LSU
Mathematics

# Calendar

Time interval:   Events:

Wednesday, September 3, 2003

Posted September 2, 2003

12:40 pm - 1:30 pm Lockett 381

Gestur Olafsson, Mathematics Department, LSU
Square Integrable Representations and Frames

Wednesday, September 10, 2003

Posted September 5, 2003

12:40 pm - 1:30 pm Lockett 381

Gestur Olafsson, Mathematics Department, LSU
Square Integrable Representations and Frames II

Wednesday, September 17, 2003

Posted September 8, 2003

12:40 pm - 1:30 pm Lockett 381

Gestur Olafsson, Mathematics Department, LSU
Square Integrable Representations and Frames III

Monday, September 22, 2003

Posted September 18, 2003

12:40 pm - 1:30 pm Lockett 381

Karl Heinrich Hofmann, Darmstadt University, Germany Professor Emeritus
How did the adjoint functor theorem get into Lie theory?

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

Wednesday, October 8, 2003

Posted October 6, 2003

12:40 pm - 1:30 pm Lockett 381

Shijun Zheng, LSU
The Perturbation of the Fourier Transform and Schroedinger Operators (continued)

Wednesday, October 22, 2003

Posted October 17, 2003

12:40 pm - 1:30 pm Lockett 381

Shijun Zheng, LSU
The wavelet decomposition for operator multiplication

Wednesday, October 29, 2003

Posted October 24, 2003

12:40 pm - 1:30 pm Lockett 381

Yongdo Lim, Kyungpook National University
Best Approximation in Riemannian Geodesic submanifolds of Positive Definite Matrices

Wednesday, November 5, 2003

Posted October 30, 2003

12:40 pm - 1:30 pm Lockett 381

Mark Davidson, Mathematics Department, LSU
Generating Functions and Representation Theory

Wednesday, November 12, 2003

Posted November 7, 2003

12:40 pm - 1:30 pm

Mark Davidson, Mathematics Department, LSU
Generating Functions and Representation Theory

Monday, February 2, 2004

Posted January 27, 2004

3:00 am - 3:50 am Lockett 381

Jimmie Lawson, Mathematics Department, LSU
The symplectic group, the symplectic semigroup, and the Ricatti Equation

Monday, February 9, 2004

Posted January 27, 2004

3:00 am - 3:50 am Monday, February 2, 2004 Lockett 381

Jimmie Lawson, Mathematics Department, LSU
The symplectic group, the symplectic semigroup, and the Ricatti Equation II

Monday, February 16, 2004

Posted February 12, 2004

3:00 pm - 5:50 am Lockett 381

Jimmie Lawson, Mathematics Department, LSU
The symplectic group, the symplectic semigroup, and the Ricatti Equation III

Monday, March 8, 2004

Posted March 7, 2004

3:00 pm - 3:50 am Lockett 381

Gestur Olafsson, Mathematics Department, LSU
The Fuglede conjecture and related problems.

Monday, March 15, 2004

Posted March 8, 2004

3:03 pm - 3:52 pm Lockett 381

Simon Gindikin, Rutgers University
Some explicit formulas in integral geometry

Monday, March 22, 2004

Posted March 15, 2004

3:00 pm - 3:50 pm Lockett 381

Karl Heinrich Hofmann, Darmstadt University, Germany Professor Emeritus
Commuting exponential matrices and Lie theory

Monday, April 12, 2004

Posted April 1, 2004

3:00 pm - 3:50 pm Lockett 381

Shijun Zheng, LSU
Operator representation in wavelet bases and Application in PDE's. Part 2.

Abstract. We give a short review on recent development on wavelet-based numerical solution of time-dependent partial differential equations. The fundamental idea is to use wavelet to give sparse matrix representations of the solution operators involved. Thus it leads to a fast algorithm for efficient approximation of the solution to the equation. We demonstrate the general scheme by considering the anisotropic diffusion problem arising in modeling thin film image processing. Other examples are advection-diffusion equations in $CFD$, including the connection with the incompressible Navier-Stokes equations in semigroup formulation.

Monday, April 26, 2004

Posted April 20, 2004

3:00 pm - 3:50 pm Lockett 381

Boris Rubin, Lousiana State University
Zeta integrals and Radon transforms on the space of rectangular matrices

Friday, April 30, 2004

Posted April 20, 2004

2:30 pm - 3:30 pm Lockett 381

Ziemowit Rzeszotnik, University of Texas, Austin
Norm of the Fourier transform on finite abelian groups

Monday, May 3, 2004

Posted April 27, 2004

3:00 pm - 3:50 pm Lockett 381

Dave Larson, Texas A&M
Wavelet sets and Frames

Wednesday, September 15, 2004

Posted September 9, 2004

2:40 pm - 3:30 pm Lockett 381

Jens Christensen, Mathematics Department, LSU
Uncertainty principles generated by Lie-groups

Wednesday, September 22, 2004

Posted September 17, 2004

2:40 pm - 3:30 pm Lockett 381

Jens Christensen, Mathematics Department, LSU
Uncertainty principles generated by Lie-groups

Wednesday, September 29, 2004

Posted September 22, 2004

2:40 pm - 3:30 pm Lockett 381

Jens Christensen, Mathematics Department, LSU
An Uncertainty Principle Related to the Euclidean motion group

I will show that a well known uncertainty principle for functions on the circle can be derived from the generators of the Euclidean motion group.

Wednesday, October 6, 2004

Posted September 30, 2004

2:40 pm - 3:30 pm Lockett 381

Daniel Sage, Mathematics Department, LSU
Group and Hopf algebra actions on central simple algebras.

Wednesday, October 20, 2004

Posted October 18, 2004

2:40 pm - 3:30 pm Lockett 312

Daniel Sage, Mathematics Department, LSU
Group and Hopf algebra actions on central simple algebras. II

Wednesday, October 27, 2004

Posted October 22, 2004

2:40 pm - 3:30 pm Lockett 281

Daniel Sage, Mathematics Department, LSU
Group and Hopf algebra actions on central simple algebras. III

Wednesday, November 3, 2004

Posted November 3, 2004

2:40 pm - 3:30 pm Lockett 381

Daniel Sage, Mathematics Department, LSU
Group and Hopf algebra actions on central simple algebras. IV

Wednesday, November 10, 2004

Posted November 3, 2004

2:40 pm - 3:30 pm Lockett 381

Daniel Sage, Mathematics Department, LSU
Group and Hopf algebra actions on central simple algebras. V

Wednesday, January 26, 2005

Posted January 25, 2005

10:40 am - 11:30 am Lockett 282

Jimmie Lawson, Mathematics Department, LSU
Symmetric Spaces with Seminegative Curvature

Wednesday, February 2, 2005

Posted January 25, 2005

10:40 am - 11:30 am Lockett 282

Jimmie Lawson, Mathematics Department, LSU
Symmetric Spaces of Seminegative Curvature

Wednesday, February 16, 2005

Posted February 4, 2005

10:40 am - 11:30 am Lockett 282

Jimmie Lawson, Mathematics Department, LSU
Symmetric Spaces of seminegative curvature

Wednesday, February 23, 2005

Posted February 16, 2005

10:40 am - 11:30 am Lockett 282

Leticia Barchini, Oklahoma State University at Stillwater
Remarks on the characteristic cycle of discrete series of SU(p,q)

Wednesday, March 2, 2005

Posted February 22, 2005

10:40 am - 11:30 am Lockett 282

Boris Rubin, Lousiana State University
The Composite Cosine Transform on the Stiefel Manifold

Wednesday, March 9, 2005

Posted March 3, 2005

10:40 am - 11:30 am Lockett 282

Boris Rubin, Lousiana State University
The Composite Cosine Transform on the Stiefel Manifold II

Wednesday, April 6, 2005

Posted March 31, 2005

10:40 am - 11:30 am Lockett 282

Daniel Sage, Mathematics Department, LSU
Racah Coefficients and Subrepresentation Semirings

Wednesday, April 13, 2005

Posted April 11, 2005

10:40 am - 11:30 am Lockett 282

Daniel Sage, Mathematics Department, LSU
Racah Coefficients and Subrepresentation semigroups. II

