means that the perturbed model is an

unbiased random perturbation of the deterministic Navier-Stokes equation.

The quantized equation is solved in the space of generalized stochastic

processes using the Cameron-Martin version of the Wiener chaos expansion. A

solution of the quantized version is unique if and only if the uniqueness

property holds for the underlying

deterministic Navier-Stokes equation. The generalized solution is obtained

as an inverse of solutions to corresponding quantized equations. We will

also demonstrate that it could be approximated by real (non-generalized

processes). A solution of the

quantized Navier-Stokes equation turns out to be non- anticipating and

Markov.

The talk is based on a joint work with R. Mikulevicius.

There will be coffee and cookies in the lounge at 3:00.

Posted March 2, 2010

Last modified April 13, 2010

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Ben Webster, MIT

Categorification, Lie algebras and topology

Abstract:

It's a long established principle that it is interesting to think

about numbers as the sizes of sets, or as the dimensions of vector spaces,

or better yet, as the Euler characteristic of complexes. You can't have a

map between numbers, but you can have one between sets or vector

spaces. For example, the Euler characteristic of topological spaces is not

functorial, but homology is.

One can try to extend this idea to a bigger stage, by, say, taking a

vector space and trying to make a category by defining morphisms

between its vectors. This approach, interpreted suitably, has been a

remarkable success with the representation theory of semi-simple Lie

algebras (and their associated quantum groups). I'll give an

introduction to this area, with a view toward applications in

topology; in particular to replacing polynomial invariants of knots

that come from representation theory with vector-space-valued

invariants that reduce to knot polynomials under Euler characteristic.

There will be coffee and cookies in the lounge at 3:00.

Posted January 7, 2010

Last modified March 3, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 285
Alexander Its, Indiana University–Purdue University Indianapolis

Special Functions and Integrable Systems

The recent developments in the theory of integrable systems
have revealed its intrinsic relation to the theory of special functions.
Perhaps the most generally known aspects of this relation are
the group-theoretical, especially the quantum-group theoretical,
and the algebra-geometrical ones. In the talk we will discuss
the analytic side of the Special Functions-Integrable Systems
connection. This aspect of the relation between the two theories
is less known to the general mathematical community,
although it goes back to the classical works of Fuchs, Garnier and
Schlesinger on the isomonodromy deformations of the systems of linear
differential equations with rational coefficients. Indeed, the monodromy
theory of linear systems provides a unified framework for the linear
(hypergeometric type) and nonlinear (Painlevé type) special functions
and, simultaneously, builds a base for the new powerful technique of the
asymptotic analysis — the Riemann-Hilbert method.

In this survey talk, which is based on the works of many authors
spanned over more than two decades, the isomonodromy point of view
on special function will be outlined. We will also review the history of
the Riemann-Hilbert method as well as its most recent applications in the
theory of orthogonal polynomials and random matrices.

Coffee and cookies: The James E. Keisler lounge, 3:00.

Posted September 10, 2010

Last modified September 17, 2010

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Hongyu He, Department of Mathematics, LSU

Theta Correspondence and Unitary Representations

In the late 1950's and early 1960's, motivated by problems from physics

and number theory, Segal, Shale and Weil independently discovered a

projective representation of the symplectic group, often called the

oscillator representation. This representation was later used by

Kashiwara-Vergne and many others to obtain new unitary representation and

automorphic forms for other classical groups. The theory underlying these

developments is often called theta correspondence. In this talk, I will

review Howe's theory of reductive dual pair and theta correspondence. I

will then discuss how theta correspondence can be used to understand the

unitary dual of the noncompact orthogonal groups. With the exception of

rank-one groups and several higher rank groups, the unitary dual of the

noncompact orthogonal groups is not completely classified.

Posted September 8, 2010

Last modified September 17, 2010

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Kalyan B. Sinha, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore

Index Theorems in Quantum Mechanics

Abstract: The notion of the Fredholm index of an operator is extended to

pairs of projections as well as to certain pairs of (unbounded)

self-adjoint operators in a Hilbert space. This index often exhibits

some invariance properties and has applications in Quantum mechanical

problems. There is also a K-theoretic description of the problem.

Posted October 7, 2010

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Vladimir Dragovic, MI SANU, Belgrade/GFM University of Lisbon

Discriminant separability, Poncelet porisms and Kowalevski top

Abstract. A new view on the Kowalevski top and Kowalevski integration procedure is presented. It is based on a classical notion of Darboux coordinates, a modern concept of n-valued Buchstaber-Novikov groups and a new notion of discriminant separability. Unexpected relationship with the Great Poncelet Theorem for a triangle is established. Further connections between discriminant separability, geometry of pencils of quadrics and integrability are discussed.

Posted September 29, 2010

Last modified November 30, 2010

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Dimitar Grantcharov, University of Texas at Arlington

Categories of weight representations of Lie algebras

Abstract: Following works of G. Benkart, D. Britten, S. Fernando, V. Futorny, A. Joseph, F. Lemire, and others, in 2000 O. Mathieu achieved a major breakthrough in representation theory by classifying the simple weight representations of Lie algebras. The next step in the study of weight representations is to look at the indecomposable representations. In this talk we will discuss recent results related to the structure of the indecomposable weight representations and connections with quiver theory and algebraic geometry. This is a joint work with Vera Serganova.

Posted October 7, 2010

Last modified January 8, 2011

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Birgit Speh, Cornell University

Branching rules of unitary representations: Examples and applications to automorphic forms

One of the basic problems of representations theory is to understand how a representation is built up of irreducible representations. For finite dimensional representations of compact groups there exist well known combinatorial algorithms. In this talk we will examine the problem for infinite dimensional representations of semisimple Lie groups, give some examples as well as applications to automorphic forms and locally symmetric spaces.

Posted January 19, 2011

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Aaron Lauda, Columbia University

Categorifying quantum groups and link invariants

Abstract: The Jones polynomial can be understood in terms of the representation theory of the quantum group associated to the Lie algebra sl2. This description facilitated a vast generalization of the Jones polynomial to other quantum link and tangle invariants called Reshetikhin-Turaev invariants. These invariants, which arise from representations of quantum groups associated to simple Lie algebras, subsequently led to the definition of quantum 3-manifold invariants. In this talk we categorify quantum groups using a simple diagrammatic calculus that requires no previous knowledge of quantum groups. These diagrammatically categorified quantum groups not only lead to a representation theoretic explanation of Khovanov homology but also inspired Webster\'s recent work categorifying all Reshetikhin-Turaev invariants of tangles.

Posted January 21, 2011

Colloquium Questions or comments?

9:40 am – 10:30 am Lockett 233
Elizabeth Dan-Cohen, Jacobs University Bremen

Structure and representation theory of finitary Lie algebras (virtual talk from Bremen)

ABSTRACT: Finitary Lie algebras are the simplest possible infinite-dimensional version of the classical Lie algebras. These infinite-dimensional algebras arise in many contexts in mathematics. However, their structure theory was underdeveloped until recently, and the representation theory is still in its early stages. I have been part of a recent series of contributions first to the structure and then most recently to the representation theory.

Posted November 26, 2010

Last modified January 11, 2011

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Vyacheslav Futorny, University of Sao Paulo

Representations of Lie algebra of vector fields on a torus

Representation theory of Lie algebra of vector fields on a circle, centerless Virasoro algebra, is well known due to results of O. Mathieu who proved Kac's conjecture. On the other hand, representations of Lie algebra of vector fields on n-dimensional torus are far less studied. We are going to discuss certain classes of representations whose construction is based on the theory of vertex algebras. These representations provide free-fields realizations of our Lie algebra. These are joint results with Yuly Billig (Ottawa, Canada).

Posted January 20, 2011

Colloquium Questions or comments?

2:40 pm – 3:30 pm Lockett 285
John Baldwin, Princeton University

Contact structures, open books, and Khovanov homology

I will describe an important class of geometric objects on 3-manifolds called contact structures, and will survey Giroux\'s celebrated correspondence between contact structures and more topological objects called open books. I will summarize progress on some open questions related to this connection, and will explain how link invariants like Khovanov homology may be used to provide further insights.

Posted January 20, 2011

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Song Yao, University of Michigan

Optimal Stopping for Dynamic Convex Risk Measures

Abstract: We use martingale and stochastic analysis techniques to study a continuous-time optimal stopping problem, in which the decision maker uses a dynamic convex risk measure to evaluate future rewards. We also find a saddle point for an equivalent zero-sum game of control and stopping, between an agent (the "stopper") who chooses the termination time of the game, and an agent (the "controller", or "nature") who selects the probability measure.

Posted January 20, 2011

Last modified January 24, 2011

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Karl Mahlburg, Princeton University

Percolation, partitions, and probability

I will discuss the surprising connections between finite-size scaling in families of bootstrap percolation models, the combinatorics of integer partitions, and limiting entropies for Markov-type processes. A combinatorial characterization of the percolation processes relates their metastability threshold exponents to the cuspidal asymptotics of generating functions for partitions with restricted sequence conditions. These generating functions are hypergeometric q-series that are of number-theoretic interest, as in many cases they are equal to the product of modular forms and Ramanujan's famous mock theta functions. In other cases, both the percolation processes and partition functions are best understood through entropy bounds for probabilistic sequences with gap conditions. These sequences can be understood as Markov-type processes with varying transition probabilities, and techniques from the theory of linear operators are used in order to bound the dominant eigenvalue.

Posted January 20, 2011

Last modified January 24, 2011

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Nam Quang Le, Columbia University

Regularity results for the mean curvature flow

Mean curvature flow is the evolution of a hypersurface moving with normal velocity equal to its mean curvature vector. If the initial hypersurface is compact then the flow will develop singularities in finite time. In this talk, we will present some recent regularity results on the compact mean curvature flow. Our focus will be on optimal conditions for the existence of smooth solutions to the mean curvature flow. We will show that if the flow is of type I then the mean curvature controls the flow in the sense that singularities cannot occur if the mean curvature is uniformly bounded. In the case of surfaces, we will show that the mean curvature controls the flow provided that either the Multiplicity One Conjecture of Ilmanen holds or the Gaussian density is less than two. When the mean curvature of our flow blows up, i.e., when singularities occur, we will also give its (sharp) blow-up rate.

Posted January 20, 2011

Last modified February 1, 2011

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Leonardo Mihalcea, Baylor University
Candidate for Assistant Professor Position in Algebraic Geometry

Rational curves in flag manifolds and Gromov-Witten invariants

Abstract: Gromov-Witten (GW) invariants for a flag manifold count rational curves subject to certain incidence conditions. These numbers satisfy remarkable equations, equivalent to the associativity relations of the quantum cohomology algebra, and were successfully used to solve many classical problems in enumerative geometry. In connection to Mirror Symmetry and Integrable systems, Givental and his collaborators defined the equivariant and K-theoretic GW invariants, which allow the corresponding space of rational curves to be positive dimensional. The purpose of my talk is to introduce the "quantum=classical" phenomenon, which in many instances allows us to compute explicitly 3-point GW invariants (of all flavors), and to obtain algorithms for multiplication in the (generalized) quantum cohomology algebra. In the case of Grassmannians, this is surprisingly related to positroids - varieties connected to Lusztig's totally positive stratification.

Posted November 8, 2010

Last modified January 22, 2011

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Robert Stanton, Ohio State University

Symplectic methods in Lie theory

Abstract: The non-degenerate Cartan-Killing form on a semisimple Lie algebra provides a natural conformal class of pseudo-Riemannian metric in Lie theory. The resulting interplay of differential geometry and harmonic analysis on Lie groups and symmetric spaces is well known. At the other extreme is the presence of a non-degenerate skew-symmetric form. We will illustrate the usefulness of symplectic techniques by several applications ranging from classical invariant theory to exceptional Lie algebras to modern differential geometry. The talk is intended for a general audience and is based on joint work with M. Slupinski.

Posted November 11, 2010

Last modified January 18, 2011

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Joseph Wolf, University of California, Berkeley

Classical Analysis and Nilpotent Lie Groups

Classical Fourier analysis has an exact counterpart in group theory and in some areas of geometry. Here I'll describe how this goes for nilpotent Lie groups and for a class of riemannian manifolds closely related to a nilpotent Lie group structure.

Posted September 14, 2010

Last modified February 16, 2011

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Noriko Yui, Queen's University

The modularity (automorphy) of Calabi-Yau varieties over the rationals

Abstract: According to the Langlands Philosophy, every algebraic variety defined over the rationals or a number field should be modular (automorphic).

In this talk, I will concentrate on a special class of algebraic varieties, called Calabi-Yau varieties (of dimension at most three), defined over the rationals, and report on the current status of their modularity (automorphy).

Posted October 29, 2010

Last modified March 1, 2011

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Skip Garibaldi, Emory University

Did a 1-dimensional magnet detect a 248-dimensional Lie group?

Abstract: In a January 2010 article in Science, a team of physicists reported that they had "detected evidence of E8 symmetry" in a neutron scattering experiment on a quasi-1-dimensional cobalt niobate magnet. Their article details their experimental results, but it does not explain how these are connected with E8, a 248-dimensional Lie group. This talk will survey the chain of reasoning leading from those results to E8.

This talk does not require any previous knowledge of advanced physics.

Posted January 8, 2011

Last modified January 20, 2011

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Grigory Litvinov, Independent University of Moscow

Dequantization and tropical mathematics

Tropical mathematics can be treated as a result of a dequantization of the traditional mathematics over numerical fields as the Planck constant tends to zero taking imaginary values. Tropical mathematics is a part of the idempotent mathematics (calculus) in the sense of V.P. Maslov and his collaborators. The basic paradigm is expressed in terms of an idempotent correspondence principle. This principle is similar to N. Bohr's correspondence principles in Quantum Physics (and closely related to it). A systematic application of the idempotent correspondence principle leads to a variety of results, often quite unexpected (e.g., the Legendre transform is a tropical version of the Fourier transform). As a result, in parallel with the traditional mathematics, its "classical shadow" appears.

Posted March 21, 2011

Last modified October 4, 2021

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Igor Verbitsky, University of Missouri-Columbia
Curators' Professor

Hessian Sobolev inequalities, $k$-convex functions, and the fractional Laplacian

Posted March 22, 2011

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Tadele Mengesha, Louisiana State University

Global estimates for nonlinear elliptic equations over nonsmooth domains

In this talk we discuss new global regularity estimates for solutions to a class of nonlinear elliptic boundary value problems over nonsmooth domains. The nonlinear coefficients are allowed to be discontinuous but assumed to have a small mean oscillation. The boundary of the domain may be nonsmooth but sufficiently flat in the sense of Reifenberg. The main regularity estimates obtained are in weighted Lorentz spaces. As an application, these estimates will be used to obtain other regularity results in Lorentz-Morrey, Morrey, and Holder spaces. This is joint work with Phuc Nguyen.

Posted August 4, 2011

Last modified August 9, 2011

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 284
Paul Smith, University of Washington

Penrose tilings of the plane and noncommutative algebraic geometry

Abstract: The space X of Penrose tilings of the plane has a natural

topology on it. Two tilings are equivalent if one can be obtained from the

other by a translation. The quotient topological space X/~ is bad: every

point in it is dense. The doctrine of non-commutative geometry is to

refrain from passing to the quotient and construct a non-commutative

algebra that encodes some of the data lost in passing to X/~. In this

example (see Connes book for details) the relevant non-commutative algebra

is a direct limit of products of matrix algebras. We will obtain this

non-commutative algebra by treating the free algebra on two variables x

and y modulo the relation y2=0 as the homogeneous coordinate ring of a

non-commutative curve. The category of quasi-coherent sheaves on this

non-commutative curve is equivalent to the module category over a simple

von Neumann regular ring. That von Neumann regular ring is the same as the

direct limit algebra that Connes associates to X/~. We will discuss

algebraic analogues of various topological features of X/~. For example,

the non-vanishing of extension groups between simple modules is analogous

to the fact that every point in X/~ is dense (which is analogous to the

fact that any finite region of one Penrose tiling appears infinitely often

in every other tiling).

Posted August 19, 2011

Last modified August 31, 2011

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 235
Artem Pulemotov, University of Chicago

The prescribed Ricci curvature problem

We will discuss the problem of finding a Riemannian metric with given Ricci curvature on a compact manifold M. This problem has been intensively studied by mathematicians since the early 1980\'s.

It is partially resolved in the case where M is a closed manifold, but almost completely unresolved on manifolds with boundary. In the first part of the talk, we will review the history of the subject. After that, our focus will be on new results regarding the situation where M is a solid torus.

Posted August 23, 2011

Last modified September 30, 2011

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 284
Angela Pasquale, University of Metz and CNRS

Ramanujan's Master Theorem for Riemannian symmetric spaces

The abstract for this talk can be found at www.math.lsu.edu/~morales/colloquium/pasquale.pdf

Posted September 13, 2011

Last modified October 24, 2011

Colloquium Questions or comments?

3:40 pm – 4:30 pm 235 Lockett
Effie Kalfagianni, Michigan State University

Geometric and combinatorial knot invariants

Abstract: It has been known since the 80's that knot complements admit geometric decompositions and that the class of hyperbolic knots is the richest class of knots. In practice, knots are often given in terms of combinatorial topological descriptions, and it is both natural to seek for ways to deduce geometric information from these descriptions. On the other hand, in the last couple of decades ideas from physics have led to powerful and subtle combinatorial knot invariants such as the Jones knot polynomials. Understanding the relation of the combinatorial knot descriptions and invariants to the detailed structures coming from the geometric picture is an important goal of low dimensional topology that received particular attention in recent years.

In this talk I will survey some conjectured and some established such relations.

Posted January 28, 2012

Last modified February 2, 2012

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
David Vela-Vick, Columbia University

Contact structures, Legendrian and transverse knots, and Heegaard Floer homology

Abstract: The past decade has ushered in a number of major advances in the field of 3-dimensional contact geometry. Beginning with Ozsvath and Szabo's construction of an invariant of contact structures on closed 3-manifolds, a complex tapestry of generalizations and applications has emerged. In this talk, I plan to survey Heegaard Floer theory as it applies to the study of Legendrian and transverse knots. In doing so, I will discuss some surprising relationships between seemingly distinct specializations of this general theory.

Posted January 31, 2012

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Jian Song, Rutgers University

Feynman-Kac formula

Abstract: In this talk, I will review the classical Feynman-Kac formula for partial differential equations (PDEs), and explain the Feynman-Kac formulas we have obtained for stochastic partial differential equations (SPDEs).

Posted February 1, 2012

Last modified February 7, 2012

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Artem Pulemotov, University of Chicago

Geometric flows on manifolds with boundary

Abstract: Geometric flows are partial differential equations that describe evolutions of geometric objects. They are typically used to tackle problems in topology, mathematical physics, and several other fields. The canonical example of a geometric flow is the heat equation on a Riemannian manifold. In the first part of the talk, we will discuss the fundamental features of this equation. We will also speak about two estimates for its positive solutions on manifolds with boundary. A more contemporary example of a geometric flow is the Ricci flow for a Riemannian metric. It is mostly famous for its role in the proof of the Poincare conjecture. The second part of the talk will be devoted to the main features and the behavior of the Ricci flow on manifolds with boundary. Towards the end, we will give a brief overview of related problems.