Wednesday, April 20, 2005

Posted April 19, 2005

10:40 am - 11:30 am Lockett 282

Daniel Sage, Mathematics Department, LSU
Racah Coefficients and Subrepresentation Semirings III

Wednesday, April 27, 2005

Posted April 20, 2005

10:40 am - 11:30 am Lockett 282

Gestur Olafsson, Mathematics Department, LSU
The Image of the Heat Transform on Symmetric Spaces

Wednesday, September 14, 2005

Posted September 8, 2005

2:40 pm - 3:30 pm Lockett 381

Mark Davidson, Mathematics Department, LSU
Differential Recursion Relations for Laguerre Functions on Symmetric Cones

Wednesday, September 21, 2005

Posted September 18, 2005

2:40 pm - 3:30 pm Lockett 381

Mark Davidson, Mathematics Department, LSU
Differential Recursion Relations for Laguerre Functions on Symmetric Cones II

Wednesday, September 28, 2005

Posted September 21, 2005

2:40 pm - 3:30 pm Lockett 381

Gestur Olafsson, Mathematics Department, LSU
The Image of the Segal-Bargman Transform

Wednesday, October 5, 2005

Posted September 28, 2005

2:40 pm - 3:30 pm Lockett 381

Gestur Olafsson, Mathematics Department, LSU
The Image of the Segal-Bargman Transform II

Wednesday, October 12, 2005

Posted October 7, 2005

2:40 am - 3:30 am Lockett 381

Gestur Olafsson, Mathematics Department, LSU
The Image of the Segal-Bargman Transform III

Wednesday, October 19, 2005

Posted October 13, 2005

2:40 pm - 3:30 pm Lockett 381

Hongyu He, Mathematics Department, LSU
Some problems concerning positive definite functions

I will give an introduction about positive definitive function and its relation to unitary representation theory, Bochner's Theorem, Gelfand-Naimark-Segal construction etc. Then I will define positive definite distributions and introduce the extension problems, square root problems and the positivity roblem of Godement. This talk will be accessible to graduate students.

Wednesday, November 2, 2005

Posted October 20, 2005

2:40 pm - 1:30 pm Lockett 381

Boris Rubin, Lousiana State University
The generalized Busemann-Petty problem on sections of convex bodies.

The generalized Busemann-Petty problem asks whether origin-symmetric convex bodies in $R^n$ with smaller $i$-dimensional central sections necessarily have smaller volume. This problem has a long history. For $i=2$ and $3$, the answer is still unknown if $n>4$. The problem is intimately connected with the spherical Radon transform. I am planning to give a survey of known results and methods, discuss some generalizations and difficulties.

Wednesday, November 9, 2005

Posted October 27, 2005

2:40 pm - 3:30 pm 381 Lockett Hall

Boris Rubin, Lousiana State University
The generalized Busemann-Petty problem on sections of convex bodies.II

The generalized Busemann-Petty problem asks whether origin-symmetric convex bodies in $R^n$ with smaller $i$-dimensional central sections necessarily have smaller volume. This problem has a long history. For $i=2$ and $3$, the answer is still unknown if $n>4$. The problem is intimately connected with the spherical Radon transform. I am planning to give a survey of known results and methods, discuss some generalizations and difficulties.

Wednesday, November 16, 2005

Posted October 28, 2005

2:40 pm - 3:30 pm 338 Johnston Hall

Eric Todd Quinto, Mathematics Department, Tufts University
LIMITED DATA TOMOGRAPHY AND MICROLOCAL ANALYSIS

In this talk, we will describe limited data tomography problems that come up in applications, including electron microscopy and diagnostic radiology. In each of these tomography problems, certain singularities (boundaries, cracks, etc.) of the object are easily visible in the reconstruction and others are not. We will show how this phenomenon is reflected in the singular functions for the corresponding tomographic problems. A theoretical framework, microlocal analysis, will be given to explain the phenomenon, and we will include an elementary introduction to this idea. If time, we will outline our basic algorithm.

Wednesday, November 30, 2005

Posted October 28, 2005

2:40 pm - 3:30 pm 381 Lockett Hall

Average local behavior of functions and Fourier Series

Wednesday, February 8, 2006

Posted January 24, 2006

3:40 pm - 4:30 pm Lockett 381

Gestur Olafsson, Mathematics Department, LSU
Determining Intertwining Operators

Wednesday, February 15, 2006

Posted February 8, 2006

3:40 pm - 4:30 pm Lockett 381

Gestur Olafsson, Mathematics Department, LSU
Determining Intertwining Operators II

Friday, February 24, 2006

Posted February 2, 2006

until Sunday, February 26, 2006

See program announcement
Workshop in Harmonic Analysis and Fractal Geometry

http://www.math.lsu.edu/~olafsson/workshop06.html

Wednesday, March 22, 2006

Posted March 14, 2006

3:40 pm - 4:30 pm Lockett 381

Gestur Olafsson, Mathematics Department, LSU
Determining Intertwining Operators III

Wednesday, March 29, 2006

Posted March 24, 2006

3:40 pm - 4:30 pm Lockett 381

Boris Rubin, Lousiana State University
On MATH 7390-1: Applied Harmonic Analysis (Fall 2006)

Abstract. I am planning to review a tentative content of this course which will be suggested to graduate students in Fall 2006. This is an introductory course in the theory of the Radon transform, one of the main objects in integral geometry and modern analysis. Topics to be studied include fractional integration and differentiation of functions of one and several variables, Radon transforms in the n-dimensional Euclidean space and on the unit sphere, selected aspects of the Fourier analysis in the context of its application to integral geometry and tomography. The talk will be illustrated by examples of mathematical problems that fall into the scope of this course.

Wednesday, April 5, 2006

Posted March 28, 2006

3:40 pm - 4:30 pm Lockett 381

Genkai Zhang, Department of Mathematics, Gothenburg University, Sweden
Radon, cosine and sine transforms on Grassmannians.

Friday, April 28, 2006

Posted April 25, 2006

2:40 pm - 3:30 pm Lockett 381

Tomasz Przebinda, University of Oklahoma
Orbital Integrals and Howe's Correspondence

Abstract. In this talk I shall explain the construction of the invariant eigendistributions in more detail. In particular, we’ll show how it relates Harish-Chandra’s orbital on the Lie algebras via and the moment maps.

Wednesday, September 13, 2006

Posted September 11, 2006

3:40 pm - 4:30 pm Lockett 381

Hongyu He, Department of Mathematics, LSU
Complementary series of the Universal Covering of the Symplectic Group

Abstract: Complementary series arise as pertubation of the (degenerate) principal series. I will first discuss Sahi's classification. I will then show that complementary series restricted to a symplectic subgroup "half" of its original size are unitarily equivalent to the corresponding restriction of the principal series. The equivalence is given by the "square" root of the intertwining operator expressed in terms of the mixed model, which I will define. This talk is closely related to G. Olafsson's talk last semester in which he discussed the intertwining operator expressed in terms of the compact model.

Wednesday, September 20, 2006

Posted September 14, 2006

3:40 pm - 4:30 pm Lockett 381

Hongyu He, Mathematics Department, LSU
Complementary series of the Universal Covering of the Symplectic Group II

Abstract: Complementary series arise as pertubation of the (degenerate) principal series. I will first discuss Sahi's classification. I will then show that complementary series restricted to a symplectic subgroup "half" of its original size are unitarily equivalent to the corresponding restriction of the principal series. The equivalence is given by the "square" root of the intertwining operator expressed in terms of the mixed model, which I will define. This talk is closely related to G. Olafsson's talk last semester in which he discussed the intertwining operator expressed in terms of the compact model.

Tuesday, October 17, 2006

Posted October 10, 2006

3:40 pm - 4:30 pm Lockett 381

Andy Sinton, Hebrew University of Jerusalem
Direct and Inverse Limits in Geometry and Representation Theory

Abstract: Direct limits (i.e. unions) of finite-dimensional groups are a natural place to look for infinite-dimensional generalizations of the finite-dimensional representation theory and related geometry. In many situations, it turns out that the appropriate analog for the regular representation is a found by letting the direct limit group act on the inverse limit of a related (quotient) space. The first half of the talk will provide an overview of the results of Olshanski, Vershik, Borodin, and others in the cases of the symmetric group and compact symmetric spaces. In the second half I will discuss the state of the art for non-compact symmetric spaces, which I am working on with Gestur Olafsson. Only a basic background in representation theory and Lie groups will be assumed.

Wednesday, November 15, 2006

Posted November 6, 2006

3:40 pm - 4:30 pm Lockett 381

Jens Christensen, Mathematics Department, LSU
Time-Frequency analysis and Gelfand triples

In the 80's Feichtinger and Groechenig found a general class of Banach spaces tied to integrable group representations. These are called coorbit spaces and they are spaces for which the representation coefficients give isometric isomorphisms into other Banach spaces (for example weighted L_p spaces). A well known example is the class of modulation spaces, but also Besov spaces are coorbit spaces (this is rather loosely claimed by Feichtinger and Groechenig). I try to generalize the concept of coorbit spaces to make this construction easier and also possible for non-integrable square integrable representations. This work has been carried out together with Prof. Olafsson.

Wednesday, December 6, 2006

Posted November 13, 2006

3:40 pm - 4:30 pm Lockett 381

Suat Namli, Louisiana State University Graduate Student
A white noise analysis idea applied to orthogonal polynomials

Wednesday, March 7, 2007

Posted February 28, 2007

3:40 pm - 4:30 pm Lockett 381

Michael Otto, University of Arizona
Poisson geometry and symmetric spaces

Abstract: Methods from Poisson/symplectic geometry can be used to study properties of Lie groups and associated symmetric spaces. A prominent example is provided by the classical symplectic convexity theorem of Atiyah and Guillemin-Sternberg and its connection with Kostant's convexity theorem for semisimple Lie groups. We will introduce several interesting Poisson structures on a symmetric space and discuss applications.

Wednesday, March 28, 2007

Posted March 14, 2007

3:40 pm - 4:30 pm Lockett 381 Originally scheduled for 3:40 pm, Wednesday, March 21, 2007

Hongyu He, Mathematics Department, LSU
Introduction to Theta Correspondence

In this talk, I will introduce Howe's dual reductive pair. I will then discuss
the basic theory of theta correspondence and its application in representation
theory. The talk will be accessible to graduate students.

Tuesday, April 17, 2007

Posted April 10, 2007

3:40 pm - 4:30 pm Lockett 285

Hongyu He, Mathematics Department, LSU
Introduction to Theta Correspondence II

In this talk, I will introduce Howe's dual reductive pair. I will then discuss the basic theory of theta correspondence and its application in representation theory. The talk will be accessible to graduate students.

Wednesday, April 25, 2007

Posted April 24, 2007

3:40 pm - 4:30 pm Lockett 381

Hongyu He, Department of Mathematics, LSU
Introduction to Theta Correspondence III

In this talk, I will introduce Howe's dual reductive pair. I will then discuss the basic theory of theta correspondence and its application in representation theory. The talk will be accessible to graduate students.

Tuesday, May 8, 2007

Posted May 4, 2007

3:40 pm - 4:30 pm Lockett 381

Raul Quiroga, Centro de Investigacion en Matematicas (cimat)
Rigidity results for pseudoRiemannian manifolds

Abstract: We will continue our discussion of compact pseudoRiemannian manifolds with a noncompact simple Lie group of isometries. It will be seen that such pseudoRiemannian manifolds have two very remarkable properties: 1) they carry large local isotropy groups, 2) they are locally homogeneous on an open dense subset. These will allow us to describe some structure results for the pseudoRiemannian manifolds considered. As a consequence, we will prove that if $M$ is an irreducible pseudoRiemannian manifold with an isometric action of $SO(p,q)$ and $dim(M) \leq dim(SO(p,q)) + p + q$, then the universal covering space of $M$ is a noncompact simple Lie group.

Wednesday, May 9, 2007

Posted May 3, 2007

3:40 pm - 4:30 pm Lockett 381

Hongyu He, Department of Mathematics, LSU
Introduction to Theta Correspondence IV

In this talk, I will introduce Howe's dual reductive pair. I will then discuss the basic theory of theta correspondence and its application in representation theory. The talk will be accessible to graduate students.

Wednesday, October 3, 2007

Posted September 28, 2007

3:40 pm - 4:30 pm Lockett 381

Jens Christensen, Mathematics Department, LSU
Bergman spaces and representations of SL_2

Graduate Students are encouraged to attend. Abstract: I will start by presenting a general framework for describing Banach spaces by use of representations. Next I will take a closer look at a specific representation leading to a characterization of Bergman spaces on the unit disc. The talk will most likely be split on two days.

Wednesday, October 17, 2007

Posted September 28, 2007

3:40 pm - 3:30 pm Lockett 381

Jens Christensen, Mathematics Department, LSU
Bergman spaces and representations of SL_2 II

Graduate Students are encouraged to attend. Abstract: I will start by presenting a general framework for describing Banach spaces by use of representations. Next I will take a closer look at a specific representation leading to a characterization of Bergman spaces on the unit disc. The talk will most likely be split on two days.

Wednesday, October 24, 2007

Posted October 17, 2007

3:40 pm - 4:30 pm Lockett 381

Jens Christensen, Mathematics Department, LSU
Bergman spaces and representations of SL_2 III

Graduate Students are encouraged to attend. Abstract: I will start by presenting a general framework for describing Banach spaces by use of representations. Next I will take a closer look at a specific representation leading to a characterization of Bergman spaces on the unit disc. The talk will most likely be split on two days.

Friday, October 26, 2007

Posted September 19, 2007

3:40 pm - 4:30 pm Lockett 381 Originally scheduled for 3:40 pm, Friday, October 12, 2007

Alex Iosevich , University of Missouri-Columbia
Bounds for discrete Radon transforms and application to problems in geometric combinatorics and additve number theory

Wednesday, October 31, 2007

Posted October 5, 2007

3:40 pm - 4:30 pm Lockett 381

Complex Structures on Principal Bundles

Holomorphic principal G-bundles over a complex manifold M can be studied using non-abelian cohomology groups H1(M,G). On the other hand, if M=\Sigma is a closed Riemann surface, there is a correspondence between holomorphic principal G-bundles over \Sigma and coadjoint orbits in the dual of a central extension of the Lie algebra C^\infty(\Sigma, \g). We review some of these results and use a Theorem of A. Borel to give more detail in the case of \Sigma having genus one. The talk is based on my diplom thesis, a short version of which is available on the ArXiv: arXiv:0708.3261v1

Wednesday, November 28, 2007

Posted November 12, 2007

3:40 pm - 4:30 pm Lockett 381 Originally scheduled for 3:40 pm, Wednesday, November 21, 2007

Boris Rubin, Lousiana State University
Spherical Means in Odd Dimensions and EPD equations.

I am planning to present a simple proof of the
Finch-Patch-Rakesh inversion formula for the spherical mean Radon
transform in odd dimensions. This transform arises in thermoacoustic
tomography. Applications will be given to the Cauchy problem for the
Euler-Poisson-Darboux equation with initial data on the cylindrical surface.
The argument relies on the idea of analytic continuation and properties of
the Erdelyi-Kober fractional integrals. Some open problem will be discussed.