Posted January 31, 2012

Last modified February 7, 2012

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Betsy Stovall, University of California Los Angeles

Counteracting flatness with affine arclength measure

Abstract: There are many operators in harmonic analysis for which the curvature of some underlying manifold plays a significant role. We will discuss recent efforts to establish uniform estimates for such operators by compensating for degeneracies of curvature with an appropriate measure. We will focus on the case when the underlying manifolds are polynomial curves.

Posted February 20, 2012

Last modified February 21, 2012

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Ling Long, Iowa State University

The arithmetic of modular forms for noncongruence subgroups

Abstract: Among all finite index subgroups of the modular group, majority of them are noncongruence, i.e. they cannot be described in terms of congruence relations. Compared to the classical theory of congruence modular forms, modular forms for noncongruence subgroups are more mysterious due to the lack of efficient Hecke theory. However, noncongruence modular forms exhibit some remarkable properties and are closely related to many topics in number theory. In this talk, we will introduce these functions and discuss some recent developments in this area. In particular, we will consider Galois representations attached to noncongruence modular forms constructed by Tony Scholl which are generalizations of Deligne's Galois representations attached to classical Hecke eigenforms. We will prove under special circumstances that these Scholl representations are automorphic in the sense that their L-functions agree with the L-functions of automorphic forms on reductive groups and then give some applications of such automorphic results.

Posted February 27, 2012

Last modified March 5, 2012

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Christopher Bremer, Mathematics Department, LSU

A new perspective on flat G-bundles and the Stokes phenomenon

Let G/C be a reductive algebraic group and let X/C be a smooth (not necessarily compact) algebraic curve. Flat G-bundles on X are a natural generalization of differential equations with algebraic coefficients, where the latter are viewed as vector bundles (having structure group GL_n) equipped with an integrable connection that is meromorphic at infinity. When the connection has an irregular singular point, the asymptotic series of a fundamental solution jumps discontinuously as it is continued across sectors in a neighborhood of that point. This behavior is known as the Stokes phenomenon. The "jumps" are encoded by elements of the group G, so one might think of the Stokes data as being an enhancement of the monodromy representation associated to a meromorphic differential equation.

The wild ramification case of the geometric Langlands conjecture suggests that there should be a natural correspondence between Stokes data arising from the irregular monodromy map, and representation theoretic data associated to the Langlands dual of the loop group. I will discuss a new approach (based on joint work with D. Sage) to the study of irregular flat G-bundles, inspired by methods from p-adic representation theory. Specifically, we have developed a geometric version of the Moy-Prasad theory of fundamental strata (aka minimal K-types). In this talk I will explain how the theory of fundamental strata applies to the study of moduli spaces of flat G-bundles, allowing one to significantly generalize results of Boalch, Jimbo et al., and others on the Stokes map and the irregular isomonodromy equations.

Posted October 27, 2011

Last modified March 15, 2012

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Abhinav Kumar, Massachusetts Institute of Technology

Lattices, sphere packings and spherical codes

Abstract: It is a classical problem in geometry to find the densest arrangement of non-overlapping spheres in Euclidean space. I will give a brief overview of sphere packing and related geometrical problems, like the kissing number problem. After surveying some of the known results, I'll describe recent approaches to these problems, involving linear programming bounds, numerical optimization using gradient descent, and deformations and the notion of rigidity. The talk will also focus on many concrete examples and explain what we are able to observe and prove using these techniques, as well as many natural open questions.

Posted January 29, 2012

Last modified February 8, 2012

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Kenneth Gross, University of Vermont

Special Functions of Matrix Argument

Classical special functions such as the gamma function, Bessel functions, and hypergeometric functions, to name a few, apply in many different contexts ranging from mathematical analysis to statistics and the physical sciences. Beginning over six decades ago, generalizations of such classical functions to spaces of matrices (and to symmetric cones more generally) arose in the context of harmonic analysis, multivariate statistics, number theory, and the representation theory of Lie groups. In this talk, expository in nature, we will discuss the origins of these generalizations, highlight the critical role of symmetry considerations, and touch upon applications which have appeared recently in multiple-input multiple-output (MIMO) models that form the basis of modern cell-phone transmission.

Posted February 23, 2012

Last modified March 7, 2012

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Aaron Welters, LSU

Dissipative Properties of Systems Composed of High-Loss and Lossless Components

Abstract: We study energy dissipation features of a system composed from lossy and lossless components. One of the principal result is that the dissipation causes modal dichotomy, i.e., splitting of the eigenmodes into two distinct classes according to their dissipative properties: high-loss and low-loss modes. Interestingly, larger losses in the lossy component make the entire composite less lossy, the dichotomy more pronounced, low-loss modes less lossy, and high-loss modes less accessible to external excitations. We also have carried out an exhaustive analytical study of the system quality factor. This is joint work with Alexander Figotin.

Posted August 23, 2011

Last modified April 9, 2012

Colloquium Questions or comments?

3:40 pm – 4:30 pm Lockett 277
Benson Farb, University of Chicago

Permutations and polynomiality in algebra and topology

Abstract: Tom Church, Jordan Ellenberg and I recently discovered that each Betti number of the space of configurations on n points on any manifold is a polynomial in n. Similarly for the moduli space of n-pointed genus g curves. Similarly for the dimensions of various spaces of homogeneous polynomials arising in algebraic combinatorics. Why? What do these disparate examples have in common? The goal of this talk will be to answer this question by explaining a simple underlying structure shared by these (and many other) examples in algebra and topology.

Posted September 10, 2012

Last modified September 13, 2012

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 277
Michael Malisoff, LSU
Roy P. Daniels Professor

Adaptive Tracking and Parameter Identification for Nonlinear Control Systems

Given a control system that has a vector of unknown constant parameters and a reference trajectory for the system, the adaptive tracking and parameter identification problem is to find a dynamic controller that forces the system to track the reference trajectory and a parameter estimator that converges to the unknown parameter vector. In the first part of this talk, I will review the necessary background on control theory. Then I will discuss my research on adaptive tracking and parameter identification for systems with unknown control gains. In the last part, I will discuss an application to marine robots and open problems. This work is joint with Frederic Mazenc and Fumin Zhang. This talk will be understandable to anyone who knows all of the material in the basic graduate ODEs course Math 7320 at LSU.

Posted August 22, 2012

Last modified October 1, 2012

Colloquium Questions or comments?

3:30 pm – 4:20 pm Locket 277
Joseph Wolf, University of California, Berkeley

Infinite Dimensional Lie Groups

Abstract: I'll survey some analytic aspects of a class of infinite dimensional

Lie groups that have a lot of properties in common with their finite

dimensional counterparts. These include smoothness, the Borel-Weil

and Bott-Borel Weil theorems, and the possibility of extending some

of Harish-Chandra's constructions from the finite dimensional case.

This leads to a number of interesting open problems.

Posted September 11, 2012

Last modified March 2, 2021

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3:30 pm – 4:20 pm Lockett 277
Barbara Rüdiger, Bergische Universität Wuppertal - Germany

Stochastic (Partial) Differential Equations with Lévy noise: Existence, uniqueness and properties of solutions

In this talk we first analyze properties of integrands when integration is done with respect to Lévy processes or compensated Poisson random measures. Then we introduce SP(D)Es with Lévy noise and show the existence, uniqueness and continuous properties of the solutions.

Posted September 11, 2012

Last modified September 17, 2021

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3:30 pm – 4:20 pm Lockett 277
Aderemi Kuku, Grambling State University
William W.S. Clayton Endowed Professor of Mathematics

Higher Algebraic K-Theory and Representations of Algebraic Groups

In this talk, I at first discuss some features of K-theory as a multidisciplinary subject that classifies various mathematical structures and objects as well ramifies into and co-ordinates several areas of mathematics. Next, I present constructions and computations of equivariant higher K-theory and profinite (continuous) equivariant higher K-theory for the action of algebraic groups with applications to twisted flag varieties.

Posted September 11, 2012

Last modified March 3, 2021

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3:30 pm – 4:20 pm Locket 277
Yang Tonghai, University of Wisconsin

The Gross-Zagier formula — history, application, and development

How do you know whether a cubic equation like $y^2 =x^3 -n^2 x$ have infinitely many solutions or not (for a fixed integer $n$)? How do you find them? The Birch and Swinnerton-Dyer conjecture would tell you that some analysis ($L$-function) would tell you a lot about this kind of questions. The Gross-Zagier formula gives a deep and direct relation between the arithmetic and analysis. In this talk, I will briefly talk about this beautiful formula, some of its significant applications, and its generalizations.

Posted November 12, 2012

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3:30 pm – 4:20 pm Lockett 277
Renzo Cavalieri, Colorado State University

Polynomiality and Wall Crossings in Hurwitz Theory

Abstract: in this talk I will present a story that began with the observation of Goulden-Jackson-Vakil that families of Hurwitz numbers tend to have interesting polynomiality or piecewise-polynomiality aspects. Cavalieri-Johnson-Markwig subsequently exploited the combinatorics suggested by tropical geometry in order to gain a good understanding of this phenomena, and to be able to describe wall crossings. The story is now evolving with an attempt of lifting these observations from the level of \"numbers\" to the level of \"cycles\". Again, the parallel with tropical geometry helps shed light on the combinatorial features of certain families of Hurwitz classes. The story is understood so far in genus 0 and becomes substantially more complicated in higher genus. The most recent work discussed is joint work with Aaron Bertram and Hannah Markwig.

Posted December 22, 2012

Last modified January 24, 2013

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3:30 pm – 4:20 pm 277 Lockett
Tai Melcher, University of Virginia

Smoothness properties for some infinite-dimensional heat kernel measures

Abstract: Smoothness is a fundamental principle in the study of measures on

infinite-dimensional spaces, where an obvious obstruction to overcome is

the lack of an infinite-dimensional Lebesgue or volume measure. Canonical

examples of smooth measures include those induced by a Brownian motion,

both its end point distribution and as a real-valued path. Heat kernel

measure is the law of a Brownian motion on a curved space, and as such is

the natural analogue of Gaussian measure there. We will discuss some

recent smoothness results for these measures on certain natural classes of

infinite-dimensional groups, including in some degenerate settings. This

is joint work with Fabrice Baudoin, Daniel Dobbs, and Masha Gordina.

Posted September 11, 2012

Last modified February 25, 2013

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3:30 pm – 4:20 pm Lockett 277
Nicholas Ercolani, University of Arizona

Conservation Laws in Random Matrix Theory

ABSTRACT: Analytical Combinatorics is an old subject (going back at least to Euler) which has received renewed impetus from several directions including complex systems theory, large scale computation, quantum gravity and non-equilibrium statistical mechanics. In particular methods from nonlinear PDE theory, both deterministic and stochastic, have begun to play a significant role in advancing this classical subject. In this talk we will discuss some very recent developments that bring to bear methods for studying systems of conservation laws on the asymptotic analysis of generating functions for "maps". A "random map" is a random topological tessellation of a Riemann surface. These are combinatorial objects that first arose in attempts to solve the four color problem but soon thereafter took on a life of their own. Subsequently, physicists working on the unification of the "strong" and "weak" forces discovered that generating functions for maps emerge naturally in the enumeration of Feynman diagrams for random unitary matrix ensembles. Our recent work provides the means for the explicit evaluation of map generating functions in terms of closed form solutions of the aforementioned conservation laws. These conservation laws are certain continuum limits of the Toda lattice differential equations in which the time variables are coupling coefficients of the random matrix ensembles. This topic brings together many areas of pure and applied mathematics and we will describe some of these bridges.

Posted December 22, 2012

Last modified March 15, 2013

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3:30 pm – 4:20 pm Lockett 285
Julian Hook, Indiana University, Bloomington

Graph Theory and the Musical Tonnetz

Note: This colloquium is held on a Friday.

Posted February 27, 2013

Last modified March 11, 2013

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3:30 pm – 4:20 pm Lockett 277
Ilya Spitkovsky, College of William and Mary

The current state of the almost periodic factorization

Factorization of matrix functions (that is, their representation as products of multiples analytic inside and outside the given closed curve) arises naturally in many applications, including those to convolution type equations on a half-line (the classical Wiener-Hopf method). As it happens, the equations on finite intervals also can be treated via the factorization method. The resulting matrix functions, however, are of oscillating type, which has not been treated until recently. The general case can be boiled down to the situation when the matrix is almost periodic, that is, its elements belong to the algebra generated by exp(iax) with real values of the parameter a. We will discuss the current state of the factorization problem for such matrices. A special attention will be paid to a (seemingly narrow) case of 2-by-2 triangular matrix functions, but even for them the factorability properties remain a mystery. in striking difference with both the scalar almost periodic case and the classical Wiener-Hopf setting.

Posted December 22, 2012

Last modified March 14, 2013

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3:30 pm – 4:20 pm Lockett 277
Sam Nelson, Claremont McKenna College

Enhancements of Counting Invariants

Abstract: Counting invariants are among the simplest computable invariants of knots and links. In this talk we will see various strategies for enhancing and strengthening counting invariants and some connections between these invariants and other knot and link invariants.

Posted May 7, 2013

Last modified May 13, 2013

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3:30 pm Lockett 277Arithmetic invariants: From finite groups to modular categories

Richard Ng, Iowa State University, visiting Cornell University

Posted May 10, 2013

Last modified May 13, 2013

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8:30 am Johnston Hall 338 (Training Room)Around Lusztig's conjecture: Arithmetic problems in representation theory, combinatorics and geometry.

Peter Fiebig, Erlangen University, Germany will present a \"virtual colloquium\"

The early time 8:30 AM is to accommodate the 7-hour time difference between here and Germany.

Posted August 28, 2013

Last modified March 2, 2021

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3:30 pm – 4:20 pm Lockett 285
Phuc Nguyen, Department of Mathematics, Louisiana State University

Capacities, nonlinear Calderón-Zygmund theory, and PDEs with power nonlinearities

We discuss the solvability of fully nonlinear and quasilinear equations of Lane-Emden type, quasilinear equations of Riccati type, and stationary Navier-Stokes equations with strongly singular external forces. Complete characterizations of existence will be given in connection with the theory of capacities, weighted norm inequalities, and a nonlinear version of Calderón-Zygmund theory for singular integrals. Removable singularities of Lane-Emden type equations will be described along with the stability of the stationary Navier-Stokes equation. This talk is based on joint work with Igor E. Verbitsky, Tadele Mengesha, and Tuoc Van Phan.

Posted August 27, 2013

Last modified September 7, 2013

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3:30 pm Lockett 285
Oliver T. Dasbach, Mathematics Department, LSU

Knots, hyperbolic volume, and q-series

There will be refreshments in the Keisler lounge at 3 pm

Posted August 27, 2013

Last modified September 16, 2013

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3:30 pm Lockett 285
Stephen Shipman, Mathematics Department, LSU

Embedded Eigenvalues and Resonance in Quantum Graphs

A locally perturbed periodic graph operator admits bands of continuous spectrum as well as discrete eigenvalues corresponding to defect states. Under generic
conditions, the eigenvalues lie in the gaps between the spectral bands. The
obstruction to eigenvalues embedded in the continuous spectrum is algebraic--namely, that a generic polynomial in several variables does not factor.
The polynomial in question comes from the Floquet transform of the operator; its
zero set, called the Fermi surface, describes the admissible quasi-momenta for a
given energy. I will show that there is an interesting class of quantum graphs
possessing symmetries that allow the Fermi surface to separate for all frequencies. This separation allows one to construct embedded eigenvalues that result in complex resonant scattering when excited by radiation states of the surrounding continuous spectrum.

There will be refreshments in the Keisler lounge at 3 pm.

Posted August 27, 2013

Last modified September 10, 2013

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3:30 pm Lockett 285
Hongchao Zhang, Louisiana State University

Recent Advances on Gradient Methods in Nonlinear Optimization

Abstract: In this talk I would briefly talk about some recently
developed gradient-based algorithms for smooth unconstrained optimization,
for smooth and nonsmooth composite optimization, for optimization
with inexact gradients, and for general smooth nonlinear programming.
The convergence properties as well as practical performance of some of
these algorithms will be discussed in this talk.

There will be refreshments in the Keisler lounge at 3pm

Posted August 27, 2013

Last modified September 13, 2013

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3:30 pm Lockett 285
Blaise Bourdin, Department of Mathematics, Louisiana State University

The variational approach to fracture: recent developments and extensions.

Most models for the fracture of brittle materials rely on an energetic argument, the celebrated Griffith criterion, combined with ad-hoc branching criteria. In addition, such models rely heavily on smoothness and regularity assumptions whose validity is debatable.
The variational approach to fracture was developed as an extension of Griffith
criterion preserving its essence, competition between surface and volume energy,
while avoiding any ad-hoc branching criterion or regularity hypothesis of fracture
sets, in space or time. It is formulated as a sequence of unilateral global
minimization problems of a free discontinuity energy.
I will first recall some elements of the mathematical analysis of this approach. I will then describe its numerical implementation, focusing on methods based on elliptic regularization. I will finally show how this approach can be used in many applications, including transverse fracture and debonding of thin films, drying of colloidal suspension, thermal shocks of glass and ceramics, dynamic fracture and hydraulic stimulation. I will describe the mathematical and algorithmic tools developed for each specific problem, and present validation and verification experiments.

There will be refreshments in the Keisler lounge at 3 pm

Posted August 27, 2013

Last modified September 23, 2013

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Lockett 285
Pallavi Dani, Department of Mathematics, LSU

Filling invariants for groups

Any finitely generated group can be endowed with a natural metric
which is unique up to maps of bounded distortion, i.e.
quasi-isometries. A central question in geometric group theory is to
classify finitely generated groups up to quasi-isometry. I will talk
about recent work on understanding quasi-isometry invariants for
groups in various settings, with an emphasis on filling invariants, a
class of invariants that are motivated by classical isoperimetric
inequalities.

There will be refreshments in the lounge at 3 pm

Posted October 14, 2013

Last modified October 20, 2013

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3:30 pm Lockett 285
Siu-hung (Richard) Ng, LSU

Frobenius-Schur indicators and exponents

ABSTRACT: Frobenius-Schur indicators were introduced a century ago for the
representations of finite groups. It has been proved recently that these
indicators are invariants of tensor categories, and so are the exponents
of finite groups. In this talk, we will discuss dimensions, exponents and
indicators for finite groups, and their generalizations in the settings of
Hopf algebras as well as tensor categories.

There will be refreshments in the Keisler lounge at 3 PM

Posted October 14, 2013

Last modified September 17, 2021

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3:30 pm – 4:20 pm Lockett 285
Tadeusz Iwaniec, Syracuse University

Sobolev mappings and energy integrals in nonlinear elasticity

Posted October 9, 2013

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 285
Gregor Masbaum, CNRS, Institut de Mathematiques de Jussieu, Paris, France

All finite groups are involved in the mapping class group

Abstract: A (finite) group G is said to be involved in an (infinite) group Gamma if G is isomorphic to a quotient of a subgroup of Gamma of finite index. I will discuss examples and non-examples of groups Gamma with the property that every finite group is involved in Gamma. I will then describe my joint work with A. Reid where we show that mapping class groups of surfaces have this property. A remarkable feature of the proof is that it involves quantum topology. In fact, the proof uses precisely the Integral TQFT-representations of mapping class groups that P. Gilmer and I constructed some years ago. However, no previous knowledge of quantum topology will be assumed in this talk.