Friday, February 8, 2008

Posted February 7, 2008

3:40 pm Lockett 381

Gestur Olafsson, Mathematics Department, LSU
Analysis on Symmetric Spaces

This is a seminar class on homogeneous symmetric spaces G/K, where G is a linear Lie group. We study the basic structure theory for G non-compact. We will then discuss the representation theory related to G/K and harmonic analysis on G/K. In particular we hope to be able to introduce the Fourier transform and, in case G is non-compact, the Radon transform on G/K related to the principal series representations. There is a link to the lecture notes on our webpage http://www.math.lsu,edu/~olafsson/teaching.html

Posted February 7, 2008

3:40 pm

Gestur Olafsson, Mathematics Department, LSU
Analysis on Symmetric Spaces

This is a seminar class on homogeneous symmetric spaces G/K, where G is a linear Lie group. We study the basic structure theory for G non-compact. We will then discuss the representation theory related to G/K and harmonic analysis on G/K. In particular we hope to be able to introduce the Fourier transform and, in case G is non-compact, the Radon transform on G/K related to the principal series representations. There is a link to the lecture notes on our webpage http://www.math.lsu,edu/~olafsson/teaching.html

Friday, February 15, 2008

Posted February 15, 2008

3:40 pm Lockett 381

Gestur Olafsson, Mathematics Department, LSU
Seminar on Symmetric Sapces

Thursday, February 21, 2008

Posted February 18, 2008

10:40 am - 11:30 am Lockett 381

Hongyu He, Mathematics Department, LSU
Associated Varieties of Irreducible Unitary Representation

Abstract: I will discuss algebraic invariants associated with Irreducible
unitary representations. These invariants will then be used to study the
restrictions of a unitary representation to its subgroups.

Friday, February 22, 2008

Posted February 22, 2008

3:40 pm Lockett 381

Gestur Olafsson, Mathematics Department, LSU
Analysis on Symmetric Spaces

This is the third lecture in the series. We will discuss the Iwasawa decomposition of the Lie algebra and the group.

Thursday, February 28, 2008

Posted February 25, 2008

10:40 am - 11:30 am Lockett 381

Dan Barbasch, Cornell University
Spherical unitary spectrum for split real and p-adic groups.

Abstract: I will give a description of the parametrization of the spherical unitary dual for split groups, and discuss the techniques used to obtain it. The spherical unitary dual is important for problems in harmonic analysis on symmetric spaces and automorphic forms.

Thursday, March 6, 2008

Posted March 5, 2008

10:40 am - 11:30 am Lockett 381

Joseph Wolf, University of California, Berkeley
Plancherel Formula for Commutative Spaces

Let $(G,K)$ be a Gelfand pair, in other words $G$ is a separable locally compact group, $K$ is a compact subgroup, and the convolution algebra $L^1(K\backslash G/K)$ is commutative. Examples include Riemannian symmetric spaces, locally compact abelian groups and homogeneous graphs. Then the natural representation of $G$ on $L^2(G/K)$ is multiplicity--free and there is a very simple analog of the Euclidean space Fourier transform. I'll describe that transform and the corresponding analog of the Fourier inversion formula.

Thursday, April 3, 2008

Posted April 1, 2008

10:40 am - 11:30 am Lockett 381

Hongyu He, Department of Mathematics, LSU
Associated Varieties of Irreducible Unitary Representation II

Abstract: I will discuss algebraic invariants associated with Irreducible unitary representations. These invariants will then be used to study the restrictions of a unitary representation to its subgroups.

Thursday, April 10, 2008

Posted April 4, 2008

10:40 am - 11:30 am Lockett 381

Hongyu He, Mathematics Department, LSU
Associated Varieties of Irreducible Unitary Representation III

Abstract: I will discuss algebraic invariants associated with Irreducible unitary representations. These invariants will then be used to study the restrictions of a unitary representation to its subgroups.

Monday, April 28, 2008

Posted April 11, 2008

3:40 pm - 4:30 pm Lockett 284

Milen Yakimov, University of California, Santa Barbara
Reality of representations of rational Cherednik algebras

The Calogero-Moser spaces are the phase spaces of the complexified CM
hamiltonian systems. Recently they also appeared in several different
contexts in representation theory. We will describe a
criterion for reality of representations of rational Cherednik algebras
of type A, which is a class of related algebras. We will then
apply it to study the real locus of a Calogero-Moser space and its
relation to the symplectic geometry of the space. We will finish with
applications to Schubert calculus. (Joint work with Iain Gordon
and Emil Horozov).

Friday, May 2, 2008

Posted April 15, 2008

3:40 pm - 4:30 pm Lockett 284

Phuc Nguyen, Purdue University
Singular quasilinear and Hessian equations and inequalities

We give complete characterizations for the solvability of the following quasilinear and Hessian equations: $$-\Delta_p u = \sigma u^q + \omega, \qquad F_k[-u] = \sigma u^q + \omega, \qquad u \ge 0$$ on a domain Omega\subset\mathbb{R}^n$. Here$\Delta_p$is the$p$-Laplacian,$F_k[u]$is the$k$-Hessian, and$\sigma$,$\omega$are given nonnegative measurable functions (or measures) on$\Omega$. Our results give a complete answer to a problem posed by Bidaut-V\'eron in the case$\sigma\equiv 1$, and extend earlier results due to Kalton and Verbitsky, Brezis and Cabr\'e for general$\sigma$to nonlinear operators. This talk is based on joint work with Igor E. Verbitsky. Wednesday, December 3, 2008 Posted November 24, 2008 3:40 am - 4:30 am Lockett 381 Jens Christensen, Mathematics Department, LSU A Wavelet Decomposition of Besov Spaces on the Forward Light Cone Abstract: We will show how the Besov spaces on the Forward Light Cone (defined for general symmetric cones by Bekolle, Bonami, Garrigos and Ricci) can be described using wavelet theory. As part of this description we will discuss work carried out by the presenter and Gestur Olafsson for constructing Banach spaces using representation theory. Friday, February 6, 2009 Posted January 29, 2009 2:40 pm - 3:30 pm Lockett 381 Dimitar Grantcharov, University of Texas, Arlington Weight Modules of Affine Lie Algebras > Abstract: The problem of classifying irreducible weight modules with finite dimensional weight spaces over affine Lie algebras has been studied actively or the last 20 years. Remarkable results include the classification of integrable modules by V. Chari, the study of parabolically induced modules by V. Futorny, and the study of weight modules with bounded weight multiplicities by D. Britten and F. Lemire. There are two important classes of irreducible weight modules with finite dimensional weight spaces: the parabolically induced modules and the loop modules. Several authors made conjectures that would imply that these exhaust all irreducible weight modules with finite dimensional weight spaces. In a joint work with I. Dimitrov we confirm that these conjectures are correct and as a result obtain the classification. In this talk we will present the main ideas and results from our joint work. Friday, March 20, 2009 Posted February 19, 2009 2:40 pm - 3:30 pm Lockett 381 Martin Laubinger, University of Muenster Groups acting on Trees The action of$SL(2,\R)$on the upper half plane is an important tool in the representation theory of$SL(2,\R)$. We explain the$p$-adic analogue, which is an action of$SL(2,Q_p)$on a tree. This tree is one of the simplest examples of a Bruhat-Tits building. We mention some applications of this action, as well as a generalization: if$K$is a field with valuation taking values in any ordered abelian group, one can still define a 'tree' associated with$SL(2,K).$Friday, March 27, 2009 Posted March 20, 2009 Last modified March 23, 2009 3:40 pm - 4:30 pm Lockett 381 Originally scheduled for 3:40 pm, Wednesday, March 25, 2009 Daniel Sage, Mathematics Department, LSU An explicit basis of lowering operators for irreducible representations of unitary groups > Abstract: It is well-known that the dimensions of irreducible > representations of unitary groups can be computed in terms of Young > tableaux. More specifically, each irreducible representation contains > a unique highest weight which may be interpreted as a Young diagram, > and the dimension of any weight space of this representation is given > by the number of semistandard Young tableaux with content determined > by the weight. In the usual Lie-theoretic construction of these > representations as highest-weight modules, it is easy to see that a > spanning set for each representation is obtained by applying lowering > operators to the highest weight vector; however, extracting a basis > from this spanning set is less straightforward. In this talk, I > describe a general method for finding such bases. In particular, I > show how to associate a monomial lowering operator to any semistandard > tableau in such a way that the lowering operators corresponding to the > semistandard tableaux of shape l and content m give rise to a basis > for the m-weight space of the irreducible representation with highest > weight l. This work is joint with Larry Smolinsky. Wednesday, April 29, 2009 Posted April 20, 2009 2:40 pm - 3:30 pm Lockett 381 Boris Rubin, Lousiana State University Comparison of volumes of convex bodies in real, complex, and quaternionic spaces The classical Busemann-Petty problem (1956) asks, whether origin-symmetric convex bodies in$R^n$with smaller hyperplane central sections necessarily have smaller volumes. The answer is known to be affirmative if$n\le 4$and negative if$n>4$. The same question for equilibrated convex bodies in the$n$-dimensional complex space$C^n$has an affirmative answer if and only if$n\le 3$. We show that the similar problem in the$n$-dimensional quaternionic space$H^n$has an affirmative answer if and only if$n=2$. Our method relies on the properties of Radon and cosine transforms on the unit sphere. Monday, May 4, 2009 Posted April 23, 2009 11:00 am - 12:00 pm Lockett 381 Huajun Huang, Auburn University On Simultaneous Isometry of Subspaces Let$(V,b)$be a metric space with a nonsingular symmetric, skew-symmetric, Hermitian, or skew-Hermitian form$b$. Witt's theorem states that an isometry between two subspaces of$V$can be extended to an isometry of the whole space$V$. In this talk, I will present several results that extend Witt's theorem to simultaneous isometries of subspaces by using matrix analysis techniques. As applications, I will illustrate some examples in isometry groups orbits and invariants. The results could be applied to isometry problems in Hilbert spaces. Thursday, November 5, 2009 Posted October 29, 2009 3:40 pm - 4:30 pm Lockett 285 Charles Conley, University of North Texas Extremal Projectors. Let g be a complex finite dimensional reductive Lie algebra. The extremal projector P(g) is an element of a certain formal extension of the enveloping algebra U(g) which projects representations in Category O to their highest weight vectors along their lower weight vectors, provided that the denominator of P(g) does not act by zero. (This denominator is a formal product in U(h), h being the chosen Cartan subalgebra.) In 1971 Asherova-Smirnov-Tolstoi discovered a noncommutative finite factorization of P(g), and in 1993 Zhelobenko discovered a commutative infinite product formula. We will discuss these results and some more recent formulas for the relative projector P(g,l), the projection to the highest l-subrepresentations, l being a Levi subalgebra. Friday, November 20, 2009 Posted November 13, 2009 3:40 pm - 4:30 pm Lockett 381 Boris Rubin, Lousiana State University Radon Transforms on the Heisenberg Group and Transversal Radon Transforms. Abstract: The notion of the Radon transform on the Heisenberg group was introduced by R. Strichartz and inspired by D. Geller and E.M. Stein. A more general transversal Radon transform integrates functions on the$m$-dimensional real Euclidean space over hyperplanes meeting the last coordinate axis. We obtain new boundedness results and explicit inversion formulas for both transforms on$L^p$functions in the full range of the parameter$p$. We also show that these transforms are isomorphisms of the corresponding Semyanistyi-Lizorkin spaces of smooth functions. In the framework of these spaces we obtain inversion formulas, which are pointwise analogues of the corresponding formulas by R. Strichartz. Wednesday, March 10, 2010 Posted March 4, 2010 1:40 pm - 2:30 pm Lockett 381 Karl Heinrich Hofmann, Darmstadt University, Germany Professor Emeritus The probability that two elements commute in a compact group The FC-center of a group$G$is the characteristic subgroup$F$of all elements those conjugacy class is finite. If$G=F$, then$G$is called an FC-group. We show that a compact group$G$is an FC-group if and only if its center$Z(G)$is open (that is,$G$is center by finite) if and only if its commutator subgroup is finite (that is,$G$is finite by commutative).Now let$G$be a compact group and let$p$denote the Haar measure of the set of all pairs$(x,y)$in$G\times G$for which$[x,y]=1$; this is the probability that two randomly picked elements commute. We prove that$p>0$if and only if the FC-center$F$of$G$is open and so has finite index. If these conditions aresatisfied, then$Z(F)$is a characteristic normal abelian open subgroup of$G$and$G$is abelian by finite. Thursday, April 22, 2010 Posted April 14, 2010 2:30 pm - 3:20 pm Lockett 381, please note that the times above are sharp Greg Muller, Cornell University Calogero-Moser Spaces and Ideals in the Weyl Algebra The classical nth Calogero-Moser system describes the motion of n-particles on a line, repelling each other proportional to the inverse-cube of their separation. This is completely integrable, and can be explicitly solved for all time; contrast this with inverse-square attraction, which is not integrable even for 3 particles. If the particles are allowed to have complex position, then the phase space for this system has a remarkable compactification, with both an elegant description and unexpected connections to far-reaches of mathematics. One such connection is that it parametrizes left ideal classes in the Weyl algebra, the ring of polynomial differential operators. Time permitting, I will mention how this observation can be used to construct related systems on other smooth algebraic curves. Friday, September 10, 2010 Posted August 30, 2010 3:40 pm - 4:30 pm Lockett 381 Ivan Dimitrov, Queen's University (Canada) Borel subalgebras of gl_\infty Abstract: The purpose of this talk is to describe all Borel (i.e. all maximal locally solvable) subalgebras of gl_\infty. Consider gl_\infty as the direct limit of gl_n. While the direct limit of Borel subalgebras of gl_n is itself a Borel subalgebra of gl_\infty, the converse is not true. It turns out that the Borel subalgebras of gl_\infty are described rather explicitly as the stabilizers of special chains of subspaces in the natural representation of gl_\infty. I will state the main result and will illustrate it with a number of examples. At the end of the talk I will discuss a couple of open problems. This talk is based on a joint work with Ivan Penkov. Friday, September 24, 2010 Posted September 14, 2010 Last modified September 20, 2010 3:40 pm - 4:40 pm Lockett 381 Karl Heinrich Hofmann, Darmstadt University, Germany Professor Emeritus On Spaces Whose Hoemeomorphism Group is Compact Abstract: This is a seminar lecture between algebra and topology. It deals with the following question: Which compact groups$G$occur as the full homeomorphism group (with the compact open topology) of a Tychonoff space? We argue that any such$G$has to be profinite, that is, totally disconnected. In fact, this is a consequence of a more general result: If a compact but not profinte group acts effectively on a Tychonoff space$X$, then its homeomorphism group$mathcal{H}(X)$contains a subgroup$H$and a closed subgroup$K$which is a normal subgroup of$H$such that$H/K$is a topological group which is homeomorphic to a separable Hilbert space$ell^2=ell^2(N)$. Moreover, the quotient map$hrightarrow H/K$has a topological cross section. Under such circumstances$mathcal{H}(X)$cannot be locally compact, let alone compact. This is a variation of a theme initiated by James Keesling 1971 by different methods. In the reverse direction we show that every monothetic (compact) profinite group is the homeomorphism group of a compact connected 1-dimensional space. We conjecture here that every profinite group is (isomorphic to) the homeomorphism group of some compact connected space. A new approach is used here which combines graph theoretical and topological methods initiated more than half a century ago by J. De Groot. All unfamiliar concepts will by explained in detail. Friday, October 29, 2010 Posted October 29, 2010 3:40 pm - 4:30 pm Lockett 381 Leonardo Mihalcea, Baylor University Spaces of rational curves in flag manifolds and the quantum Chevalley formula Abstract: Given Omega a Schubert variety in a flag manifold, one can consider two spaces: the moduli space GW_d(Omega) of rational curves of fixed degree d passing through Omega (a subvariety of the moduli space of stable maps), and the space Gamma_d(\Omega) obtained by taking the union of these curves (a subvariety of the flag manifold). I will show how some simple considerations about the geometry of these spaces leads to a new, natural, proof of the equivariant quantum Chevalley formula proved earlier by Fulton and Woodward and by the speaker. This is joint work with A. Buch. Wednesday, November 10, 2010 Posted September 20, 2010 3:40 pm - 4:30 pm Lockett 381 Mark Sepanski, Mathematics Department, Baylor University Distinguished orbits and the L-S category of simply connected compact Lie groups We show that the Lusternik-Schnirelmann category of a simple, simply connected, compact Lie group G is bounded above by the sum of the relative categories of certain distinguished conjugacy classes in G corresponding to the vertices of the fundamental alcove for the action of the affine Weyl group on the Lie algebra of a maximal torus of G. This is joint work with M. Hunziker. Friday, November 12, 2010 Posted October 6, 2010 3:40 pm - 4:30 pm Lockett 381 Yen Do, Georgia Tech Variational estimates for paraproducts Abstract: We generalize a family of variation norm estimates of Lepingle with endpoint estimates of Bourgain and Pisier-Xu to a family of variational estimates for paraproducts, both in the discrete and the continuous setting. This expands on work of Friz and Victoir, our focus being on the continuous case and an expanded range of variation exponents. Some applications in time-frequency analysis are also discussed. Joint work with Camil Muscalu and Christoph Thiele. Friday, November 19, 2010 Posted November 9, 2010 3:40 pm - 4:30 pm Lockett 381 Alexander Fish, University of Wisconsin Geometric properties of Intersection Body Operator Abstract: The notion of an intersection body of a star body was introduced by E. Lutwak: K is called the intersection body of L if the radial function of K in every direction is equal to the (d-1)-dimensional volume of the central hyperplane section of L perpendicular to this direction. The notion turned out to be quite interesting and useful in Convex Geometry and Geometric tomography. It is easy to see that the intersection body of a ball is again a ball. E. Lutwak asked if there is any other star-shaped body that satisfies this property. We will present a solution to a local version of this problem: if a convex body K is closed to a unit ball and intersection body of K is equal to K, then K is a unit ball. Friday, December 3, 2010 Posted November 12, 2010 Last modified December 3, 2010 10:40 am - 11:30 am Lockett 233 Rodolfo Torres, University of Kansas Weighted estimates for multilinear singular integrals, commutators, and maximal functions. Abstract: We will recall a theory of weights developed for multilnear Calder\'on-Zygmund operators and describe some recent related results for the multilinear commutators of singular integrals with point-wise multiplication by BMO functions, their iterations, and new multilinear maximal functions. Friday, February 11, 2011 Posted January 27, 2011 Last modified February 2, 2011 3:40 pm - 4:30 pm Lockett 381 Benjamin Harris, MIT Fourier Tranforms of Nilpotent, Coadjoint Orbits and Leading Terms of Tempered Characters. Abstract: Suppose$pi$is an irreducible, admissible representation of a reductive Lie group with character$Theta_{pi}$. By results of Barbasch-Vogan and Schmid-Vilonen, the leading term of$Theta_{pi}$at one is an integral linear combination of Fourier transforms of nilpotent coadjoint orbits. The first half of this talk will be about understanding Fourier transforms of nilpotent coadjoint orbits. I will state the most powerful theorem in the subject due to Rossmann and Wallach. Then I will explicitly write down Fourier transforms of nilpotent coadjoint orbits for$text{GL}(n,mathbb{R})$. The second half of this talk will be about understanding which orbits occur in leading terms of characters. In particular, I will state a necessary condition for an orbit to occur in the wave front cycle of a tempered representation. Then I will give an analogue of Kirillov's dimension formula for tempered representations of reductive Lie groups. Monday, March 14, 2011 Posted January 20, 2011 Last modified February 25, 2011 3:40 pm - 4:30 pm Lockett 277 Grigory Litvinov, Independent University of Moscow INTEGRAL GEOMETRY, HYPERGROUPS AND I.M. GELFAND'S QUESTION It is well known that the Radon Transform is closely related to the classical Fourier transform and harmonic analysis on the additive groups of finite dimensional real linear spaces. In this talk we discuss similar" interrelations between standard problems of Integral Geometry (in the sense of Gelfand and Graev) and harmonic analysis on certain commutative hypergroups (in the sense of J. Delsarte). These interrelations may be interpreted as an answer to an old question of I.M. Gelfand concerning algebraic foundations of Integral Geometry. Wednesday, September 14, 2011 Posted September 9, 2011 3:40 pm - 4:30 pm Lockett 244 Daniel Maier, University of Tuebingen Compact Monothetic Groups in Dynamical Systems Abstract: We give a short introduction in compact monothetic groups and show the interplay between such groups and dynamical systems. Especially, we construct measure preserving dynamical systems which are isomorphic to rotations on compact monothetic groups. Wednesday, September 21, 2011 Posted September 12, 2011 3:40 pm - 4:30 pm Lockett 244 Karl Heinrich Hofmann, Darmstadt University, Germany Professor Emeritus Compact Groups in which any Two Closed Subgroups Commute We shall discuss compact groups in which any pair of closed subgroups$M$,$N$satisfies$MN=NM$. After reviewing the existing literature we shall see that it remains to complete the classification by describing profinite metabelian$p$-groups for a prime$p$. The groups we are looking for are quotients of a semidirect product of some power$Z_p^J$of the additive group of the ring$Z_p$of$p$-adic integers by the group$Z_p$acting as the$p$-component of the group$Z_p^\times$of units of this ring under scalar multiplication. These quotients are explicitly described. This topic provides the motivation to take a closer look at some of the basic properties of the ring$Z_p$of p$-adic integers.