Posted October 9, 2013

Last modified November 20, 2013

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3:30 pm Lockett 285
Ravi Rau, Department of Physics and Astronomy, LSU

Quantum spins, real rotations, and a 1913 Ramanujan conjecture

Abstract: Quantum states are defined as complex variables and their time evolution is given by unitary transformations. For a quantum spin-1/2 or qubit of the field of quantum information, an equivalent picture of the Bloch sphere and real rotations of a unit vector from the origin to a point on the sphere has proved enormously useful. Extension of this nice geometrical view is also possible for a pair of qubits, such pairs being the fundamental objects of interest for entanglement and other quantum correlations that are used in quantum computing, key distribution and teleportation. The above results rest on the homeomorphism of SU(2)-SO(3) and SU(4)-SO(6) group pairs. These will be discussed and a hundred-year old conjecture of number theory (later the Ramanujan-Nagell Theorem) used to show that no such correspondence between unitary evolution and real rotations is available for systems of more qubits. However, the general construction of the evolution operator for SU(N) and of some of its sub-groups are likely to be of interest throughout the field of quantum information.

Posted January 9, 2014

Last modified January 15, 2014

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3:30 pm – 4:20 pm Lockett 285
Karl-Hermann Neeb, Universität Erlangen-Nürnberg

Reflection positivity and unitary Lie group representations

Reflection positivity (sometimes called Osterwalder-Schrader positivity) was introduced by Osterwalder and Schrader in the context of axiomatic euclidean field theories. On the level of unitary representations, it provides a passage from representations of the euclidean isometry group to representations of the Poincaré group. In our talk we shall explain how these ideas can be used to obtain a natural context for the passage from representations of Lie groups with an involutive automorphism (symmetric Lie groups) to representations of their dual Lie group. Already the case of one-parameter groups is of considerable analytic interest.

Posted January 9, 2014

Last modified January 15, 2014

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3:30 pm – 4:20 pm Lockett 285
Tom Lenagan, University of Edinburgh

Totally nonnegative matrices

A real matrix is totally nonnegative if each of its minors is nonnegative, and is totally positive if each minor is greater than zero. We will outline connections between the theory of total nonnegativity and the torus invariant prime spectrum of the algebra of quantum matrices, and will discuss some new and old results about total nonnegativity which may be obtained using methods derived from quantum matrix methods. Most of the material is joint work with Stephane Launois and Ken Goodearl. (You don't need to know anything about quantum matrices to follow this talk.)

Posted January 9, 2014

Last modified February 24, 2014

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3:30 pm – 4:20 pm Lockett 285
Satyan Devadoss, Williams College and Stanford University

Combinatorics of Surface Deformations

In the 1970s, Deligne and Mumford constructed a way to keep track of particle collisions using Geometric Invariant Theory. These spaces were then utilized by Gromov and Witten as invariants arising from string field theory and quantum cohomology. Later, Kontsevich and Fukaya generalized these ideas when studying deformation quantization to include surfaces with boundary and marked points. Such spaces appear in open-closed string theory, particularly with respect to holomorphic curves with Lagrangian submanifolds.

We consider a combinatorial framework to view the compactification of this space based on the pair-of-pants decomposition of the surface, relating it to the well-known phenomenon of bubbling. Our main result classifies all such spaces that can be realized as convex polytopes. A new polytope is introduced based on truncations of cubes, and its combinatorial and algebraic structures are related to generalizations of the classical associahedron.

Posted March 19, 2014

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 285
Stefan van Zwam, LSU

Beyond Total Unimodularity

A matrix is totally unimodular if the determinant of each square submatrix is in {-1, 0, 1}. Such matrices are a cornerstone of the theory of integer programming, and they have been studied extensively. In the late \'90s, Whittle introduced several classes of matrices with similar properties: the determinants of the submatrices are restricted to a certain set. In this talk I will discuss some results from the theory of totally unimodular matrices, and outline which of those results will, won\'t, or might generalize to Whittle\'s classes. The natural context for these problems is matroid theory, but prior knowledge of matroids is not required for this talk.

Posted February 25, 2014

Last modified April 5, 2014

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3:30 pm Business Education Complex (BEC) 1510
Jean-Pierre Serre, College de France, emeritus
Fields Medal recipient, Abel Prize recipient, and many more

Trace Forms

Refreshments will be served in the Keisler lounge from 2:45 to 3:15 pm. It takes approximately ten minutes to walk from Lockett Hall to BEC where the colloquium will be given.

Posted January 13, 2014

Last modified April 21, 2014

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3:30 pm – 4:20 pm Lockett 285
Melvin Leok, University of California, San Diego

Lie group and homogeneous variational integrators and their applications to geometric optimal control theory

The geometric approach to mechanics serves as the theoretical underpinning of innovative control methodologies in geometric control theory. These techniques allow the attitude of satellites to be controlled using changes in its shape, as opposed to chemical propulsion, and are the basis for understanding the ability of a falling cat to always land on its feet, even when released in an inverted orientation.

We will discuss the application of geometric structure-preserving numerical schemes to the optimal control of mechanical systems. In particular, we consider Lie group variational integrators, which are based on a discretization of Hamilton's principle that preserves the Lie group structure of the configuration space. In contrast to traditional Lie group integrators, issues of equivariance and order-of-accuracy are independent of the choice of retraction in the variational formulation. The importance of simultaneously preserving the symplectic and Lie group properties is also demonstrated.

Recent extensions to homogeneous spaces yield intrinsic methods for Hamiltonian flows on the sphere, and have potential applications to the simulation of geometrically exact rods, structures and mechanisms. Extensions to Hamiltonian PDEs and uncertainty propagation on Lie groups using noncommutative harmonic analysis techniques will also be discussed.

We will place recent work in the context of progress towards a coherent theory of computational geometric mechanics and computational geometric control theory, which is concerned with developing a self-consistent discrete theory of differential geometry, mechanics, and control.

This research is partially supported by NSF CAREER Award DMS-1010687 and NSF grants CMMI-1029445, DMS-1065972, and CMMI-1334759.

Posted August 26, 2014

Last modified September 2, 2014

Colloquium Questions or comments?

3:30 pm Lockett 241 (refreshments in the lounge at 3:00)
Ling Long, LSU

Hypergeometric functions and some recent applications in number theory

Abstract: Hypergeometric functions, which are solutions of ordinary differential equations with only 3 regular singularities, are an important class of special functions. Hypergeometric functions are very useful for several subjects such as combinatorial identities, triangle groups, modular forms, and algebraic varieties. In this talk, we will give a general introduction to hypergeometric functions and discuss some recent applications in number theory.

Posted August 26, 2014

Last modified September 8, 2014

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3:30 pm – 4:20 pm 235 Lockett
Xiaoliang Wan, Louisiana State University

Some numerical issues about quantifying the effects of uncertainty

Abstract: The role of uncertainty in mathematical models has received more attention in the last two decades due to the quick development of algorithms and computation capability. In this talk I will discuss numerical computation for three cases to quantify the effects of uncertainty, including parametric uncertainty, stochastic modeling based on Wick product and minimum action method for large deviation principle, where I will focus on the last case. I will describe some recent progress of minimum action method and its application to model the nonlinear instability of wall-bounded shear flows as a rare event.

Posted August 26, 2014

Last modified September 15, 2014

Colloquium Questions or comments?

3:30 pm – 4:30 pm Lockett 241
Scott Baldridge, Louisiana State University

Using a discrete set of transformations to prove smooth invariants of embedded tori in R^n

Abstract: It has been well known for over 80 years that knots in R^3 can be represented by 2-dimensional knot diagrams, and that three Reidemeister moves on those diagrams could be used to understand and prove invariants of knots. The problem with 2-dimensional diagrams is that smooth invariants that rely on metrics, differential forms, curvature forms, differential equations, etc. cannot be defined nor checked using the 2-dimensional moves. Motivated by search for a "Reidemeister-like" set of moves for embedded surfaces in R^4 (using actual embeddings, not projections), I recently developed the notion of a cube diagram to represent embedded n-tori in R^(n+2). In the case of knots in R^3, we proved that any cube diagram of a knot can be obtained from any other cube diagram of the knot using 2 types of cube diagram moves; this theorem allows us to prove differential topology invariants of knots in R^3 using a discrete set of transformations of the knot. In this talk, I will describe some differential topology applications of cube diagrams to Chern-Simons theory, and (if time) to higher dimensional embeddings.

Posted August 27, 2014

Last modified March 3, 2021

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 241
Yaniv Almog, Department of Mathematics, LSU

A Rigorous Proof of the Maxwell-Clausius-Mossotti Formula

We consider a large number of identical inclusions (say spherical), in a bounded domain, with conductivity different than that of the matrix. In the dilute limit, with some mild assumption on the first few marginal probability distribution (no periodicity or stationarity are assumed), we prove convergence in H1 norm of the expectation of the solution of the steady state heat equation, to the solution of an effective medium problem, which for spherical inclusions is obtained through the Maxwell-Clausius-Mossotti formula. Error estimates are provided as well.

Posted September 15, 2014

Last modified October 4, 2021

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3:30 pm – 4:20 pm Lockett 241
Susan Montgomery, University of Southern California

Orthogonal Representations: From Groups to Hopf Algebras

Posted September 4, 2014

Last modified October 8, 2014

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3:30 pm – 4:20 pm Lockett 241
Victor Moll, Department of Mathematics, Tulane University

The evaluation of definite integrals

Abstract: The question of finding out a closed-form of a definite integral has its origin with Calculus itself. In spite of many advances, the algorithmic question is still not completely resolved. This talk will present a variety of such evaluations that lead to interesting connections to Number Theory, Combinatorics and (naturally) Special Functions.

Posted October 30, 2014

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 241
James Zhang, University of Washington

Zariski cancellation problem for noncommutative algebras

Abstract: The famous Zariski cancellation problem asks if the polynomial algebra of more than two variables is cancellative. We consider a noncommutative version of the cancellation problem and obtain an affirmative answer for many classes of noncommutative algebras. Joint work with Jason Bell.

Posted October 2, 2014

Colloquium Questions or comments?

3:30 pm – 4:40 pm
F. Alberto Grünbaum, University of California, Berkeley

Time and band limiting and the bispectral problem: motivation, applications and open problems.

Abstract: The problem of how to best use (noisy) partial spectral information to determine a signal of finite duration starts with the work of C. Shannon, D. Slepian, H. Landau and H. Pollak at Bell Labs back around 1960. I will tell parts of this story as well as the way in which it led to the bispectral problem. The mathematics behind the bispectral problem is much richer than one may suspect: integrable systems like KdV, isomonodromic deformations, characters of representations of certain groups, certain non-commutative algebras of matrix valued differential operators, contacts with random matrix theory and other gems are all pieces of this puzzle. I am very interested in making this a two-way street in terms of finding bispectral situations that lead to the \"time-band limiting\" miracle of Slepian, Landau and Pollak in terms of finding a commuting differential operator for a naturally appearing integral one. This ongoing effort involves the work of several people, including Milen Yakimov.

Posted January 9, 2015

Last modified January 22, 2015

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 285
Arnab Ganguly, University of Louisville

Limit theorems in stochastic analysis

Limit theorems for stochastic processes have a variety of applications in diverse fields ranging from statistics to biology. While the weak convergence results often help to identify approximate continuous models from discrete-time ones, the results in the form of moderate and large deviations help to assess the quality of the approximations by studying various 'rare events'. The talk will focus on some systematic approaches to limit theorems for stochastic differential equations, which are particularly beneficial in infinite-dimensional settings. Several specific examples will be discussed illustrating the usefulness of these approaches.

Posted January 9, 2015

Last modified January 22, 2015

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 285
Lingjiong Zhu, University of Minnesota

Self-Exciting Point Processes

Self-exciting point processes are simple point processes that have been widely used in neuroscience, sociology, finance and many other fields. In many contexts, self-exciting point processes can model the complex systems in the real world better than the standard Poisson processes. We will discuss the Hawkes process, the most studied self-exciting point process in the literature. We will talk about the limit theorems and asymptotics in different regimes. Extensions to Hawkes processes and other self-exciting point processes will also be discussed.

Posted January 9, 2015

Last modified January 22, 2015

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 285
Eunghyun Lee, Centre de recherches mathématiques, Montréal

The Coordinate Bethe Ansatz Solvable Interacting Particle Systems

Interacting particle systems on the integer lattice that the coordinate Bethe Ansatz method can be applied to have connections to the random matrix theory and the symmetric function theory. Based on the coordinate Bethe Ansatz method, the exact formulas of the transition probabilities of the finite systems are given by the multi-dimensional contour integrals. In particular, we consider two initial configurations of the systems that are related to the GUE Tracy-Widom distribution and the GOE Tracy-Widom distribution. The combinatorial properties arising from our models are also discussed.

Posted January 9, 2015

Last modified January 22, 2015

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3:30 pm – 4:20 pm Lockett 285
Grace Wang, Duke University

Data Analysis with Low-dimensional Structures

Analyzing data collected from different fields is a challenge facing scientists and engineers. The property of being high-dimensional makes these data sets hard to investigate. Fortunately, in many cases, data locally concentrate along a low-dimensional subspace, which makes it possible to analyze. This talk will demonstrate different objectives where low-dimensional structures can be utilized for various data analysis purposes.

A major part of the talk will introduce a solution to the high-dimensional regression problem. More precisely, given a set of high-dimensional predictors {xi} and the corresponding high-dimensional responses {yi}, the high-dimensional regression problem seeks a function f such that f(xi) is sufficiently close to yi for all i. An algorithm with piecewise linear mappings built on a tree structure is proposed. It is designed to handle high-dimensional predictors and responses, and in particular, cases where closeness of predictors is inconsistent with closeness of responses. Experimental results demonstrate the excellent performance of our method.

Additional problems in the area will be discussed briefly, including the consistency analysis of a subspace-based classification algorithm and an automated J wave (anomaly) detection in heart (electrocardiography) signals.

Posted February 10, 2015

Last modified October 4, 2021

Colloquium Questions or comments?

3:30 pm – 4:20 pm 285 Lockett
Tadele Mengesha, The University of Tennessee, Knoxville

The variational convergence of some nonlocal convex functionals

In this talk, I will discuss a class of variational problems associated with nonlocal elastic energy of peridynamic-type which result in nonlinear nonlocal systems of equations with various volumetric constraints. The well-posedness of variational problems is established via careful studies of the related energy spaces which are made up of vector-valued functions. In the event of vanishing nonlocality we establish the convergence of the nonlocal energy to a corresponding local energy via Gamma convergence. For some convex energy functionals we will explicitly find the corresponding limit energy. As a special case the classical Navier–Lamé potential energy will be realized as a limit of linearized peridynamic energy offering a rigorous connection between the nonlocal peridynamic model to classical mechanics for small uniform strain.

Posted March 6, 2015

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 285
Aaron Lauda, University of Southern California

Traces and diagrams on the annulus

In this talk we will explore the notion of \"trace\" and how it can be generalized and given a graphical description in terms of diagrams on an annulus. Extending these notions beyond vector spaces and linear maps, we will show how the notion of trace can be generalized to any situation in which an algebraic object admits a graphical description in terms of planar diagrammatics. We will explain how these ideas provide a simple and intuitive way to understand sophisticated constructions arising in \"categorification\" and geometric representation theory.

Posted September 9, 2014

Last modified March 20, 2015

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3:30 pm – 4:40 pm TBA
Peter Kuchment, Texas A&M

Can one hear the heat of a body? Mathematics of hybrid imaging

Medical/industrial/geophysical imaging has been for decades an amazing area of applications of mathematics, providing a bonanza of beautiful and hard problems with real world applications. One can find almost any area of math being involved there. In this talk, I will survey a recent trend of designing the so called coupled physics (or hybrid) imaging methods and mathematical problems arising there. No prior knowledge of mathematics of imaging is assumed.

Posted October 20, 2014

Last modified March 31, 2015

Colloquium Questions or comments?

3:30 pm Lockett 285
Henri Cohen, University of Bordeaux I

Number Fields, Class Groups, and Regulators.

Abstract: In this talk, I will present the main results and conjectures related to the enumeration of number fields, and to their class groups and regulators. I will in particular give the known results on the enumeration of number fields of small degree, general conjectures on the subject, known theorems on the size of the class number, and the main heuristic predictions concerning class groups. I will also mention the computational tools used to obtain data concerning these objects.

Posted April 18, 2015

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 285
Ruth Charney, Brandeis University

Hyperbolic-like Boundaries

Boundaries of hyperbolic spaces play an important role in the study of hyperbolic groups and hyperbolic manifolds. Analogous boundaries exist for simply connected spaces of non-positive curvature (CAT(0) spaces) but they are not as well behaved and hence less effective. I will discuss the differences between these two settings and then introduce a new boundary for a very general class of metric spaces, designed to capture hyperbolic-like behavior in non-hyperbolic spaces. (Joint work with Harold Sultan and Matt Cordes)

Posted August 31, 2015

Last modified October 4, 2021

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 277
Pramod Achar, Mathematics Department, LSU

Intersection cohomology and representation theory

Intersection cohomology (a variant of the usual singular cohomology) has been a powerful tool in representation theory for over 35 years, playing a role in such major developments as the proof of the Kazhdan–Lusztig conjectures. After discussing some classical applications of complex intersection cohomology through examples, I will focus on new developments that have made intersection cohomology with Z/pZ coefficients more accessible, along with applications such as the proof of the Mirković–Vilonen conjecture.

Posted August 31, 2015

Last modified September 4, 2015

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 277
Shawn Walker, LSU

Numerical Analysis For Multi-Physics Moving Interface Problems

Moving interface and free boundary problems play a critical role in many areas of mathematics, physics, and engineering (examples are surface tension/curvature-driven flows and other geometric flows). Many new computational methods have been developed in recent years to tackle these problems in the presence of other physical effects. In this talk, I will discuss modeling and numerical analysis for three areas: two-phase problems, shape optimization, and liquid crystals. I will highlight theoretical results for modeling and simulating these problems, as well as show numerical results and simulations to illustrate the methods.

Posted September 23, 2015

Last modified October 18, 2015

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3:30 pm – 4:20 pm Lockett 277
Geoffrey Mason, UC Santa Cruz

The unbounded denominator conjecture

The unbounded denominator conjecture (UBD) is a very general statement that includes as special cases both the main modular-invariance conjecture in rational conformal field theory and an old conjecture of Atkin-Swinnerton-Dyer about noncongruence modular forms. We will explain the origins of UBD, its connections with the Fuchsian theory of linear differential equations, and how it can be proved in low dimensions. The talk is aimed at a general audience -- no special expertise required.

Posted January 29, 2016

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 285
Brian Street, University of Wisconsin-Madison

The Frobenius Theorem

We present a quantitative version of the classical Frobenius theorem from differential geometry. This theorem can be seen as providing scaling maps which can be used to study a range of problems in analysis. We present two such applications: a theory of singular Radon transforms (joint with E. M. Stein) and a theory of multi-parameter singular integrals which has applications to PDEs and several complex variables.

Posted January 29, 2016

Last modified January 30, 2016

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1:30 pm – 2:20 pm Lockett 241
Jing Tao, University of Oklahoma

Stable commutator lengths in right-angled Artin groups

The commutator length of an element g in the commutator subgroup [G,G] of a group G is the smallest k such that g is the product of k commutators. When G is the fundamental group of a topological space, then the commutator length of g is the smallest genus of a surface bounding a homologically trivial loop that represents g. Commutator lengths are notoriously difficult to compute in practice. Therefore, one can ask for asymptotics. This leads to the notion of stable commutator length (scl) which is the speed of growth of the commutator length of powers of g. It is known that for n > 2, SL(n,Z) is uniformly perfect; that is, every element is a product of a bounded number of commutators, and hence scl is 0 on all elements. In contrast, most elements in SL(2,Z) have positive scl. This is related to the fact that SL(2,Z) acts naturally on a tree (its Bass-Serre tree) and hence has lots of nontrivial quasimorphisms. In this talk, I will discuss a result on the stable commutator lengths in right-angled Artin groups. This is a broad family of groups that includes free and free abelian groups. These groups are appealing to work with because of their geometry; in particular, each right-angled Artin group admits a natural action on a CAT(0) cube complex. Our main result is an explicit uniform lower bound for scl of any nontrivial element in any right-angled Artin group. This work is joint with Talia Fernos and Max Forester.

Posted January 29, 2016

Last modified January 30, 2016

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3:30 pm – 4:20 pm Lockett 277
Roi Docampo Álvarez, Instituto de Ciencias Matemáticas (ICMAT)

The Nash problem for arc spaces

Algebraic varieties (zeros of polynomial equations) often present singularities: points around which the variety fails to be a manifold, and where the usual techniques of calculus encounter difficulties. The problem of understanding singularities can be traced to the very beginning of algebraic geometry, and we now have at our disposal many tools for their study. Among these, one of the most successful is what is known as resolution of singularities, a process that transforms (often in an algorithmic way) any variety into a smooth one, using a sequence of simple modifications.

In the 60's Nash proposed a novel approach to the study of singularities: the arc space. These spaces are natural higher-order analogs of tangent spaces; they parametrize germs of curves mapping into the variety. Just as for tangent spaces, arc spaces are easy to understand in the smooth case, but Nash pointed out that their geometric structure becomes very rich in the presence of singularities.

Roughly speaking, the Nash problem explores the connection between the topology of the arc space and the process of resolution of singularities. The mere existence of such a connection has sparked in recent years a high volume of activity in singularity theory, with connections to many other areas, most notably birational geometry and the minimal model program.

The objective of this talk is to give an overview of the Nash problem. I will give a precise description of the problem, and discuss the most recent developments, including a proof of the Nash conjecture in dimension two, and a partial solution to the Nash problem in arbitrary dimension.

Posted January 29, 2016

Last modified January 30, 2016

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3:30 pm – 4:20 pm Lockett 277
Jiuyi Zhu, LSU

Nodal geometry of Steklov eigenfunctions

The eigenvalue and eigenfunction problem is fundamental and essential in mathematical analysis. The Steklov problem is an eigenvalue problem with its spectral parameter at the boundary of a compact Riemannian manifold. Recently the study of Steklov eigenfunctions has been attracting much attention. We consider the quantitative properties: Doubling inequality and nodal sets. We obtain the sharp doubling inequality for Steklov eigenfunctions on the boundary and interior of manifolds using delicate Carleman estimates. We can ask Yau's type conjecture for the Hausdorff measure of nodal sets of Steklov eigenfunctions on the boundary and interior of the manifold. I will describe some recent progress about this challenging direction. Part of work is joint with C. Sogge and X. Wang.

Posted January 29, 2016

Last modified September 17, 2021

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3:30 pm – 4:20 pm Lockett 243
Fang-Ting Tu, National Center for Theoretical Sciences, Taiwan

Modular Forms on Shimura Curves and Hypergeometric Functions

Posted February 11, 2016

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 277
Thomas Parker, Michigan State University

Holomorphic curves, strings, and the GV conjecture

This is a talk on counting solutions of non-linear elliptic PDEs. After presenting the basic idea, I will explain, from two completely different perspectives, how the search for simple examples leads -- rather surprisingly -- to considering holomorphic maps into Calabi-Yau 3-folds X. Such maps are counted by the Gromov-Witten invariants of X, which are an infinite set of rational numbers. In 1998, physicists R. Gopakumar and C. Vafa conjectured that these Gromov-Witten invariants have a hidden structure: they are obtained, by a specific transform, from a set of more fundamental \"BPS numbers\", which are integers. The talk will conclude with a pictorial proof of the GV conjecture (joint work with E. Ionel) based on the idea of using deformations of almost complex structures to count the contributions of \"clusters of curves\".

Posted March 14, 2016

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 277
Barbara Rüdiger, Bergische Universität Wuppertal, Germany

Exponential ergodicity of jump-diffusion CIR processes

We analyze exponential ergodicity properties of affine term structure models. For some (Jump -) diffusion CIR processes, in particular the CIR process and the basic affine jump diffusion process (BAJD), we prove Harris recurrence properties. This permits in particular to calibrate the parameters of the model. These results are based on joint articles with V. Mandrekar, Peng Jin, Chiraz Trabelsi, and Jonas Kremer . All models will be introduced in this talk.

Posted April 15, 2016

Last modified March 3, 2021

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4:00 pm – 4:50 pm Lockett 277
Adimurthi, TIFR Bangalore

Structure of Entropy solutions of Hyperbolic conservation laws in space dimension

Hyperbolic Conservation laws in one space dimension has been studied for quite a long time starting from Lax and Oleinik. Main questions related to the existence of numbers shocks and nature of the solutions was not well understood. In this talk I will discuss these questions when the flux is convex.

Posted September 7, 2016

Last modified September 21, 2016

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3:30 pm – 4:20 pm Lockett 9
Karl Mahlburg, Department of Mathematics, LSU

Classical partition identities and automorphic forms

I will discuss my work on the deep connections between automorphic forms and partition identities. These include classical results such as the famous "sum-product" identities of Euler, Rogers-Ramanujan, Andrews-Gordon, and Schur, as well as more recent identities arising from affine Lie algebras and Lepowsky-Wilson's program of vertex operator algebras; a prominent example is due to Capparelli. Although these identities are of interest due to their intrinsic combinatorics and algebraic applications, they also often display automorphic properties, with examples of theta functions, modular functions, mock modular forms, and false theta functions. Some of these connections were only discovered recently, and have led to applications including asymptotic formulas, algebraicity, congruences, and probabilistic interpretations.

Posted October 25, 2016

Last modified November 29, 2016

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3:30 pm – 4:20 pm Lockett 9
Christof Geiss, UNAM Mexico

Geometric construction of semisimple Lie algebras - the non simply laced cases.

Abstract: After an unpublished manuscript by Schofield it is nowadays folklore

how to construct the universal enveloping algebra U(n) of the positive part

n of a symmetric Kac-Moody Lie algebra in terms of constructible functions

on the complex representation spaces of a quiver of the corresponding type.

This fails in the non-symmetric cases because the corresponding weighted quivers

can not be realized over an algebraically closed field.

In joint work with B. Leclerc and J. Schroeer we remedy this situation partially

by considering constructible functions on the representations of projective

dimension on certain 1-Iwanaga-Gorenstein algebras. However, for now this

approach works only in the Dynkin cases since we need to construct explicitly

for each root vector a non-vanishing constructible function.

Posted January 13, 2017

Last modified January 18, 2017

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 277
Priyam Patel, UC Santa Barbara, Department of Mathematics

Quantitative methods in hyperbolic geometry

Abstract: Peter Scott\'s famous result states that the fundamental groups of hyperbolic surfaces are subgroup separable, which has many powerful consequences. For example, given any closed curve on such a surface, potentially with many self-intersections, there is always a finite cover to which the curve lifts to an embedding. It was shown recently that hyperbolic 3-manifold groups share this separability property, and this was a key tool in Ian Agol\'s resolution to the Virtual Haken and Virtual Fibering conjectures for hyperbolic 3-manifolds.

I will begin this talk by giving some background on separability properties of groups, hyperbolic manifolds, and these two conjectures. There are also a number of interesting quantitative questions that naturally arise in the context of these topics. These questions fit into a recent trend in low-dimensional topology aimed at providing concrete topological and geometric information about hyperbolic manifolds that often cannot be gathered from existence results alone. I will highlight a few of them before focusing on a quantitative question regarding the process of lifting curves on surfaces to embeddings in finite covers.

Posted January 15, 2017

Last modified January 17, 2017

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 277
Vu Hoang, Rice University, Department of Mathematics

Singularity formation for equations of fluid dynamics

The basic equations of fluid mechanics were written down about 200 years

ago by Euler. To this day, they present a challenge for mathematical analysis and

many basic questions are still unsolved. One of these basic concerns the issue

of finite-time singularity formation versus global regularity. A great obstacle

for mathematical analysis is the fact that these equations involve both nonlinear

and non-local interactions. In my talk, I will describe recent efforts to understand the

mechanisms that are behind the singularity formation in fluid equations, starting from simple

model equations.

Posted January 16, 2017

Last modified March 2, 2021

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 277
Aynur Bulut, Princeton University, Department of Mathematics

Recent developments on deterministic and probabilistic well-posedness for nonlinear Schrödinger and wave equations.

Dispersive equations such as nonlinear Schrödinger and wave equations arise as mathematical models in a variety of physical settings, including models of plasma physics, the propagation of laser beams, water waves, and the study of many-body quantum mechanics. They also serve as model equations for studying fundamental issues in many aspects of nonlinear partial differential equations. Key questions in the analysis of these equations include issues of well-posedness (for instance, existence of solutions, uniqueness of these solutions, and their continuous dependence on initial data in appropriate topologies) locally in time, long-time existence and behavior of solutions, and, conversely, the possible existence of solutions which blow-up in finite time. In this talk, we will give an overview of several recent results concerning the local and global (long-time) theory, including some results where probabilistic tools are used to obtain estimates for randomly chosen initial data which are not available in deterministic settings. A recurring theme (and oftentimes obstacle) is the notion of supercriticality arising from the natural scaling of the equation — seeking to characterize long-time behavior of solutions when the relevant scale-invariant norms are not controlled by the conserved energy, or for initial data of very low regularity. The techniques involved include input from several areas of mathematics, including ideas arising in many areas of PDE, harmonic analysis, and probability.

Posted January 16, 2017

Last modified January 18, 2017

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 277
Marcel Bischoff, Vanderbilt

Subfactors, Fusion Categories and Conformal Nets

Abstract: Von Neumann algebras were mainly introduced to understand quantum theory and group representations. A factor is a von Neumann algebra with trivial center and an inclusion of two factors is called a subfactor. Finite index subfactors, in some sense, describe quantum symmetries which generalize finite groups. Similarly, fusion categories generalize the representation categories of finite groups. One can use von Neumann algebras to study chiral conformal field theory via so-called conformal nets. It turns out that conformal nets are a natural source of subfactors and fusion categories. It is an exciting open question if all fusion categories and subfactors come from conformal nets. I will introduce these three concepts and their interaction and discuss some recent results on the structure of inclusions of conformal nets and their representation theory.

Posted January 19, 2017

Last modified October 4, 2021

Colloquium Questions or comments?

3:30 pm – 4:20 pm TBA
Tian Yang, Stanford University

Mapping class group action on character varieties and the ergodicity

Character varieties of a surface are central objects in several branches of mathematics, such as low dimensional topology, algebraic geometry, differential geometry and mathematical physics. On the character varieties, there is a tautological action of the mapping class group - the group of symmetries of the surface, which is expected to be ergodic in certain cases. In this talk, I will review related results toward proving the ergodicity and introduce two long standing and related conjectures: Goldman's Conjecture and Bowditch's Conjecture. It is shown by Marche and Wolff that the two conjectures are equivalent for closed surfaces. For punctured surfaces, we disprove Bowditch's Conjecture by giving counterexamples, yet prove that Goldman's Conjecture is still true in this case.

Posted January 15, 2017

Colloquium Questions or comments?

3:30 pm – 4:20 pm
Cheng Yu, UT Austin

Weak solutions to the compressible Navier-Stokes equations

In this talk, we will discuss the construction of weak solutions for 3D compressible Navier-Stokes equations with degenerate viscosity. The method is based on the Bresch and Desjardins entropy conservation. The main contribution is to derive the Mellet-Vasseur type inequality for the weak solutions, even if it is not verified by the first level of approximation. This provides existence of global solutions for the compressible Navier-Stokes equations with large data. This is a joint work with A. Vasseur.

Posted January 30, 2017

Last modified March 2, 2021

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 277
Anton Zeitlin, Columbia University

Generalized Teichmüller Spaces, Spin Structures and Ptolemy Transformations

Teichmüller space, which parameterizes surfaces, is a fundamental space that is important in many areas of mathematics and physics. In recent times generalizations of this space have been intensely studied. The examples of such higher Teichmüller spaces are given by the so-called super-Teichmüller spaces. These appear as a natural object when studying a combinatorial approach to spin structures on Riemann surfaces and the generalization to supermanifolds. Super means that the structure sheaf is Z/2Z graded and contains odd or anti-commuting coordinates. The super-Teichmüller spaces are higher Teichmüller spaces corresponding to supergroups, which play an important role in geometric topology, algebraic geometry and mathematical physics. There the anti-commuting variables correspond to Fermions. After the introduction of these spaces, I will provide the solution of a long-standing problem of describing the analogue of Penner coordinates on super-Teichmüller space and its generalizations. The importance of these coordinates is justified by two remarkable properties: the action of the mapping class group, expressed via the so-called Ptolemy transformations, is rational, and the Weil-Petersson form is given by a simple explicit formula. I will end outlining some of the many emerging applications of this theory.

Posted October 11, 2016

Last modified February 13, 2017

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 277
Christopher Sogge, Johns Hopkins University
J. J. Sylvester Professor of Mathematics

On the concentration of eigenfunctions

I shall present some results in global harmonic analysis that concern properties of eigenfunctions on compact Riemannian manifolds. Using local arguments we can show that $L^p$ norms of eigenfunctions over the entire manifold are saturated if and only if there are small balls (if $p$ is large) or small tubular neighborhoods of geodesics (if $p$ is small) on which the eigenfunctions have very large $L^p$ mass. Neither can occur on manifolds of nonpositive curvature, or, more generally, on manifolds without conjugate points.

Posted January 31, 2017

Last modified February 15, 2017

Colloquium Questions or comments?

3:30 pm – 4:20 pm Locket 277
Barbara Rüdiger, Bergische Universität Wuppertal - Germany

The Enskog process and its relation to the Boltzmann equation

Abstract: The existence of a weak solution to a McKean-Vlasov type stochastic differential system corresponding to the Enskog equation of the kinetic theory of gases is established under suitable hypotheses. The distribution of any solution to the system at each fixed time is shown to be unique when the density exists. The existence of a probability density for the time-marginals of the velocity is verified in the case where the initial condition is Gaussian, and is shown to be the density of an invariant measure. This is a joint work with S. Albeverio and P. Sundar.

Posted October 25, 2016

Last modified March 6, 2017

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 277
Qing Xiang, University of Delaware

Applications of Linear Algebraic Methods in Combinatorics and Finite Geometry

Abstract: Most combinatorial objects can be described by incidence, adjacency, or some other (0,1)-matrices. So one basic approach in combinatorics is to investigate combinatorial objects by using linear algebraic parameters (ranks over various fields, spectrum, Smith normal forms, etc.) of their corresponding matrices. In this talk, we will look at some successful examples of this approach; some examples are old, and some are new. In particular, we will talk about the recent bounds on the size of partial spreads of H(2d-1,q^2) and on the size of partial ovoids of the Ree-Tits octagon.

Posted March 7, 2017

Last modified March 22, 2017

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 277
Peter Jorgensen, Newcastle University

SL_2-tilings, infinite triangulations, and continuous cluster categories

Abstract: An SL_2-tiling is an infinite grid of positive integers such that each adjacent 2x2-submatrix has determinant 1. These tilings were introduced by Assem, Reutenauer, and Smith for combinatorial purposes. We will show that each SL_2-tiling can be obtained by a procedure called Conway--Coxeter counting from certain infinite triangulations of the circle with four accumulation points. We will see how properties of the tilings are reflected in the triangulations. For instance, the entry 1 of a tiling always gives an arc of the corresponding triangulation, and 1 can occur infinitely often in a tiling. On the other hand, if a tiling has no entry equal to 1, then the minimal entry of the tiling is unique, and the minimal entry can be seen as a more complex pattern in the triangulation. The infinite triangulations also give rise to cluster tilting subcategories in a certain cluster category with infinite clusters related to the continuous cluster categories of Igusa and Todorov. The SL_2-tilings can be viewed as the corresponding cluster characters. This is a report on joint work with Christine Bessenrodt and Thorsten Holm.

Posted August 24, 2017

Last modified September 13, 2017

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 277
Shea Vela-Vick, Louisiana State University

Contact geometry and Heegaard Floer theory

Abstract: Among the most transformative applications of Ozsvath and Szabo's Heegaard Floer theory is to the study of contact structures on 3-manifolds. Ozsvath and Szabo first identified an invariant of contact structures taking values in their Heegaard Floer homology in 2002. Since its definition, this invariant has been responsible for a tremendous amount of progress in our understanding of tight contact structures. A steady stream of evidence suggests subtle links between geometric characteristics of contact structures and the algebraic formalism present in Heegaard Floer theory. In this talk, we will discuss how the flow of ideas between contact geometry and Floer theory can be leveraged to establish significant results in each context.

Posted August 22, 2017

Last modified August 27, 2017

Colloquium Questions or comments?

3:30 pm – 4:20 pm
Hongyu He, Department of Mathematics, LSU

Branching Laws and Interlacing Relation

Abstract: Let $H$ be a subgroup of a compact group $G$. Then any irreducible unitary representation of $G$, when restricted to $H$, decomposes into a direct sum of irreducible representations of $H$. A description of such a decomposition is often called a branching law. They are important in harmonic analysis, quantum mechanics and number theory. In this talk, I shall discuss the branching laws of the discrete series of the noncompact unitary groups and the recent progress towards the local Gan-Gross-Prasad conjectures. Discrete series representations were classified by Harish-Chandra in the sixties and played a fundamental role in Langland's program.

Posted August 25, 2017

Last modified September 19, 2017

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 277
John Etnyre, Georgia Institute of Technology

Curvature and contact topology

Abstract: Contact geometry is a beautiful subject that has important interactions with topology in dimension three. In this talk I will give a brief introduction to contact geometry and discuss its interactions with Riemannian geometry. In particular I will discuss a contact geometry analog of the famous sphere theorem and more generally indicate how the curvature of a Riemannian metric can influence properties of a contact structure adapted to it. This is joint work with Rafal Komendarczyk and Patrick Massot.

Posted September 1, 2017

Last modified November 8, 2017

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 277
Yifan Yang, National Taiwan University

Some evaluations of hypergeometric functions from theory of Shimura curves

Abstract: Shimura curves are generalizations of classical modular curves. Because of the lack of cusps on Shimura curves, most of the methods for classical modular curves cannot possibly be extended to the case of Shimura curves. However, in recent years, there have been some methods for Shimura curves emerging in literature. Among them, one method is to realize modular forms in terms of solutions of the so-called Schwarzian differential equations. In the case where a Shimura curve has genus 0 and exactly three elliptic points, this means that modular forms can be expressed in terms of hypergeometric functions. In this talk, we will explain how this idea, together with arithmetic properties and CM theory of Shimura curves, leads to some beautiful evaluations of hypergeometric functions.