Joint work with Francesco Russo, Universit\'a degli studi, Palermo

Wednesday, October 19, 2011

Posted September 26, 2011

3:40 pm - 4:30 pm Lockett 244

Angela Pasquale, University of Metz and CNRS
Reductive dual pairs and orbital integrals on symplectic spaces

Abstract: We present a Weyl integration formula on the symplectic space for a real reductive dual pair. The formula is motivated by the study of the regularity properties of the intertwining distributions of irreducible admissible representations occurring in the Howe correspondence of a reductive dual pair. This is a joint work with M. McKee and T. Przebinda.

Wednesday, October 26, 2011

Posted October 23, 2011

3:40 pm - 4:30 pm Lockett 244

Benjamin Harris, LSU
Limit Formulas for Reductive Lie Groups.

Abstract: Limit formulas for reductive Lie groups were first studied by Gelfand-Graev and Harish-Chandra in connection with the Plancherel formula for reductive Lie groups. Limit formulas for nilpotent orbits are closely related to character theory and invariants of irreducible representations. In this talk, we will discuss some of the things that are known and some of the things that are not known about limit formulas for reductive Lie groups.

Wednesday, November 2, 2011

Posted October 12, 2011

3:40 pm - 4:30 pm Lockett 244

Boris Rubin, Lousiana State University

Abstract. In 1927 Philomena Mader derived elegant inversion formulas for the hyperplane Radon transform on $R^n$. These formulas differ from the original ones by Johann Radon (1917) and do not need Abel's integral equation or fractional powers of the minus-Laplacian. Surprisingly, these remarkable formulas have been forgotten. We generalize Mader's formulas to totally geodesic Radon transforms in any dimension on arbitrary constant curvature space. This is a joint work with Yuri Antipov.

Wednesday, November 16, 2011

Posted September 26, 2011

3:40 pm - 4:30 pm Lockett 244

Gestur Olafsson, Mathematics Department, LSU
Limits of spherical representations and spherical functions for inductive limits of compact symmetric spaces

Abstract: Spherical representations and functions are the building blocks for harmonic analysis on Riemannian symmetric spaces. We will give a short overview of injective limits of compact symmetric spaces $G_infty/K_infty = varinjlim G_n/K_n$ and limits of spherical representations. We will then describe what happens to the limits of the related spherical $varphi_n (x) = langle e_n, pi_n (x)e_nrangle$ where $e_n$ is a $K_n$--fixed unit vector for $pi_n$. The main result is that the limit $lim_{nto infty} varphi_n(x)$ defines a spherical function on $G_infty /K_infty$ if and only if the rank of $G_n/K_n$ is bounded.