Posted December 17, 2017

Last modified October 4, 2021

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 277
Susanna Dann, Technical University of Vienna

Bounding marginal densities via affine isoperimetry

Let $\mu$ be a probability measure on $R^n$ with a bounded density $f$. We prove that the marginals of $f$ on most subspaces are well-bounded. For product measures, studied recently by Rudelson and Vershynin, our results show there is a trade-off between the strength of such bounds and the probability with which they hold. Our proof rests on new affinely-invariant extremal inequalities for certain averages of $f$ on the Grassmannian and affine Grassmannian. These are motivated by Lutwak's dual affine quermassintegrals for convex sets. We show that key invariance properties of the latter, due to Grinberg, extend to families of functions. The inequalities we obtain can be viewed as functional analogues of results due to Busemann–Straus, Grinberg and Schneider. As an application, we show that without any additional assumptions on $\mu$, any marginal $\pi_E(\mu)$, or a small perturbation thereof, satisfies a nearly optimal small-ball probability.

Posted January 12, 2018

Last modified October 4, 2021

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 241
Jake Fillman, Virginia Tech

Spectral properties of quasicrystals

Discovered in the early 1980s by Dan Shechtman, quasicrystals are solids that simultaneously exhibit aperiodicity (a lack of translation symmetries) and long-range order (quantified by the presence of Bragg peaks in their diffraction patterns). We will discuss almost-periodic Schrödinger operators, which supply a rich family of operator-theoretic models of quasicrystals. Our discussion will center around the spectral properties of the underlying operator and transport properties of the associated quantum dynamics. We will discuss how some of our results may be viewed as an inverse spectral theoretic obstruction to solving Deift's conjecture for the KdV equation with current technology. We will conclude with a discussion of results in higher dimensions that are motivated by the Bethe–Sommerfeld conjecture.

Posted January 13, 2018

Last modified January 19, 2018

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 241
Galyna Dobrovolska, Columbia University

A geometric Fourier transform, noncommutative resolutions, and Hilbert schemes

Abstract: I will start by defining and computing an example of a geometric Fourier transform for constructible functions, and more generally for constructible sheaves. Next I will explain how geometric representation theory can be used to study categories of modules over Lie algebras and more general algebras which quantize symplectic resolutions. Lastly I will apply the above techniques in the case of the Hilbert scheme of points in the plane. (This talk is based on a joint work in progress with R. Bezrukavnikov and I. Loseu and on my Ph.D. thesis)

Posted January 12, 2018

Last modified January 17, 2018

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 241
Christine Lee, University of Texas at Austin

Understanding quantum link invariants via surfaces in 3-manifolds

Abstract: Quantum link invariants lie at the intersection of hyperbolic geometry, 3-dimensional manifolds, quantum physics, and representation theory, where a central goal is to understand its connection to other invariants of links and 3-manifolds. In this talk, we will introduce the colored Jones polynomial, an important example of quantum link invariants. We will discuss how studying properly embedded surfaces in a 3-manifold provides insight into the topological and geometric content of the polynomial. In particular, we will describe how relating the definition of the polynomial to surfaces in the complement of a link shows that it determines boundary slopes and bounds the hyperbolic volume of many links, and we will explore the implication of this approach on these classical invariants.

Posted January 19, 2018

Last modified January 21, 2018

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 241
Shawn X. Cui, Stanford, Institute for Theoretical Physics

Four Dimensional Topological Quantum Field Theories

Abstract: We give an introduction to topological quantum field theories (TQFTs), which have wide applications in low dimensional topology, representation theory, and topological quantum computing. In particular, TQFTs provide invariants of smooth manifolds. We give an explicit construction of a family of four dimensional TQFTs. The input to the construction is a class of tensor categories called $G$-crossed braided fusion categories where $G$ is any finite group. We show that our TQFTs generalize most known examples such as Yetter's TQFT and the Crane-Yetter TQFT. It remains to check if the resulting invariant of 4-manifolds is sensitive to smooth structures. It is expected that the most general four dimensional TQFTs should arise from spherical fusion 2-categories, the proper definition of which has not been universally agreed upon. Indeed, we prove that a $G$-crossed braided fusion category corresponds to a 2-category which does not satisfy the criteria to be a spherical fusion 2-category as defined by Mackaay. Thus the question of what axioms properly define a spherical fusion 2-category is open.

Posted January 22, 2018

Last modified October 4, 2021

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 241
Larry Rolen, Trinity College Dublin & Georgia Tech

Jensen–Pólya Criterion for the Riemann Hypothesis and Related Problems

In this talk, I will summarize forthcoming work with Griffin, Ono, and Zagier. In 1927 Pólya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann's $\Xi$-function. This hyperbolicity has been proved for degrees $d <= 3$. We obtain an arbitrary precision asymptotic formula for the derivatives $\Xi^{(2n)}(0)$, which allows us to prove the hyperbolicity of 100 % of the Jensen polynomials of each degree. We obtain a general theorem which models such polynomials by Hermite polynomials. In the case of Riemann's $\Xi$-function, this proves the GUE random matrix model prediction for the distribution of zeros in derivative aspect. This general condition also confirms a conjecture of Chen, Jia, and Wang on the partition function.

Posted January 21, 2018

Last modified October 4, 2021

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 241
Chun-Hung Liu, Princeton University

Graph minors and topological minors

Minors and topological minors are two closely related graph containment relations that have attracted extensive attentions in graph theory. Though giant breakthroughs have been made over decades, several questions about these two relations remain open, especially for topological minors. This talk addresses part of our recent work in this direction, including a proof of Robertson's conjecture about well-quasi-ordering graphs by the topological minor relation, a complete characterization of the graphs having the Erdős–Pósa property with respect to topological minors which answers a question of Robertson and Seymour, and a proof of Thomas's conjecture on half-integral packing. More open questions, such as Hadwiger's conjecture on graph coloring and its variations and relaxations, will be discussed in this talk.

Posted January 30, 2018

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 241
Andrew Zimmer, William and Mary

Non-positive curvature in several complex variables

Abstract: In this talk I will discuss how to use ideas from the theory of metric spaces of non-positive curvature to understand the behavior of holomorphic maps between bounded domains in complex Euclidean space. Every bounded domain has an metric, called the Kobayashi metric, which is distance non-increasing with respect to holomorphic maps. Moreover, sometimes this metric satisfies well-known non-positive curvature type conditions (for instance, Gromov hyperbolicity) and one can then use these conditions to understand holomorphic maps.

Posted February 22, 2018

Last modified February 25, 2018

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 284
Wen-Ching Winnie Li, Pennsylvania State University

Distribution of primes

Abstract: The distribution of prime numbers has been one of the central topics in number theory. It has a deep connection with the zeros of the Riemann zeta function. The concept of "primes" also arises in other context. For example, in a compact Riemann surface, as introduced by Selberg, primitive closed geodesic cycles play the role of primes; while in a finite quotient of a finite-dimensional building, for each positive dimension, there are primes of similar nature. In this talk we shall discuss the distributions of such primes and their connections with the analytic behavior of the associated zeta and L-functions.

Posted December 17, 2017

Last modified October 4, 2021

Colloquium Questions or comments?

3:00 pm – 3:50 pm Lockett 277
Guozhen Lu, University of Connecticut

Harmonic analysis on hyperbolic spaces and Sharp geometric inequalities

Sharp geometric and functional inequalities play an important role in modern analysis, geometry and partial differential equations. We will begin the talk with some overall review on the best constants and maximizers (aka optimizers) for geometric inequalities such as Sobolev inequalities, Hardy–Trudinger–Moser inequalities, and Caffarelli–Kohn–Nirenberg inequalities, etc. Then we will briefly explain why such inequalities are useful in analysis and geometry. We will then review the Poincaré model of hyperbolic spaces. Next, we will describe some recent works using the techniques of harmonic analysis on hyperbolic spaces to establish optimal geometric inequalities. These include the sharp Hardy-Adams inequalities on hyperbolic balls and Hardy–Sobolev–Maz'ya inequalities on upper half spaces or hyperbolic balls. Using the Fourier analysis on hyperbolic spaces, we will be able to establish sharper inequalities than those known classical inequalities in the literature. This talk in intended for general audience including graduate students.

Posted December 26, 2017

Last modified April 2, 2018

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 277
Stefan Kolb, Newcastle University

Quantum symmetric pairs

Abstract: Since they arose in the 1980s, quantum groups have become an integral part of representation theory with many deep applications, including canonical bases, link invariants and quantum integrable systems. Quantum symmetric pairs were originally invented to perform quantum group analogs of harmonic analysis on symmetric spaces. Over the past five years it has become increasingly clear that many of the applications and constructions for quantum groups allow analogs for quantum symmetric pairs. In this talk I will give an introduction to the theory of quantum symmetric pairs and explain some of the recent developments, illustrating them, where possible, in the simplest example of quantum sl(2).

Posted January 12, 2018

Last modified April 15, 2018

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 277
Birge Huisgen-Zimmermann, University of California, Santa Barbara

Representations of Quivers with Relations

Abstract: It is well known that the representations of a finite dimensional algebra may be viewed as representations of a certain quiver (insider jargon for a finite directed graph). We will start by outlining the ultimate goals pursued in the theory of such representations and illustrate them with classical results and examples. In particular, we will discuss and exemplify the notions of tame and wild representation type.

Then we will zero in on a geometric approach to classifying representations of a quiver. A priori, "degenerations" of a given representation appear to constitute an obstacle on the road to classification. We will explain how this negative can be turned into a positive by harnessing degeneration theory towards the classification goal.

Posted January 30, 2018

Last modified April 23, 2018

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 277
Luis Silvestre, University of Chicago

Integro-differential equations

Abstract: Integro-differential equations have been a very active area of research in recent years. In this talk we will explain what they are and in what sense they are similar to more classical elliptic partial differential equations. We will discuss connections with problems in probability, fluids and statistical mechanics.

Posted September 26, 2018

Last modified October 4, 2021

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Alexey Karapetyants, Southern Federal University, Russia and SUNY Albany

On Bergman type spaces of functions of nonstandard growth and related questions.

We study various Banach spaces of holomorphic functions on the unit disc and half plane. As a main question we investigate the boundedness of the corresponding holomorphic projection. We exploit the idea of V.P. Zaharyuta, V.I. Yudovich (1962) where the boundedness of the Bergman projection in Lebesgue spaces was proved using Calderón–Zygmund operators. We treat the cases of variable exponent Lebesgue space, Orlicz space, Grand Lebesgue space and variable exponent generalized Morrey space. The major idea is to show that the approach can be applied to a wide range of function spaces. This opens a door in a sense for introducing and studying new function spaces of Bergman type in complex analysis. We also study the rate of growth of functions near the boundary in spaces under consideration and their approximation by mollifying dilations.

Posted August 29, 2018

Last modified October 1, 2018

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Guillermo Goldsztein , School of Mathematics, Georgia Institute of Technology

Synchronization of pendula clocks and metronomes

Abstract: In 1665, Huygens discovered that, when two pendulum clocks

hanged from a same wooden beam supported by two chairs, they synchronize

in anti-phase mode. Metronomes provides a second example of oscillators

that synchronize. As it can be seen in many YouTube videos, metronomes

synchronize in-phase when oscillating on top of the same movable surface.

In this talk, we will review these phenomena, introduce a mathematical

model, and analyze the the different physical effects. We show that, in a

certain parameter regime, the increase of the amplitude of the

oscillations leads to a bifurcation from the anti-phase synchronization

being stable to the in-phase synchronization being stable. We argue that

this may explain the experimental observations.

Posted August 14, 2018

Last modified October 23, 2018

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Hongjie Dong, Brown University

Interior and boundary regularity for the incompressible Navier-Stokes Equations

Abstract: The regularity theory for fluid equations is at the heart of fluid flow modeling and leads to many new ideas in scientific computing, differential equations, and statistics, and has been characterized as one of the outstanding technical challenges in this era. For sufficiently regular data, the local strong solvability of the incompressible Navier-Stokes equations is well understood. Such solution is unique and locally smooth in both spatial and time variables. On the other hand, the global in time strong solvability is an outstanding open problem for d \geq 3. Another important type of solutions are called Leray-Hopf weak solutions. In the pioneering works of Leray and Hopf, it is shown that evolving from any divergence-free vector field in $L_2$, there exists at least one Leray-Hopf weak solution to the Navier-Stokes Equations. Although the problems of uniqueness and regularity of Leray-Hopf weak solutions are still open, solutions are known to be partially regular in certain cases, and fully regular under certain criteria. In this talk, I will first review some previous results on the conditional regularity of solutions to the incompressible Navier-Stokes equations in the critical Lebesgue spaces, and then discuss a recent work which mainly addressed the boundary regularity issue.

Posted October 11, 2018

Last modified November 3, 2018

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Scott Ahlgren, University of Illinois at Urbana-Champaign

Kloosterman sums, Maass cusp forms, and partitions.

Abstract: Kloosterman sums are exponential sums which appear naturally in a wide range of applications in number theory. Maass cusp forms are real-analytic automorphic forms which are eigenfunctions of the Laplace operator; they encode information about a variety of arithmetical problems. The partition function is the basic function of additive number theory: it counts the number of ways to break a natural number into parts. I will describe these objects and their history, discuss connections between them, and discuss some deep conjectures about their properties. I will also describe some recent applications to the theory of partitions.

Posted September 5, 2018

Last modified November 26, 2018

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Ko Honda, UCLA

Convex hypersurface theory in higher-dimensional contact topology

Abstract: Contact 3-manifolds occupy a central role in low-dimensional topology due to their interactions with Floer-theoretic invariants. Convex surface theory and bypasses are extremely powerful tools for analyzing contact 3-manifolds and in particular have been successfully applied to many classification problems. After reviewing convex surface theory in dimension three, we explain how to generalize many of their properties to higher dimensions. This is joint work with Yang Huang.

Posted December 4, 2018

Last modified January 7, 2019

Colloquium Questions or comments?

3:30 pm – 4:30 pm Lockett 232
Ruoyu Wu, University of Michigan

Weakly interacting particle systems on random graphs: from dense to sparse

Abstract: We consider the asymptotic behaviors of weakly interacting (mean-field) particle systems on random graphs that could be dense or sparse. The system consists of a large number of nodes in which the state of each node is governed by a stochastic process that has a mean-field type interaction with the neighboring nodes. Such systems arise in many areas, including but not limited to neuroscience, queueing theory and social sciences, which we will discuss in this talk.

In the dense graph case, the limiting system is described by the classic McKean--Vlasov equation. Law of large numbers, propagation of chaos, and central limit theorems are established and turn out to be the same as those in the complete graph case.

In the sparse case, the limiting system is quite different and depends heavily on the graph structure. We obtain an autonomous characterization of the local dynamics of a typical node and its neighbors when the limiting graph is a D- regular tree or a Galton--Watson tree.

If time permits, certain queueing systems with non-mean-field interactions will be discussed.

Posted January 11, 2019

Last modified January 14, 2019

Colloquium Questions or comments?

3:30 pm – 4:30 pm Lockett 232
William Feldman, University of Chicago

Interfaces in inhomogeneous media: pinning, hysteresis, and facets

Abstract: I will discuss some models for the shape of liquid droplets on rough solid surfaces. The framework of homogenization theory allows to study the large scale effects of small scale surface roughness, including interesting physical phenomena such as contact line pinning, hysteresis, and formation of facets.

Posted January 22, 2019

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Wenqing Hu, Missouri University of Science and Technology

Stochastic Approximations, Diffusion Limit and Randomly Perturbed Dynamical Systems - a probabilistic approach to machine learning

Abstract: The remarkable empirical performance of the Stochastic Gradient Descent (SGD) algorithm with constant learning rate has led to the effective training of many large-scale machine learning models in modern data science.

In this talk, we provide a theoretical understanding of the effectiveness of the SGD algorithm from a probabilistic approach. As the learning rate tends to zero, a stochastic differential equation is introduced to describe the diffusion limit of the discrete recursive scheme used in SGD. Based on this diffusion limit, we connect SGD with a randomly perturbed gradient system. This connection enables us to understand the stochastic dynamics of SGD via delicate probabilistic techniques in stochastic analysis (stochastic calculus). As examples, we will discuss several important theoretical problems around SGD that are raised by the daily practice of data scientists: How does SGD escape from stationary points (including saddle points and local minima)? Does SGD finally choose a local minimum point that agrees with the global minimum point of the loss function? How does SGD's noise covariance structure implicitly affect the regularization properties of its solution path? Our probabilistic approach provides insights into these problems through a unified mathematical framework that can also be carried to many other stochastic approximation algorithms.

Posted January 26, 2019

Last modified October 4, 2021

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Huanchen Bao, University of Maryland

From Schur-Weyl duality to quantum symmetric pairs

The classical Schur-Weyl duality relates the representation theory of general linear Lie algebras and symmetric groups. Drinfeld and Jimbo independently introduced quantum groups in their study of exactly solvable models, which leads to a quantization of the Schur duality relating quantum groups of general linear Lie algebras and Hecke algebras of symmetric groups.

In this talk, I will explain the generalization of the (quantized) Schur-Weyl duality to other classical types. This new duality leads to a theory of canonical bases arising from quantum symmetric pairs generalizing Lusztig's canonical bases on quantum groups.

Posted January 28, 2019

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Andrei Tarfulea, University of Chicago

Existence, continuation, and lower mass bounds for the Landau equation.

Abstract: Kinetic equations model gas and particle dynamics, specifically focusing on the interactions between the micro-, meso-, and macroscopic scales.
Mathematically, they demonstrate a rich variety of nonlinear phenomena,
such as hypoellipticity through velocity-averaging and Landau damping. The
question of well-posedness remains an active area of research.

In this talk, we look at the Landau equation, a mathematical model for plasma physics arising from the Boltzmann equation as so-called grazing
collisions dominate. Previous results are in the perturbative regime, or
in the homogeneous setting, or rely on strong a priori control of the
solution (the most crucial assumption being a lower bound on the density,
as this prevents the elliptic terms from becoming degenerate).

We prove that the Landau equation has local-in-time solutions with no
additional a priori assumptions; the initial data is even allowed to
contain regions of vacuum. We then prove a "mass spreading" result via a
probabilistic approach. This is the first proof that a density lower bound
is generated dynamically from collisions. From the lower bound, it follows
that the local solution is smooth, and we establish the mildest (to date)
continuation criteria for the solution to exist for all time.

Posted November 21, 2018

Last modified February 3, 2019

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Daniel Nakano, University of Georgia

On Donkin's Conjectures

Abstract: Let $G$ be a simple, simply connected algebraic group over an algebraically closed field of prime characteristic. Recent work of Kildetoft and Nakano and of Sobaje has shown close connections between two long-standing conjectures of Donkin: one on tilting modules and the lifting of projective modules for Frobenius kernels of $G$ and another on the existence of certain filtrations of $G$-modules. In this talk, I will survey recent results in this area and present new results where we verify the one direction of Donkin's $(p,r)$ Filtration Conjecture for rank 2 groups for all primes. I will also show a recently discovered counterexample to the Tilting Module Conjecture. These results represent joint work with Christopher Bendel, Cornelius Pillen and Paul Sobaje.

Posted August 29, 2018

Last modified March 3, 2021

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Iván Angiono, Universidad Nacional de Cordoba (National University of Cordoba)

Finite-dimensional pointed Hopf algebras over abelian groups

Hopf algebras appeared in the 50s associated to algebras of functions of algebraic groups, with a well-known bijection when we restrict to commutative Hopf algebras over the complex numbers. Other classical examples of Hopf algebras are the enveloping algebras of Lie algebras, which in turn are co-commutative. In the 80s Drinfeld and Jimbo introduced quantized versions of these enveloping algebras, which are examples of non-commutative nor co-cocommutative Hopf algebras with several applications to Mathematics and Physics. Related with them, Lusztig introduced in the 90s finite-dimensional versions, the so-called small quantum groups. All these Hopf algebras are pointed: the coradical is a group algebra, which dominates the structure of the Hopf algebra and becomes a first invariant to describe it. In this talk we present the classification of finite-dimensional Hopf algebras whose group is abelian. We start from the basic definitions with the first known examples. Then we recall the Lifting Method introduced by Andruskiewitsch and Schneider, a fundamental step for their classification result when the order of the group is coprime with 210. We describe Nichols algebras of diagonal type (some kind of algebras closely related to our problem) and the classification obtained by Heckenberger. Next we give a presentation by generators and relations and the first consequences of this result. Finally we give a generalization of the Lifting Method to obtain deformations of graded Hopf algebras and the end of the classification, a result by myself, Andruskiewitsch and Garcia Iglesias, and some consequences about tensor categories attached to these Hopf algebras.

Posted August 22, 2018

Last modified February 17, 2019

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Nicolas Andruskiewitsch, Universidad Nacional de Cordoba (National University of Cordoba)

On the classification of finite-dimensional Hopf algebras

Abstract: A gentle overview of the status of the classification of finite-dimensional Hopf algebras with emphasis on the relations with Lie theory.

Posted October 18, 2018

Last modified February 17, 2019

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Matthew Hedden, Michigan State University

Knot theory and algebraic curves

Abstract: The modern study of knots and links has important roots in the theory of algebraic curves, where links encode subtle features of singularities. This thread was taken in interesting new directions in the 20th century, and the interaction between links in 3-dimensional manifolds and algebraic curves in complex surfaces continues to be a rich and beautiful area. In this talk I will survey the subject, from its seeds in the work of Newton to interesting advances which have occurred in the past decade.

Posted January 9, 2019

Last modified March 18, 2019

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Amarjit Budhiraja, UNC Chapel Hill

On Some Calculus of Variations Problems for Rare Event Asymptotics

The theory of large deviations gives decay rates of probabilities of rare events in terms of certain optimal control problems. In general these control problems do not admit simple form solutions and one needs numerical methods in order to obtain useful information. In this talk I will present some large deviation problems where one can use methods of calculus of variations to give explicit solutions to the associated optimal control problems. These solutions then yield explicit asymptotic formulas for probability decay rates in several settings.

Posted January 17, 2019

Last modified April 22, 2019

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Jasson Vindas, Ghent University, Belgium

Complex Tauberian theorems for Laplace transforms

Abstract: Complex Tauberian theorems have been strikingly useful tools in diverse areas of mathematics such as number theory and spectral theory for differential operators. Many results in the area from the last three decades have been motivated by applications in operator theory and semigroups. In this talk we shall discuss some developments in complex Tauberian theory for Laplace transforms. We will focus on two groups of statements, usually labeled as Ingham-Karamata theorems and Wiener-Ikehara theorems. We will present sharp versions of such theorems, including results with minimal boundary requirements on the Laplace transforms, computation of optimal Tauberian constants, and error terms. Several classical applications will be discussed in order to explain the nature of these Tauberian theorems.

Posted September 13, 2019

Last modified October 4, 2021

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Marta Lewicka, University of Pittsburgh

Geometry and Elasticity: the Mathematics of Shape Formation

We discuss some mathematical problems combining geometry and analysis, that arise from the description of elastic objects displaying heterogeneous incompatibilities of strains. These strains may be present in bulk or in thin structures, may be associated with growth, swelling, shrinkage, plasticity, etc. We will describe the effect of such incompatibilities on the singular limits of bidimensional models, in the variational description pertaining to the "non-Euclidean elasticity" and discuss the interaction of nonlinear PDEs, geometry and mechanics of materials in the prediction of patterns and shape formation.

Posted September 13, 2019

Last modified October 28, 2019

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Selim Esedoglu , University of Michigan

Algorithms for motion by mean curvature of networks.

Many applications in science and engineering call for simulating the evolution of interfaces (curves in the plane, or surfaces in space), including networks of them, under motion by mean curvature and related geometric flows. These dynamics arise as gradient descent for energies that contain the sum of (sometimes weighted and anisotropic) surface areas of the interfaces in the network. The applications include image processing, computer vision, machine learning, and materials science. There are a plethora of algorithms for simulating motion by mean curvature, especially in the challenging multiphase setting. I will review some of the simplest and most elegant: those that attempt to generate the evolution, including any necessary topological changes, by alternating just a few very efficient operations. They include the threshold dynamics algorithm of Merriman, Bence, and Osher, and the Voronoi implicit interface method of Saye and Sethian. Unfortunately, not all of these extremely streamlined, closely related methods converge to their advertised limit. I will discuss how recent developments in our understanding of some of these algorithms have allowed us to fix their lack of convergence.

Posted October 21, 2019

Last modified November 2, 2019

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Ignacio Nahuel Zurrian, Universidad Nacional de Cordoba (National University of Cordoba)

Bispectrality and commuting operators

Abstract: We will discuss the role of bispectrality in the commuting operators phenomenon. We will also consider situations of different nature, e.g. continuous and discrete variables or a matrix/valued setup. Finally, I would like to explore some very recent results as well as the notion of reflecting operators.

Posted October 21, 2019

Last modified November 12, 2019

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Nathan Glatt-Holtz, Tulane University

A Bayesian Approach to Quantifying Uncertainty in Divergence Free Flows

We treat the statistical regularization of the ill-posed inverse problem of estimating a divergence free flow field u from the partial and noisy observation of a passive scalar θ. Our solution is a Bayesian posterior distribution, that is a probability measure μ which precisely quantifies uncertainties in u once one specifies models for measurement error and a prior knowledge for u. We present some of our recent work which analyzes μ both analytically and numerically. In particular we discuss a posterior contraction (consistency) result as well as some Markov Chain Monte Carlo (MCMC) algorithms which we have developed and refined and rigorously analyzed to effectively sample from μ. This is joint work with Jeff Borggaard, Justin Krometis and Cecilia Mondaini.

Posted September 24, 2019

Last modified November 2, 2019

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
John Voight, Dartmouth College

Heuristics for units in number rings

Units in number rings are gems of arithmetic, the most famous being the golden ratio and the integer solutions x,y to Pell's equation x^2 - D*y^2 = +/-1 for D > 0. Like gems, they are embedded deeply within. Refined questions about the structure of units remain difficult to answer, for example: how often does it happen that Pell's equation has a solution to the -1 equation? More generally, how often in a number ring is it that all totally positive units are squares? Absent theorems, we may still try to predict the answer to these questions. In this talk, we present heuristics (and some theorems!) for signatures of unit groups inspired by the Cohen-Lenstra heuristics for class groups, but involving an lustrous structure of number rings we call the 2-Selmer signature map. This is joint work with David S. Dummit and Richard Foote and with Ben Breen, Noam Elkies, and Ila Varma.

Posted September 10, 2019

Last modified March 2, 2021

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Leonid V. Berlyand, Department of Mathematics, Pennsylvania State University

PDE/Analysis techniques in deep learning: convergence & stability of neural net classifiers

While algorithms based on deep neural networks (DNNs) have been recently used in a wide range of practical problems (e.g., object, speech, and pattern recognition), there is no rigorous understanding for why and when DNNs may fail. Thus, there is a pressing need for a mathematical understanding of the behavior of DNNs to improve existing algorithms and develop new ones. The power of DNNs lies in their ability to "learn" how to solve problems via training, the iterative minimization of a loss (error) function. We use modern tools from PDE/ODE analysis to address the convergence and stability of DNN training algorithms. First, using entropy-entropy dissipation estimates, we study the convergence of DNNs, and establish a striking feature: the DNN mathematically diverges as the number of gradient descent iterations goes to infinity, but this divergence is very slow (logarithmic), with the loss function vanishing polynomially, leading to "practical convergence." Second (work in progress) we established conditions of the structure and dimensionality of data sets and DNN architecture under which an DNN algorithm is stable, that is, training cannot lead to a significant drop in accuracy. In particular, we demonstrated a connection between stability and distribution of misclassified objects in the training set. This is a joint work with P-E Jabin (U. of Maryland) and A. Safsten ( Ph.D. student at PSU).

Posted November 25, 2019

Last modified March 2, 2021

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Alexander Shapiro, UC Berkeley

Modular functor from higher Teichmüller theory

Abstract: Quantized higher Teichmüller theory, as described by Fock and Goncharov, assigns an algebra and its representation to a surface and a Lie group. This assignment is equivariant with respect to the action of the mapping class group of the surface, and is conjectured to give an analog of a modular functor, that is it should respect the operation of cutting and gluing of surfaces. In this talk I will outline a proof of the above conjecture, and explain how it is related to representation theory of quantum groups. This talk will be mostly based on joint works with Gus Schrader.

Posted November 26, 2019

Last modified November 27, 2019

Colloquium Questions or comments?

3:30 pm – 4:20 pm 232 Lockett
Yulong Lu, Duke University

Understanding and Accelerating Statistical Sampling via PDEs and Deep Learning

Abstract: A fundamental problem in Bayesian inference and statistical machine learning is to efficiently sample from probability distributions. Standard Markov chain Monte Carlo methods could be prohibitively expensive due to various complexities of the target distribution, such as multimodality, high dimensionality, large datasets, etc. To improve the sampling efficiency, several new interesting ideas/methods have recently been proposed in the community of machine learning, whereas their theoretical analysis are very little understood. In the first part of the talk, I will show how PDE analysis can be useful to understand some recently proposed sampling algorithms. Specifically, I will focus on the Stein variational gradient descent (SVGD), which is a popular particle sampling algorithm used in the machine learning community. I justify rigorously SVGD as a sampling algorithm through a mean field analysis. Then I will introduce a new birth-death dynamics, which can be used as a universal strategy for accelerating existing sampling algorithms. The acceleration effect of the birth-death dynamics is examined carefully when applied to the classical Langevin diffusion. For both SVGD dynamics and the birth-death dynamics, I will emphasize the (Wasserstein) gradient flow structure and the convergence to the equilibrium of the underlying PDE dynamics. The second part of the talk devotes to learning implicit generative models for sampling. Generative model such as Generative Adversarial Network (GAN) provides an important framework for learning and sampling from complex distributions. Despite the celebrated empirical success, many theoretical questions remain unsolved. A fundamental open question is: how well can deep neural networks express distributions? I will answer this question by proving a universal approximation theorem of deep neural networks for generating distributions.

Posted September 13, 2019

Last modified November 25, 2019

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Eric Rowell, Texas A&M

Mathematical Problems in Topological Quantum Computation

Abstract: Two-dimensional topological states of matter offer a route to quantum computation that would be topologically protected against the nemesis of the quantum circuit model: decoherence. In this talk I will give a mathematicians' perspective on some of the advantages and challenges of this model, focusing on the interplay of condensed matter physics, representation theory, low-dimensional topology and category theory. We will discover some compelling mathematical questions inspired by foundational problems in topological information theory along the way, and I will present a few results and ongoing projects with collaborators.

Posted November 27, 2019

Last modified December 5, 2019

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Rui Han, LSU

Spectral gaps in graphene structures

Abstract: In 1976 a celebrated butterfly was plotted by Douglas Hofstadter, describing (as a function of the magnetic flux $\alpha$) the spectrum of a tight-binding model of the motion of electrons on the square lattice under a perpendicular magnetic field (known as Harper's model). Proving the spectrum is a Cantor set for the Harper's model for any irrational flux was dubbed the "Ten Martini Problem" after Kac and Simon. This problem was solved completely about 10 years ago by Avila and Jitomirskaya. After a brief introduction to the Harper's model, I will talk about a model of graphene in magnetic fields, which has an underlying hexagonal lattice structure. I will discuss some recent results, including Cantor spectrum, spectral decomposition, Hausdorff dimension of the spectrum, Dirac points, and Bethe-Sommerfeld conjecture.

Posted November 25, 2019

Last modified November 26, 2019

Colloquium Questions or comments?

3:30 pm – 4:20 pm 232 Lockett
Krystal Guo, Université de Montreal

Applying algebraic graph theory to quantum computing

Abstract: The interplay between the properties of graphs and the eigenvalues of their adjacency matrices is well-studied. Important graph invariants, such as diameter and chromatic number, can be understood using these eigenvalue techniques. In this talk, we use classical techniques in algebraic graph theory to study quantum walks. A system of interacting quantum qubits can be modelled by a graph. The evolution of the quantum system can be completely encoded as a quantum walk in a graph, which can be seen, in some sense, as a quantum analogue of random walk. This gives rise to a rich connection between algebraic graph theory, linear algebra and quantum computing. In this talk, I will present recent results on the average mixing matrix of a graph; a quantum walk has a transition matrix which is a unitary matrix with complex values and thus will not converge, but we may speak of an average distribution over time, which is modelled by the average mixing matrix.

Posted December 19, 2019

Last modified January 8, 2020

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Rebecca Winarski, University of Michigan

Polynomials, branched covers, and trees

Abstract: Thurston proved that a branched cover of the plane that satisfies certain finiteness conditions is either equivalent to a polynomial (that is: conjugate via a mapping class) or it has a topological obstruction. We use topological techniques - adapting tools used to study mapping class groups - to produce an algorithm that determines when a branched cover is equivalent to a polynomial, and if it is, determines which polynomial a branched cover is equivalent to. Our approach gives a new, topological solution to Hubbard's twisted rabbit problem, as well as generalizations of this problem. This is joint work with Jim Belk, Justin Lanier, and Dan Margalit.

Posted January 19, 2020

Last modified January 21, 2020

Colloquium Questions or comments?

3:30 pm – 4:20 pm 232 Lockett
Kevin Schreve, University of Chicago

Group actions on contractible manifolds and L^2-cohomology

Abstract: The action dimension of a finitely generated group G is the smallest dimension of contractible manifold with proper action by G. I will describe a conjectured homological obstruction to such actions, and how this fits in with a conjecture of Hopf about Euler characteristics of closed, nonpositively curved manifolds. I will then describe some classes of groups where we can show this conjecture holds. This is based on joint work with Grigori Avramidi, Mike Davis, Giang Le, and Boris Okun.

Posted January 20, 2020

Colloquium Questions or comments?

3:30 pm – 4:20 pm 232 Lockett
Yu Pan, MIT

Augmentations and exact Lagrangian surfaces.

Abstract: A major theme in symplectic and contact topology is the study of Legendrian knots and the study of exact Lagrangian surfaces that connecting the knots. In the talk, we will talk about some rigidity results of exact Lagrangian surfaces using augmentation, a Floer type invariant of Legendrian knots.

Posted January 21, 2020

Last modified January 24, 2020

Colloquium Questions or comments?

3:30 pm – 4:20 pm 232 Lockett
Christin Bibby, University of Michigan

Combinatorics and topology of arrangements

Abstract: A hyperplane arrangement is a finite set of hyperplanes in a vector space. The way in which these hyperplanes intersect has a rich combinatorial structure (known as a matroid). A topologist may be more interested in the complement of their union. A motivating example is an ordered configuration space of distinct complex numbers, which is the complement to an arrangement whose underlying combinatorial structure is the lattice of set partitions. In this talk, we will explore some classical questions in the field of hyperplane arrangements, and what changes when more general varieties (or manifolds) play the role of the vector spaces. That is, we consider arrangements of smooth codimension-one subvarieties in a smooth algebraic variety, which intersect like hyperplanes, and examine the interplay between combinatorics, topology, and algebra.

Posted January 21, 2020

Last modified January 26, 2020

Colloquium Questions or comments?

3:30 pm – 4:20 pm 232 Lockett
Martin Friesen, University of Wuppertal

Ergodicity and regularity of affine processes

Abstract: In this talk, we address convergence to equilibrium as well as regularity of transition densities for affine processes on the canonical state space. First, we introduce and review different characterizations of affine processes through their Generator, corresponding Riccati equations, and semi-martingales. Then we prove that each affine process is the unique strong solution to a system of stochastic differential equations. As a particular application of this result, we investigate the convergence of transition probabilities in Wasserstein distances towards their unique invariant measure. Afterward, we study the regularity of transition probabilities (smoothness, Besov, strong Feller property). By combining this regularity with a coupling argument we also deduce exponential ergodicity in total variation. This talk is based on several works joint with: Peng Jin and Barbara Rüdiger.

Posted January 21, 2020

Last modified January 23, 2020

Colloquium Questions or comments?

3:30 pm – 4:20 pm 232 Lockett
Li Chen, LSU

On several functional inequalities for Markov semigroups and their applications

Abstract: Markov semigroups lie at the interface of analysis, PDEs, probability and geometry. Markov semigroup techniques, from both analytic and probabilistic viewpoints, have important applications in the study of functional inequalities coming from harmonic analysis, PDEs and geometry. In this talk, we discuss regularization properties of heat semigroups and their applications to the study of Sobolev type inequalities, isoperimetric inequalities and the $L^p$ boundedness of Riesz transforms in different geometric settings. Fractal examples without differential structures will be emphasized. Besides, we also discuss sharp and dimension-free $L^p$ bounds of singular integral operators via the martingale transform method.

Posted January 22, 2020

Last modified January 29, 2020

Colloquium Questions or comments?

3:00 pm – 3:50 pm Lockett 276
Khai Nguyen, NCSU

Differential Game Models of Optimal Debt Management

Abstract: In this talk, I will present recent results on game theoretical formulation of optimal debt management problems in infinite time horizon with exponential discount, modeled as a noncooperative interaction between a borrower and a pool of risk-neutral lenders. Here, the yearly income of the borrower is governed by a stochastic process and bankruptcy instantly occurs when the debt-to-income ratio reaches a threshold. Since the borrower may go bankrupt in finite time, the risk-neutral lenders will charge a higher interest rate in order to compensate for this possible loss of their investment. Thus, a "solution" must be understood as a Nash equilibrium, where the strategy implemented by the borrower represents the best reply to the strategy adopted by the lenders, and conversely. This leads to highly nonstandard optimization processes.

Posted February 3, 2020

Last modified October 4, 2021

Colloquium Questions or comments?

3:30 pm – 4:20 pm 232 Lockett
Felix Janda, University of Michigan & IAS Princeton

Enumerative geometry: old and new

Ever since people have studied geometry, they have counted geometric objects. For example, Euclid's Elements start with the postulate that there is exactly one line passing through two distinct points in the plane. The kinds of counting problems we are able to pose and to answer has grown significantly since then. Today enumerative geometry is a rich subject with connections to many fields, including combinatorics, physics, representation theory, number theory and integrable systems.

In this talk, I will show how to solve several classical counting questions. I will then move to a more modern problem with roots in string theory which has been the subject of intense study for the last three decades: The computation of the Gromov–Witten invariants of the quintic threefold, an example of a Calabi–Yau manifold.