Wednesday, February 8, 2012

Posted January 30, 2012

3:40 pm - 4:30 pm Lockett 244

Inversion Formulas for the Spherical Means in Constant Curvature Spaces

Abstract: This is a talk on recent work by Boris Rubin, Yuri Antipov and
Ricardo Estrada on inversion formulas for the spherical means. For details
see: arXiv:1107.5992

Wednesday, February 15, 2012

Posted February 6, 2012

3:40 pm - 4:30 pm Lockett 244

Yongdo Lim, Kyungpook National University
Deterministic approaches to the Karcher mean on Hadamard spaces

Abstract: Means on positive matrices and operators have received considerable attention in recent years, particular multivariable and weighted means. Applications have arisen in a variety of areas: approximations, interpolation, filtering, estimation, and averaging, diffusion tensor-MRI, sensor networks, radar signal processing. It has become clear that geometric and metric notions are a vital tool, and the Cartan centroid (least squares mean) on non-positive curved metric spaces plays a key role in metric-based computational algorithms. We discuss some deterministic (i.e., probability-free) approaches to the Cartan centroid on Hadamard spaces.

Wednesday, February 22, 2012

Posted January 25, 2012

3:40 pm - 4:30 pm Lockett 244

Kubo Toshihisa, Oklahoma State University, Stillwater
Conformally Invariant systems of differential operators of non-Heisenberg parabolic type

The wave operator $square$ in Minkowski space $mathbf{R}^{3,1}$ is a classical example of a conformally invariant differential operator. The Lie algebra $mathfrak{so}(4,2)$ acts on $mathbf{R}^{3,1}$ via a multiplier representation $sigma$. When acting on sections of an appropriate bundle over $mathbf{R}^{3,1}$, the elements of $mathfrak{so}(4,2)$ are symmetries of the wave operator $square$; that is, for $X in mathfrak{so}(4,2)$, we have begin{equation*} [sigma(X), square] = C(X) square end{equation*} noindent with $C(X)$ a smooth function on $mathbf{R}^{3,1}$. The notion of conformal invariance for a differential operator appears implicitly and explicitly in the literature. The conformality of one operator has been generalized by Barchini-Kable-Zierau to systems of differential operators. Such systems yield homomrophisms between generalized Verma modules. In this talk we build such systems of first and second-order differential operators in the maximal non-Heisenberg parabolic setting. We also discuss the corresponding homomorphisms between generalized Verma modules.

Wednesday, February 29, 2012

Posted February 15, 2012

3:40 pm - 4:30 pm Lockett 244

Angela Pasquale, University of Metz and CNRS
Estimates for the hypergeometric functions associated with root systems

Wednesday, April 25, 2012

Posted January 20, 2012

2:40 pm - 3:30 pm Lockett 244

Andreas Seeger, University of Wisconsin, Madison
Singular Integrals and a Problem on Mixing

Tuesday, August 14, 2012

Posted August 8, 2012

3:40 pm - 4:30 pm Lockett Hall 285

The Fourier and Gegenbauer analysis of fundamental solutions for Laplace's equation on Riemannian spaces of constant curvature

Due to the isotropy of $d$-dimensional hyperbolic and hyperspherical spaces, there exist spherically symmetric fundamental solutions for their corresponding Laplace-Beltrami operators. The $R$-radius hyperboloid model of hyperbolic geometry with $R>0$ represents a Riemannian manifold with negative-constant sectional curvature and the $R$-radius hypersphere emedded in Euclidean space represents a Riemannian manifold with positive-constant sectional curvature. We obtain spherically symmetric fundamental solutions for Laplace's equation on these manifolds in terms of their geodesic radii. We give several matching expressions for these fundamental solutions including definite integral results, finite summation expressions, Gauss hypergeometric functions, and associated Legendre and Ferrers function of the second kind representations. On the $R$-radius hyperbolid we perform Fourier and Gegenbauer analysis for a fundamental solution of Laplace's equation. For instance, in rotationally-invariant coordinate systems, we compute the azimuthal Fourier coefficients for a fundamental solution of Laplace's equation. For $d\ge 2$, we compute the Gegenbauer polynomial expansion in geodesic polar coordinates for a fundamental solution of Laplace's equation on the $R$-radius hyperboloid. In three-dimensions, an addition theorem for the azimuthal Fourier coefficients for a fundamental solution of Laplace's equation is obtained through comparison with its corresponding Gegenbauer expansion. Generalization of this work on the rank one symmetric spaces will be discussed. Short Bio: Dr Howard Cohl obtained a B.S. in Astronomy and Astrophysics from Indiana University, a M.S. and Ph.D. in Physics from Louisiana State University, and a Ph.D. in Mathematics from the University of Auckland in New Zealand. He has worked as a research scientist at various research institutions including the National Solar Observatory in Sunspot, New Mexico; Naval Oceanographic Office Major Shared Resource Center in Stennis Space Center, Mississippi; Lawrence Livermore National Laboratory in Livermore, California; and the School of Physics, University of Exeter in Exeter, United Kingdom. Howard started in December 2010, as a National Research Council Postdoctoral Research Associate in the Applied and Computational Mathematics Division at the National Institute of Standards and Technology. Dr Cohl is currently interested in the special functions associated with fundamental solutions for linear partial differential equations on Riemannian manifolds.

Wednesday, September 5, 2012

Posted August 21, 2012

3:30 pm - 4:20 pm Lockett 284

Boris Rubin, Lousiana State University
Weighted norm inequatlities for $k$-plane transforms

Abstract.

We obtain sharp weighted norm inequalities for the $k$-plane transform,

the $j$-plane to $k$-plane'' transform, and the corresponding dual

transforms, acting on $L^p$ spaces with a radial power weight. These

transforms are well known in integral geometry and harmonic analysis.

The operator norms are explicitly evaluated. Some generalizations and

open problems will be discussed.

The paper is available in arXiv:1207.5180v1.

Wednesday, September 19, 2012

Posted September 14, 2012

2:30 pm - 3:20 pm Lockett 284

Raul Quiroga, Centro de Investigacion en Matematicas (cimat)
Commutative C*-algebras of Toeplitz operators and their geometric aspects, Part 1

Abstract Our main object of interest are the Toeplitz operators on weighted Bergman spaces over complex bounded domains. Such operators are given by a ultiplication operator (by a measurable bounded symbol) followed by the Bergman projection. These operators generalize those considered in Hardy spaces and also naturally appear in Berezin's quantization procedure. We will explain a rather unexpected fact: the existence of large and rich families of symbols that define commutative C*-algebras of Toeplitz operators on weighted Bergman spaces. It is also found that Berezin's quantization implies that any such commutative C*-algebra always carries a distinguished geometric structure. We will see how the use of such geometric structure allows to classify the symbols that define commutative C*-algebras on all weighted Bergman spaces on the unit disk. These geometric tools have also provided some interesting constructions for Reinhardt domains and the n-dimensional unit ball.

Wednesday, September 26, 2012

Posted September 14, 2012

2:30 pm - 3:20 pm Lockett 284

Raul Quiroga, Centro de Investigacion en Matematicas (cimat)
Commutative C*-algebras of Toeplitz operators and their geometric aspects, Part 2

Abstract Our main object of interest are the Toeplitz operators on weighted Bergman spaces over complex bounded domains. Such operators are given by a ultiplication operator (by a measurable bounded symbol) followed by the Bergman projection. These operators generalize those considered in Hardy spaces and also naturally appear in Berezin's quantization procedure. We will explain a rather unexpected fact: the existence of large and rich families of symbols that define commutative C*-algebras of Toeplitz operators on weighted Bergman spaces. It is also found that Berezin's quantization implies that any such commutative C*-algebra always carries a distinguished geometric structure. We will see how the use of such geometric structure allows to classify the symbols that define commutative C*-algebras on all weighted Bergman spaces on the unit disk. These geometric tools have also provided some interesting constructions for Reinhardt domains and the n-dimensional unit ball.

Wednesday, October 3, 2012

Posted September 26, 2012

2:30 pm - 3:20 pm Lockett 284

Square roots of elliptic systems

We study a system $A$ of $2$nd-order elliptic differential equations on the whole space. We realize $A$ as a maximal-accretive operator on $L^2$. It turns out that in this setting, $A$ admits a unique maximal-accretive square root $A^{\frac{1}{2}}$ that shares an astonishing regularity property: Its domain allows for one weak derivative although the domain of the full operator $A$ does not allow for the expected two weak derivatives in general. As a consequence, the Riesz transform $\nabla A^{-\frac{1}{2}}$ is a bounded operator on $L^2$. Finally, we study the Riesz transform on the $L^p$-scale ($p \in (1,\infty)$) culminating in Auscher's characterization of those $p$ for which the Riesz transform extends to a bounded operator on $L^p$.