Posted January 10, 2020

Last modified February 2, 2020

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Christopher Sogge, Johns Hopkins University
J. J. Sylvester Professor of Mathematics

The wave equation and Fourier analysis

Abstract: Many problems in harmonic analysis involve the wave equation, and one can use Fourier analysis and Fourier integral operators to solve wave equations. We shall discuss several of these problems, including spherical maximal estimates, local smoothing bounds and Kakeya problems. We shall also go over recent decoupling estimates of Bourgain and Demeter that were inspired by the work of Wolff on regularity estimates for the wave equation.

Posted January 22, 2020

Last modified March 3, 2021

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Jeffrey Danciger, UT Austin

Affine geometry and the Auslander Conjecture

The Auslander Conjecture is an analogue of Bieberbach's theory of Euclidean crystallographic groups in the setting of affine geometry. It predicts that a complete affine manifold (a manifold equipped with a complete torsion-free flat affine connection) which is compact must have virtually solvable fundamental group. The conjecture is known up to dimension six, but is known to fail if the compactness assumption is removed, even in low dimensions. We discuss some history of this conjecture, give some basic examples, and then survey some recent advances in the study of non-compact complete affine manifolds with non-solvable fundamental group. Tools from the deformation theory of pseudo-Riemannian hyperbolic manifolds and also from higher Teichmüller theory will enter the picture.

Posted February 9, 2020

Last modified March 2, 2020

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Thang Le, Georgia Tech

Knot invariants and algebraic structures based on knots

Abstract: Knot theory plays an important role in topology and has interesting relations to many remote branches of mathematics and physics, like number theory and non-commutative algebras. In this talk we discuss the an algebra of surfaces defined by knots (skein algebra) which has connections to many important objects including hyperbolic structures of surfaces, cluster algebra, and quantum groups. The talk is elementary, and no prior knowledge of knot theory or quantum groups is required.

Posted February 22, 2021

Last modified February 23, 2021

Colloquium Questions or comments?

12:00 pm – 1:00 am Zoom
Bahar Acu, ETH Zürich

Contact Topology and Geometry in High Dimensions

Posted February 25, 2021

Colloquium Questions or comments?

3:30 pm – 4:20 pm Zoom
Amita Malik, American Institute of Mathematics

Asymptotic analysis for certain arithmetic objects

Abstract: Counting problems lie at the heart of number theory, be it the study of primes, class numbers or the number of partitions of a positive integer. One of the most difficult underlying questions here pertains to the distribution of the zeros of L-functions. This goes back to the seminal paper on the study of the (Riemann) zeta function by Riemann in 1859. After an overview of the distribution of zeros of these functions, we discuss asymptotic behavior of the partition function with parts concerning a Chebotarev condition. In special cases, we obtain partitions into primes in arithmetic progressions. The error term present here gives an equivalent formulation of the generalized Riemann Hypothesis. Monotonicity of this partition function is established explicitly via an asymptotic formula in connection to a result of Bateman and Erdos.

Posted March 1, 2021

Last modified October 4, 2021

Colloquium Questions or comments?

12:30 pm – 1:20 pm Zoom
Gene Kopp, University of Bristol

Complex equiangular lines and the Stark conjectures

How many lines can you draw through the origin in $d$-dimensional space with all pairwise angles equal? The complex version of this question in an open problem with applications in quantum information theory. Zauner conjectured based on numerical evidence that a configuration of $d^2$ complex equiangular lines—also called a SIC-POVM (symmetric, informationally complete, positive operator-valued measure)—always exists, but such configurations are known only in finitely many dimensions. We discuss an unexpected connection between Zauner's conjecture and number theory, specifically, class field theory. In known examples, a set of numbers called overlap phases that characterize a SIC-POVM are Galois conjugate to square roots of Stark units—algebraic units in class fields predicted to arise from special values of $L$-functions. We discuss a conjectural construction of SIC-POVMs from special values of $L$-functions and some results toward proving that construction.

Posted September 28, 2021

Last modified October 4, 2021

Colloquium Questions or comments?

3:30 pm – 4:20 pm 232 Lockett
Mikhail Khovanov, Columbia University

Evaluating foams

Explicit constructions of link homology groups are based on foams, which can be thought of as cobordisms in 3D between planar graphs. We will explain a particular example of foams and their evaluations that relates to both the 4-color theorem and to the link homology that categorifies the quantum SL(3) link invariant (the Kuperberg bracket).

Posted January 7, 2022

Last modified January 20, 2022

Colloquium Questions or comments?

3:30 pm – 4:20 pm Zoom
Fan Yang, LSU

Localization for quasi-periodic quantum systems

Abstract: I will introduce models for quantum-mechanical particles where the Hamiltonian has random disorder (Anderson model) or is quasi-periodic. These models arise in condensed matter physics and are expected to exhibit insulator-like behavior (Anderson localization), when the disorders are large enough. These phenomena are proved using analytic methods, ranging from spectral theory and Fourier analysis to Diophantine approximations. I will discuss new analytical developments for studying the Maryland model (unbounded aperiodic potential) and quantum walks in magnetic and electric fields. I will give an outlook of various problems that one could study using our methods.

Posted January 10, 2022

Last modified January 12, 2022

Colloquium Questions or comments?

3:30 pm – 4:20 pm Zoom
Zhiyu Wang, Georgia Tech

Polynomial $\chi$-binding functions for graph classes

A graph class is called polynomially $\chi$-bounded if there is a function $f$ such that $\chi(G) \leq f(\omega(G))$ for every graph $G$ in this class, where $\chi(G)$ and $\omega(G)$ denote the chromatic number and clique number of $G$ respectively. A $t$-broom is a graph obtained from $K_{1,t+1}$ by subdividing an edge once. A fork is a graph obtained from $K_{1,4}$ by subdividing two edges. We show two conjectures: (1) we show that for graphs $G$ without induced $t$-brooms, $\chi(G) = o(\omega(G)^{t+1})$, answering a question of Schiermeyer and Randerath. For $t=2$, we strengthen the bound on $\chi(G)$ to $7.5\omega(G)^2$, confirming a conjecture of Sivaraman. (2) We show that any {triangle, fork}-free graph $G$ satisfies $\chi(G)\leq \omega(G)+1$, confirming a conjecture of Randerath.

Posted January 21, 2022

Colloquium Questions or comments?

12:00 pm – 12:50 pm Zoom
Benjamin Fehrman, University of Oxford

Rare events in interacting particle systems and stochastic PDE

Abstract: Interacting particle systems arise in diverse contexts, including to model magnetism in statistical mechanics, opinion and disease dynamics in society, competition between species, and neural networks in machine learning. For large populations evolving over a long period of time, the random fluctuations in these systems average out and their evolution can be described deterministically. However, the particle process will exhibit large deviations away from its mean. These events though rare can have substantial effects---such as a large concentration of energy or the appearance of a vacuum---and they are therefore important to understand and simulate. The purpose of this talk is to introduce a continuum model that replicates the far-from-equilibrium behavior in interacting particle systems. While such approximations have long been used to model out-of-equilibrium behavior in fluctuating hydrodynamics, their use has lacked a rigorous mathematical justification due to the supercriticality and degeneracy of the associated stochastic equations. We will first introduce a general probabilistic framework for describing fluctuations in random systems. Beginning with the simple random walk, which is a random process generated by flipping a coin, we will describe the law of large numbers, the central limit theorem, and a large deviations principle. We will then explain how these ideas are applied to interacting particle systems, and how such considerations provide a formal link to certain stochastic PDE. We have developed a robust existence and uniqueness theory for such equations, and shown that along appropriate scaling limits the solutions correctly describe the random fluctuations and rare events in particle systems. These results make rigorous the longstanding formal connection above, and provide new techniques to understand and model particle processes.

Posted January 11, 2022

Last modified January 12, 2022

Colloquium Questions or comments?

3:30 pm – 4:20 pm Zoom
Daniel Massatt, University of Chicago

Electronic Structure of Incommensurate 2D Heterostructures with Mechanical Relaxation

Since their discovery by Geim, 2D heterostructures have become a hotbed of research due to their novel structure. Stacking varying materials on top of each other allows for a vast range of possible materials and corresponding properties. However, this stacking typically leads to an aperiodic, or incommensurate, material due to lattice mismatch or rotational misalignment. This incommensuration increases the potential for tunability of electronic and mechanical properties, leading for example to the famous discovery of unconventional superconductivity of twisted bilayer graphene at the magic angle.

Experiment is vastly more expensive than numerical simulation, so much work is needed on building analysis and algorithms for understanding and computing efficiently these incommensurate materials. In this work, we discuss configuration and momentum space methodologies to build algorithms to compute observables of incommensurate heterostructures. We exploit the ergodicity of the misalignment, and for appropriate materials (such as those with conic or parabolic bands) we use carefully selected perturbative expansions in momentum space with a thorough error analysis.

Posted January 12, 2022

Last modified January 25, 2022

Colloquium Questions or comments?

11:30 am – 12:20 pm Zoom
Guangqu Zheng, University of Edinburgh

Wiener chaos, Gaussian analysis and Stochastic partial differential equations

Abstract: This talk goes around the concept of Wiener chaos, which was first introduced by N. Wiener (1938) and later modified by K. Ito (1951, 56). It has been recurrently brought up in recent years, as it arises naturally in the study of stochastic partial differential equations, parameter estimation, nodal statistics of a Gaussian random fields and stochastic geometry, to name a few. Notably, Nualart-Peccati’s fourth moment theorem (2004) and Nourdin-Peccati’s Malliavin-Stein approach (2008) further push Wiener chaos to the center of Gaussian analysis, and it has turned out to be very effective in obtaining quantitative limit theorems in practice. In this talk, we will focus on the central limit theorem for the stochastic wave equation driven by Gaussian noise. We will present how the Wiener chaos enters the picture, and then highlight the key ideas and sketch main steps for obtaining relevant limit theorems. If time permits, we will talk about how this line of research (ideas, techniques) may lead to some other interesting results, for example: (i) extending random field solution theory for nonlinear SPDEs driven by colored noise, (ii) obtaining Gaussian fluctuations for (renormalized) singular stochastic dispersive/parabolic PDEs.

Posted January 25, 2022

Colloquium Questions or comments?

3:30 pm – 4:20 pm Zoom
Ana Balibanu, Harvard University

Poisson transversals in representation theory

Abstract: Geometric representation theory studies groups and algebras by realizing their representations geometrically, through actions on associated algebraic varieties. Symplectic and Poisson structures appear naturally in this setting, and give key insights into the geometry of the spaces that carry them. In turn, these spaces provide foundational examples for new research directions in Poisson geometry. The purpose of this talk is to illustrate this interplay in the framework of transversal structures. We will begin by introducing the notion of Poisson transversality, and by giving examples of several well-known representation-theoretic algebraic varieties that arise as Poisson transversals. Motivated by multiplicative analogues of these varieties, we will then define a general class of transversal slices for quasi-Poisson structures. This construction is based on the algebraic data that comes from an associated complex semisimple group, and can be used to produce canonical compactifications of these spaces which have interpretations in the setting of mathematical physics.

Posted January 26, 2022

Colloquium Questions or comments?

3:30 pm – 4:20 pm Zoom
Michael Lindsey, Courant Institute, NYU

A sampling of Monte Carlo methods

Abstract: In this talk I discuss recent work in several different areas involving Monte Carlo sampling. In the first part of the talk, I consider generic sampling problems, especially in low to moderate dimension, for which I introduce an ensemble Markov chain Monte Carlo (MCMC) method that overcomes the difficulty of slow transitions between isolated modes. In the second part, I consider the problems of computing ground states and excited states of quantum many-body systems, which are eigenpairs of exponentially high-dimensional Hermitian operators. I present a new optimization method within the framework of variational Monte Carlo (VMC). The VMC framework approaches these problems by stochastic optimization over a parametric class of wavefunctions. Of particular interest are recently introduced neural-network-based parametrizations for which this approach yields state-of-the-art results. In the last part, I consider a lattice model of quantum critical metals that captures a plausible mechanism for high-temperature superconductivity. This model can be studied numerically by Monte Carlo methods, but previous approaches cannot reach the large-volume limit needed to reveal critical scaling properties due to cubic computational cost in the lattice volume. I present recent work toward a linear-scaling approach.

Posted January 25, 2022

Colloquium Questions or comments?

3:30 pm – 4:20 pm Zoom
Spencer Leslie, Duke University

Periods, L-values, and stabilization

Abstract: The study of period integrals of automorphic forms originates in deep questions about cohomology of locally symmetric spaces. A particularly powerful tool for studying periods is a relative trace formula, which often allows one to relate these integrals to other arithmetic objects like L-functions. In this talk, I review some of this story, discuss this modern approach to relating period integrals to L-functions, and introduce an important case of active research: unitary Friedberg-Jacquet periods. These periods are conjecturally related to central values of certain L-functions and are thus connected to deep conjectures on the cohomology of the associated locally symmetric spaces. To prove these conjectural relationships, a promising approach is to use a relative trace formula. However, new problems (known as instability) arise in this setting that must be overcome if one is to prove this relation. I will discuss my work on a theory of endoscopy and a stable relative trace formula to overcome these problems. This gives a refinement of the relative trace formula amenable to proving this conjecture.

Posted January 31, 2022

Last modified February 1, 2022

Colloquium Questions or comments?

3:30 pm – 4:20 pm Zoom
Mee Seong Im, United States Naval Academy

One-dimensional topological theories with zero-dimensional defects and finite state automata

Abstract: I will explain a surprising relation between topological theories for one-dimensional manifolds with decorations and values in the Boolean semiring and finite-state automata and their generalizations. This is joint work with Mikhail Khovanov.

Posted February 8, 2022

Colloquium Questions or comments?

3:30 pm – 4:20 pm Zoom
Yuwen Li, The Pennsylvania State University

Preconditioning in H(curl) and H(div)

Abstract: Preconditioning is widely used for efficiently solving large-scale systems of linear equations. In the first part of my talk, I will present a novel application of preconditioning and derive a posteriori error estimates of finite element methods in H(curl) and H(div) using preconditioning theory. In the second part, I will show new user-friendly preconditioners for solving discrete H(curl) and H(div) systems on triangulated surfaces. Part of this talk is based on joint work with Ludmil Zikatanov.

Posted February 14, 2022

Colloquium Questions or comments?

1:30 pm – 2:20 pm Zoom
Mikhail Khovanov, Columbia University

Universal construction and its uses in link homology and beyond.

Abstract: Universal construction starts with an evaluation of closed n-dimensional manifolds and extends that to a weak TQFT (topological quantum field theory) for n-dimensional cobordisms. We explain its role in a combinatorial approach to SL(N) link homology and review other applications, including in construction of new rigid tensor categories.

Posted February 12, 2022

Last modified April 7, 2022

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Andrey Smirnov, UNC-Chapel Hill

3-dimensional mirror symmetry and elliptic cohomology

Theoretical physicists predict the existence of a large class of symplectic varieties that appear in pairs (X, X'). The enumerative invariants of the varieties X and X' are expected to be closely related. In recent years it has been understood that equivariant elliptic cohomology provides a natural language to mathematically define and study this phenomenon. My talk is an elementary introduction to equivariant elliptic cohomology and the theory of the elliptic stable envelopes. I will define 3d mirror symmetry in this language and consider some simple explicit examples. If time permits, we discuss applications to enumerative geometry, quantum cohomology, and K-theory. The talk is aimed at a broad audience, no prior knowledge of equivariant cohomology theories is assumed.

Posted April 21, 2022

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Michael Falk, Northern Arizona University

Hypergraphs and Orlik-Solomon algebras

We define the Orlik-Solomon algebra of a hypergraph and establish an elementary decomposition theorem. This is sufficient to allow a classification of Orlik-Solomon algebras of matroids, and consequently, cohomology rings of complements of complex hyperplane arrangements, using the hypergraphic generalization of Whitney's 2-isomorphism theorem for graphs, due to Vertigan and Whittle (1997). The talk is based on joint work with Geoff Whittle, still in progress.

Posted September 6, 2022

Last modified October 12, 2022

Colloquium Questions or comments?

3:30 pm – 4:30 pm Lockett 232
Peter Schauenburg, Université de Bourgogne

Prime factorization(s) of modular categories

The raison d'être of modular categories is in their rôle (or rather in their many rôles) in mathematics and mathematical physics; the most direct perhaps in topological quantum field theory. Forgetting about all the exciting applications they are (finite, in a sense) algebraic structures with a categorical but not overly complicated definition. They are impossible to classify, but there are ongoing and very active investigations into partial (but still rich, hard, and interesting) classification results. We will humbly study an early general structural result by Michael Müger: A (semisimple) modular category can be written as a Deligne product of prime factors. This immediately begs the question whether the factors are unique. Along with the existence of the factorization it was already proved that it is in fact unique if the category does not contain invertible objects. If all the simples of the category are invertible on the other hand, the factorization was already known to be non-unique before modular categories were ever defined (sic). In joint work with Michael Müger we try to find out what exactly can happen between these extreme cases. This is work in progress, since between partial positive uniqueness results and some counterexamples there remains uncharted territory.

Posted October 20, 2022

Last modified October 21, 2022

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Hal Schenck, Auburn University

Calabi-Yau threefolds in P^n and Gorenstein rings

In physics, String Theory posits a symmetry between certain pairs of geometric objects: Calabi-Yau threefolds. In algebra, Gorenstein rings are objects possessing an intrinsic, internal symmetry. We’ll discuss connections between the two sets of objects, going from the basic algebraic construction to computation, culminating in smooth Calabi-Yau threefolds with new Hodge numbers. The talk will start from ground zero, and anything we need along the way will be illustrated with examples. Joint work with M. Stillman and B. Yuan.

Posted January 17, 2023

Last modified February 14, 2023

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Naga Manasa Vempati, Georgia Tech

Weighted Inequalities for Singular Integral Operators

Weighted inequalities for singular integral operators are central in the study of non-homogeneous harmonic analysis. Two weight inequalities for singular integral operators, in particular attracted attention as they can be essential in the perturbation theory of unitary matrices, spectral theory of Jacobi matrices and PDE's. In this talk, I will discuss several results concerning the two weight inequalities for various Calderón-Zygmund operators in both Euclidean setting and in the more generic setting of spaces of homogeneous type in the sense of Coifman and Weiss. The two-weight conjecture for singular integral operators T was first raised by Nazarov, Treil and Volberg on finding the real variable characterization of the two weights u and v so that T is bounded on the weighted L^2 spaces. This conjecture was only solved completely for the Hilbert transform on R until recently. In this talk, I will describe our result that resolves a part of this conjecture for any Calderón-Zygmund operator on the spaces of homogeneous type by providing a complete set of sufficient conditions on the pair of weights. We will also discuss the existence of similar analogues for multilinear Calderón-Zygmund operators.

Posted January 11, 2023

Last modified January 18, 2023

Colloquium Questions or comments?

2:30 pm – 3:20 pm Lockett 285
Zhongjian Wang, University of Chicago

Deep learning of multi-scale PDEs based on data generated from particle methods

Abstract: Solving multi-scale PDEs is difficult in high-dimensional and/or convection-dominant cases. The interacting particle methods (IPM) are shown to outperform solving PDEs directly. Examples include computing effective diffusivities, KPP front speed, and asymptotic transport properties in topological insulators. However, the particle simulation takes a long time before convergence and is lack of surrogate models for physical parameters. In this regard, we introduce the DeepParticle methods, which learn the pushforward map from arbitrary distribution to IPM-generated distribution by minimizing the Wasserstein distance. In particular, we formulate an iterative scheme to find the transport map and prove the convergence. On the application side, in addition to KPP invariant measures, our method also applies to investigate the blow-up behavior in chemotaxis models.

Posted January 18, 2023

Last modified February 14, 2023

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Oishee Banerjee, IAS, Princeton

Cohomology and arithmetic of Mapping spaces

How do we describe the topology of the space of all nonconstant holomorphic (respectively, algebraic) maps F: X->Y from one complex manifold (respectively, variety) to another? What is, for example, its cohomology? Such problems are old but difficult, and are nontrivial even when the domain and range are Riemann spheres. In this talk I will explain how these problems relate to other parts of mathematics such as spaces of polynomials, arithmetic (e.g. the geometric Batyrev-Manin type conjectures) and algebraic geometry (e.g. moduli spaces of elliptic fibrations, of smooth sections of a line bundle, etc). I will show how one can fruitfully attack such problems by incorporating techniques from homotopy theory to the holomorphic/algebraic world (e.g. by constructing a new spectral sequence). Most of this talk should be understandable to first year graduate students.

Posted January 18, 2023

Last modified January 23, 2023

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Nick Rozenblyum, University of Chicago

Langlands duality and higher traces

A central theme in arithmetic geometry over finite fields is the passage from geometric invariants to arithmetic information by taking the trace of Frobenius. I will describe a higher version of this procedure with a particular focus on applications to the Langlands correspondence over function fields. In this case, this procedure relates the geometric Langlands correspondence with the classical one. Specifically, we obtain that the space of automorphic functions is the categorical trace (aka Hochschild homology) of Frobenius acting on an appropriate version of the automorphic category. This leads to a localization of the space of automorphic functions on a moduli space of Langlands parameters, giving a refinement of V. Lafforgue's spectral decomposition. This is based on joint works with Arinkin, Gaitsgory, Kazhdan, Raskin, and Varshavsky.

Posted January 25, 2023

Last modified January 27, 2023

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Yeonjong Shin, KAIST

Towards Trustworthy Scientific Machine Learning: Theory, Algorithms, and Applications

Abstract: Machine learning (ML) has achieved unprecedented empirical success in diverse applications. It now has been applied to solve scientific problems, which has become an emerging field, Scientific Machine Learning (SciML). Many ML techniques, however, are very complex and sophisticated, commonly requiring many trial-and-error and tricks. These result in a lack of robustness and interpretability, which are critical factors for scientific applications. This talk centers around mathematical approaches for SciML, promoting trustworthiness. The first part is about how to embed physics into neural networks (NNs). I will present a general framework for designing NNs that obey the first and second laws of thermodynamics. The framework not only provides flexible ways of leveraging available physics information but also results in expressive NN architectures. The second part is about the training of NNs, one of the biggest challenges in ML. I will present an efficient training method for NNs - Active Neuron Least Squares (ANLS). ANLS is developed from the insight gained from the analysis of gradient descent training.

Posted January 23, 2023

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Xingting Wang, Howard University

Quantum groups and their invariants in positive characteristic

Abstract: The theory of quantum groups originated in algebraic topology and algebraic geometry in mid-twentieth century. Nowadays, it has applications in a wide rage of settings across mathematics and physics, including quantum invariants of links and 3-manifolds and solutions of the quantum Yang-Baxter equation. In this talk, I will discuss the theory of quantum groups and their invariants in positive characteristic. As an application, I will investigate their connections to different areas of mathematics and physics.

Posted January 31, 2023

Last modified February 6, 2023

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Nadia Drenska, Johns Hopkins University

A PDE Interpretation of Prediction with Expert Advice

Abstract: We study the problem of prediction of binary sequences with expert advice in the online setting, which is a classic example of online machine learning. We interpret the binary sequence as the price history of a stock, and view the predictor as an investor, which converts the problem into a stock prediction problem. In this framework, an investor, who predicts the daily movements of a stock, and an adversarial market, who controls the stock, play against each other over N turns. The investor combines the predictions of n ≥ 2 experts in order to make a decision about how much to invest at each turn, and aims to minimize their regret with respect to the best-performing expert at the end of the game. We consider the problem with history-dependent experts, in which each expert uses the previous d days of history of the market in making their predictions. The prediction problem is played (in part) over a discrete graph called the d-dimensional de Bruijn graph. We focus on an appropriate continuum limit. Using methods from optimal control, graph theory, and partial differential equations, we discuss strategies for the investor and the adversarial market. We prove that the value function for this game, rescaled appropriately, converges as N goes to infinity at a rate of O(N^−1/2) (for C^4 payoff functions) to the viscosity solution of a nonlinear degenerate parabolic PDE. It can be understood as the Hamilton-Jacobi-Issacs equation for the two-person game. As a result, we are able to deduce asymptotically optimal strategies for the investor. This is joint work with Robert Kohn and Jeff Calder.

Posted February 8, 2023

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Xiaoqi Huang, University of Maryland

Spectral Cluster and Weyl Remainder Estimates for the Laplacian and on compact manifolds

Abstract: In the first part of the talk, we shall discuss generalizations of classical versions of the Weyl formula involving Schrödinger operators on compact boundaryless Riemannian manifolds with critically singular potentials V. In particular, we extend the classical results of Avakumović, Levitan and Hörmander by obtaining sharp bounds for the error term in the Weyl formula when we merely assume that V belongs to the Kato class, which is the minimal assumption to ensure that H_V is essentially self-adjoint and bounded from below or has favorable heat kernel bounds. In the second part, we will discuss the problem of trying to obtain improved eigenfunction estimates under geometric assumptions, such as the presence of negative sectional curvatures, which is based on the use of microlocal Kakeya-Nikodym estimates along with a combination of local and global harmonic analysis that further exploit geometric assumptions. We will also discuss some new sharp estimates involving the tori eigenfunctions.

Posted February 13, 2023

Last modified February 14, 2023

Colloquium Questions or comments?

1:00 pm – 2:00 pm Zoom
Benjamin Moore, Charles University, Prague

Quantum field theory, colouring, and the Strong Nine Dragon Tree Conjecture

Abstract: I'll discuss various aspects of structural graph theory which I have worked on. In particular, I'll describe how certain nice Feynman Integrals correspond with a minor closed class, allowing structural graph theory to show up in Quantum field theory. I'll also discuss the Strong Nine Dragon Tree Conjecture, and Sebastian Mies and my recent progress has led to towards the thin tree conjecture, and finish with a counterexample to a conjecture of Galvin and Rodl from 1977, by describing graphs with arbitrarily large chromatic number, have no clique of size four, and all triangle-free induced subgraphs have chromatic number at most 4.

Posted January 19, 2023

Last modified January 20, 2023

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Irina Markina, University of Bergen

On rolling of manifolds

In the talk, we will introduce the notion of rolling one manifold over another. The idea of the rolling map originated as a simple mathematical model of rolling a ball over a plate with the constraints of no-slip and no-twist motion in the works of S. Chaplygin (1897), K. Nomizu (1978), R.Bryan and L.Hsu (1993). The geometric features are closely related to the distributions of E.Cartan type (1910). Later this idea was extended to the rolling of Riemannian manifolds of any dimension, as an isometry map preserving the parallelism of vector fields. After a historical overview and necessary definitions, we also mention some applications in the interpolation on nonlinear spaces and construction of stochastic processes on manifolds.

Posted February 9, 2023

Last modified February 24, 2023

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Melody Chan, Brown University

Moduli spaces of graphs

My goal is to share with you a broad view on the topic of moduli spaces of graphs. A metric graph is a graph—a finite network of vertices and edges—together with a prescription of a positive real length on each edge. I’ll use the term “moduli space of graphs’’ to refer to certain combinatorial spaces—think simplicial complexes—that furnish parameter spaces for metric graphs. There are different flavors of spaces depending on some additional choices of decorations on the graphs, but roughly, each cell parametrizes all possible metrizations of a fixed combinatorial graph. Many flavors of these moduli spaces have been in circulation for a while, starting with the work of Culler-Vogtmann in the 1980s on Outer Space. They have also recently played an important role in some recent advances using tropical geometry to study the topology of moduli spaces of curves and other related spaces. These advances give me an excuse to give what I hope will be an accessible introduction to moduli spaces of graphs and their connections with geometry.

Posted August 8, 2023

Last modified August 27, 2023

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
John Etnyre, Georgia Institute of Technology

Contact geometry and Legendrian knots

In this talk, I will discuss the current state of our understanding of contact structures on 3-manifolds and the related problem of Legendrian knots in contact manifolds. We will begin the talk by recalling basic definitions and examples in contact geometry, discuss natural occurrences of contact structures in everyday life, and then survey the long history of contact geometry and its connections to many areas of math and science. After this, we will focus on recent advances in our understanding of Legendrian knots in contact manifolds and how this is relevant to trying to classify the types of contact structures a 3-manifold might support.

Posted August 8, 2023

Last modified August 28, 2023

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Emanuele Delucchi, University of Applied Sciences and Arts of Southern Switzerland (SUPSI)

On polytopes associated to metric spaces

In 2010, Vershik proposed a new combinatorial invariant of metric spaces given by a class of polytopes that arise in the theory of optimal transport, called “Wasserstein polytopes” or “Kantorovich-Rubinstein polytopes” (KRP). Recently such polytopes have been shown to play an important role in a host of contexts, from machine learning to computational biology. However, little is known in general to date about their structure. In this talk I will state the definitions in the case of finite metric spaces and by outlining some context. I will then show some examples and survey some recent results and open questions about the combinatorial structure of such polytopes. The talk includes joint work with Linard Hössly, Alessio D'Alì, Mateusz Michalek, Lukas Kühne and Leonie Mühlherr.

Posted September 18, 2023

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Wilhelm Schlag, Yale University

Lyapunov exponents, Schrödinger cocycles, and Avila’s global theory

In the 1950s Phil Anderson made a prediction about the effect of random impurities on the conductivity properties of a crystal. Mathematically, these questions amount to the study of solutions of differential or difference equations and the associated spectral theory of self-adjoint operators obtained from an ergodic process. With the arrival of quasicrystals, in addition to random models, nonrandom lattice models such as those generated by irrational rotations or skew-rotations on tori have been studied over the past 30 years. By now, an extensive mathematical theory has developed around Anderson’s predictions, with several questions remaining open. This talk will attempt to survey certain aspects of the field, with an emphasis on the theory of SL(2,R) cocycles with an irrational or Diophantine rotation on the circle as base dynamics. In this setting, Artur Avila discovered about a decade ago that the Lyapunov exponent is piecewise affine in the imaginary direction after complexification of the circle. In fact, the slopes of these affine functions are integer valued. This is easy to see in the uniformly hyperbolic case, which is equivalent to energies falling into the gaps of the spectrum, due to the winding number of the complexified Lyapunov exponent. Remarkably, this property persists also in the non-uniformly hyperbolic case, i.e., on the spectrum of the Schrödinger operator. This requires a delicate continuity property of the Lyapunov exponent in both energy and frequency. Avila built his global theory (Acta Math. 2015) on this quantization property. I will present some recent results with Rui HAN connecting Avila’s notion of acceleration (the slope of the complexified Lyapunov exponent in the imaginary direction) to the number of zeros of the determinants of finite volume Hamiltonians relative to the complex toral variable. This connection allows one to answer questions arising in the supercritical case of Avila’s global theory concerning the measure of the second stratum, Anderson localization on this stratum, as well as settle a conjecture on the Hölder regularity of the integrated density of states.

Posted January 25, 2024

Colloquium Questions or comments?

3:30 pm – 4:30 pm Lockett 232
Changhong Mou, University of Wisconsin-Madison

Reduced Order Modeling in the Age of Data

Abstract: Data-driven modeling of complex dynamical systems is becoming increasingly popular across various domains of science and engineering. In this talk, I will introduce a systematic multiscale data-driven closure reduced order model (ROM) framework for complex systems with strong chaotic or turbulent behavior. I will utilize available data to construct novel ROM closure terms, thereby capturing the interaction between resolved and unresolved modes. Next, I will explain how the new data-driven closure ROM can be integrated with a conditional Gaussian data assimilation framework that employs cost-effective, conditionally linear functions to capture the statistical features of the closure terms. This leads to the stochastic data-driven closure ROM that facilitates an efficient and accurate scheme for nonlinear data assimilation (DA), the solution of which is provided by closed analytic formulae that do not require ensemble methods. It also allows the ROM to avoid many potential numerical and sampling issues in recovering the unobserved states from partial observations. Furthermore, I will introduce a hybrid DA algorithm for complex dynamical systems with partial observations. The method exploits cheap stochastic parameterized ROMs for filtering the observed state variables, significantly reducing the computational cost. It also uses machine learning to build a nonlinear map between observed and unobserved state variables, which enables the efficient computation of the ensemble members of the unobserved states. The hybrid DA algorithm is successfully applied to a precipitating quasi-geostrophic (PQG) model, which includes the effects of water vapor, clouds, and rainfall beyond the classical two-level QG model.

Posted January 25, 2024

Colloquium Questions or comments?

3:30 pm – 4:30 pm Lockett 232
Michael Novack, Carnegie Mellon

Soap films, Plateau's laws, and the Allen-Cahn equation

Abstract: Plateau's problem of minimizing area among surfaces with a common boundary is the basic model for soap films and leads to the theory of minimal surfaces. In this talk we will discuss a modification of Plateau's problem in which surfaces are replaced with regions of small but positive volume. The model captures features of real soap films that cannot be described by minimal surfaces, and the corresponding analysis requires the development of new ideas in geometric measure theory. We will also discuss the PDE approximation of this problem via the Allen-Cahn equation and its relation to Plateau's laws, which govern singularities in soap films.

Posted January 31, 2024

Colloquium Questions or comments?

2:30 pm – 3:30 pm Zoom
Bruno Poggi, Universitat Autònoma de Barcelona

Two problems in the mathematical physics of the magnetic Schrödinger operator and their solutions via the landscape function.

Abstract. In two papers in the 90's, Zhongwei Shen studied non-asymptotic bounds for the eigenvalue counting function of the magnetic Schrödinger operator, as well as the localization of eigenfunctions. But in dimensions 3 or above, his methods required a strong scale-invariant quantitative assumption on the gradient of the magnetic field; in particular, this excludes many singular or irregular magnetic fields, and the questions of treating these later cases had remained open, giving rise to a problem and a conjecture. This strong assumption on the gradient of the magnetic field has appeared also in the harmonic analysis related to the magnetic Schrödinger operator. In this talk, we present our solutions to these questions, and other new results on the exponential decay of solutions (eigenfunctions, integral kernels, resolvents) to Schrödinger operators. We will introduce the Filoche-Mayboroda landcape function for the (non-magnetic) Schrödinger operator, present its pointwise equivalence to the classical Fefferman-Phong-Shen maximal function (also known as the critical radius function in harmonic analysis literature), and then show how one may use directionality assumptions on the magnetic field to construct a new landscape function in the magnetic case. We resolve the problem and the conjecture of Z. Shen (and recover other results in the irregular setting) by putting all these observations together.

Posted February 1, 2024

Last modified February 2, 2024

Colloquium Questions or comments?

3:30 pm – 4:30 pm Lockett 232
Narek Hovsepyan, Rutgers University

On the lack of external response of a nonlinear medium in the second-harmonic generation process.

Abstract: Second Harmonic Generation (SHG) is a process in which the input wave (e.g. laser beam) interacts with a nonlinear medium and generates a new wave, called the second harmonic, at double the frequency of the original input wave. We investigate whether there are situations in which the generated second harmonic wave does not scatter and is localized inside the medium, i.e., the nonlinear interaction of the medium with the probing wave is invisible to an outside observer. This leads to the analysis of a semilinear elliptic system formulated inside the medium with non-standard boundary conditions. More generally, we set up a mathematical framework needed to investigate a multitude of questions related to the nonlinear scattering problem associated with SHG (or other similar multi-frequency optical phenomena). This is based on a joint work with F. Cakoni, M. Lassas and M. Vogelius.

Posted January 25, 2024

Last modified January 31, 2024

Colloquium Questions or comments?

3:30 pm Lockett 232
Ali Kara, University of Michigan

Reinforcement Learning in Non-Markovian Environments under General Information Structures

Abstract: For decision-making under uncertainty, typically only an ideal model is assumed, and the control design is based on this given model. However, in reality, the assumed model may not perfectly reflect the underlying dynamics, or there might not be an available mathematical model. To overcome this issue, one approach is to use the past data of perceived state, cost and control trajectories to learn the model or the optimal control functions directly, a method also known as reinforcement learning. The majority of the existing literature has focused on methods structured for systems where the underlying state process is Markovian and the state is fully observed. However, there are many practical settings where one works with data and does not know the possibly very complex structure under which the data is generated and tries to respond to the environment. In this talk, I will present a convergence theorem for stochastic iterations, particularly focusing on Q-learning iterates, under a general, possibly non-Markovian, stochastic environment. I will then discuss applications of this result to the decision making problems where the agent's perceived state is a noisy version of some hidden Markov state process, i.e. partially observed MDPs, and when the agent keeps track of a finite memory of the perceived data. I will also discuss applications for a class of continuous-time controlled diffusion problems.

Posted February 12, 2024

Colloquium Questions or comments?

3:30 pm – 4:30 pm 232 Lockett Hall
Ke Chen, University of Maryland

Towards efficient deep operator learning for forward and inverse PDEs: theory and algorithms

Abstract: Deep neural networks (DNNs) have been a successful model across diverse machine learning tasks, increasingly capturing the interest for their potential in scientific computing. This talk delves into efficient training for PDE operator learning in both the forward and inverse PDE settings. Firstly, we address the curse of dimensionality in PDE operator learning, demonstrating that certain PDE structures require fewer training samples through an analysis of learning error estimates. Secondly, we introduce an innovative DNN, the pseudo-differential auto-encoder integral network (pd-IAE net), and compare its numerical performance with baseline models on several inverse problems, including optical tomography and inverse scattering. We will briefly mention some future works at the end, focusing on the regularization of inverse problems in the context of operator learning.

Posted September 29, 2023

Last modified January 29, 2024

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Jacob Rasmussen, University of Illinois Urbana-Champaign

The L-space conjecture for 3-manifolds

The L-space conjecture of Boyer-Gordon-Watson and Juhasz relates three very different properties that a closed 3-manifold M can possess. One of these properties is algebraic: is \pi_1(M) left orderable? The second is geometric: does the M admit a coorientable taut foliation? The third is analytic: is the Heegaard Floer homology M as simple as it can be, given the size of H_1(M). If the conjecture is true, it would reveal the existence of a striking dichotomy for rational homology 3-spheres. In this talk, I'll explain what each of the three conditions appearing in the L-space conjecture mean, and then discuss efforts to prove and disprove it, and why we should care.

Posted January 20, 2024

Last modified February 5, 2024

Colloquium Questions or comments?

3:30 pm – 4:20 pm Lockett 232
Chongying Dong, UC Santa Cruz

Monstrous moonshine and orbifold theory

This introductory talk will survey the recent development of the monstrous moonshine. Conjectured by McKay-Thompson-Conway-Norton and proved by Borcherds, the moonshine conjecture reveals a deep connection between the largest sporadic finite simple group Monster and genus zero functions. From the point of view of vertex operator algebra, moonshine is a connection among finite groups, vertex operator algebras and modular forms. This talk will explain how the moonshine phenomenon can be understood in terms of orbifold theory.