Monday, October 29, 2012

Posted September 10, 2012

2:30 pm - 3:20 pm Lockett 284

Angela Pasquale, University of Metz and CNRS
Resonances and meromorphic continuation of the resolvent of the Laplace operator on Riemannian symmetric spaces of the noncompact type

Abstract: Let $Delta$ be the Laplace-Beltrami operator on a symmetric space of the noncompact type $G/K$, and let $sigma(Delta)$ denote its spectrum. The resolvent $R(z)=(Delta-z)^{-1}$ is a holomorphic function on $mathbb C setminus sigma(Delta)$, with values in the space of bounded operators on L^2(G/K)$. We study the meromorphic continuation of$R$as distribution valued map on a Riemann surface above$mathbb C setminus sigma(Delta)$. If such a meromorphic continuation is possible, then the poles of the meromorphically extended resolvent are called the resonances. If$dim X$is odd and all Cartan subgroups of$G$are conjugate, then there are no resonances. This can be seen as a consequence of Huygens'principle for the modified wave equation on$X$. In other examples the resonances exist and can be explicitly determined. This is a work in progress with Joachim Hilgert and Tomasz Przebinda. Monday, November 5, 2012 Posted October 9, 2012 Last modified October 12, 2012 2:30 pm - 3:20 pm Lockett 284 Pierre Clare, Penn State University C*-algebraic normalised intertwiners Normalised intertwining integrals related to principal series are central objects in representation theory. The aim of this talk is to describe how to construct and study analogous objects at the level of Hilbert modules and C*-algebras that arise when considering the (reduced) dual of a Lie group from the point of view of noncommutative geometry. Some results appear to carry a similar flavour to recent advances in the classical geometric approach. Monday, March 25, 2013 Posted March 19, 2013 3:30 pm - 4:20 pm Lockett 284 Karl Heinrich Hofmann, Darmstadt University, Germany Professor Emeritus Transitive actions of a compact group on a locally contractible space: a Theorem by Janos Szenthe revisited and recast Abstract: In 1974, J. Szenthe published a theorem according to which a compact group which acts faithfully and transitively on a locally contractible space is a Lie group. This theorem was widely used in the sequel. In 2011, Sergey Antonyan discovered that one lemma claimed and essentially used in the original presentation was irretrievably false. So Szenthe's important theorem was open again. I shall report how in a joint paper with Linus Kramer of the University of Munster (Germany) a proof of Senthe's theorem recovers the result as originally stated. Antonyan and Dobrowolski submitted a paper a few days ago which presents a proof di erent from ours. A preprint of A. A George Michael surfaced in november of last year with another proof similar theirs. Wednesday, April 3, 2013 Posted January 24, 2013 3:30 pm - 4:20 pm tba Matthew Dawson, Department of Mathematics, LSU Graduate Student TBA Wednesday, April 10, 2013 Posted January 24, 2013 Last modified April 10, 2013 3:30 pm - 4:20 pm Lockett 284 Joachim Hilgert, Paderborn University Fock spaces and small representations Monday, July 15, 2013 Posted June 20, 2013 3:30 pm Lockett 381 Toshihisa Kubo Construction of explicit homomorphisms between generalized Verma modules Abstract: In this talk we study constructions of explicit homomorphisms between generalized Verma modules(equivalently, to construct explicit covariant differential operators between homogeneous vector bundles). We in particular show that there is a certain connection between constructions of such homomorphisms and a classic work of Wallach on the analytic continuation of holomorphic discrete series representations. Wednesday, September 18, 2013 Posted September 10, 2013 Last modified September 11, 2013 3:30 pm - 4:20 pm Lockett 235 Matthew Dawson, Department of Mathematics, LSU Graduate Student Conical Representations of Direct-Limit Groups Abstract: Motivated in part by physics, infinite-dimensional Lie groups have been studied more deeply over the past few decades. Due partially to the fact that they are not locally compact and thus do not possess Haar measures, there is currently no general theory of representations and harmonic analysis for infinite-dimensional Lie groups. However, much progress has been made in specific cases. In particular, direct limits of (finite-dimensional) Lie groups provide the simplest examples of infinite-dimensional Lie groups. We overview of some of some of the surprising properties of direct-limit groups and present some recent results related to the classification of conical and spherical representations for direct limits of compact Riemannian symmetric spaces. Wednesday, December 4, 2013 Posted November 19, 2013 3:30 pm - 4:20 pm Lockett 235 Boris Rubin, Lousiana State University On the Overdeterminicity in Integral Geometry Abstract: A simple example of an$n$-dimensional admissible complex of planes is given for the overdetermined$k$-plane transform in$\bbr^n$. Existence of the corresponding restricted$k$-plane transform on$L^p$functions and explicit inversion formulas are discussed. Similar questions are studied for overdetermined Radon type transforms on the sphere and the hyperbolic space. A theorem describing the range of the restricted$k$-plane transform on the space of rapidly decreasing smooth functions is proved. Wednesday, January 22, 2014 Posted January 9, 2014 Last modified January 20, 2014 3:30 pm - 4:20 pm Lockett 235 Palle Jorgensen, University of Iowa Cross roads of stochastic processes, representations of Lie groups, and their applications in physics Wednesday, March 19, 2014 Posted March 17, 2014 3:30 pm - 4:20 pm Lockett 235 Raul Quiroga, Centro de Investigacion en Matematicas (cimat) Commutative algebras generated by Toeplitz operators Abstract: For a bounded symmetric domain$D$we define the (weighted) Bergman spaces and their Toeplitz operators. The latter are given by multiplication operators followed by an orthogonal projection. We will also exhibit non-trivial and large commutative algebras generated by spaces of Toeplitz operators. All our examples will be seen to be closely related to the geometry of$D$and to the holomorphic discrete series of the group of biholomorphisms of$D$. Wednesday, November 5, 2014 Posted October 22, 2014 Last modified October 23, 2014 3:30 pm - 4:20 pm Lockett 277 Originally scheduled for 3:30 pm, Wednesday, October 29, 2014 Boris Rubin, Lousiana State University Gegenbauer-Chebyshev Integrals and Radon Transforms We suggest new modifications of Helgason's support theorems and related characterizations of the kernel (the null space) for the classical hyperplane Radon transform and its dual, the totally geodesic transforms on the sphere and the hyperbolic space, the spherical slice transform, and the spherical mean transform for spheres through the origin. The assumptions for functions are close to minimal and formulated in integral terms. The proofs rely on projective equivalence of these transforms and new facts for the Gegenbauer-Chebyshev fractional integrals. Wednesday, March 4, 2015 Posted February 27, 2015 3:30 pm - 4:20 pm Lockett 277 Eli V Roblero-Mendez, LSU Rigidity of actions of simple Lie groups, I ABSTRACT: In this talk we'll give an introduction to Rigidity Theory and the study of actions of simple Lie groups on manifolds which preserve some geometric structure. We'll also give an overview on some recent results obtained in Zimmer's Program and some techniques of how these results have been obtained. Wednesday, March 11, 2015 Posted February 27, 2015 Last modified March 4, 2015 3:30 pm - 4:30 pm Lockett 277 Eli V Roblero-Mendez, LSU Rigidity of actions of simple Lie group, II ABSTRACT: In this talk we'll give an introduction to Rigidity Theory and the study of actions of simple Lie groups on manifolds which preserve some geometric structure. We'll also give an overview on some recent results obtained in Zimmer's Program and some techniques of how these results have been obtained. Thursday, November 12, 2015 Posted November 5, 2015 3:30 pm - 4:20 pm Lockett 277 Amer Darweesh, LSU Wavletes, Coorbit Theory, and Projective Representation. Wednesday, March 8, 2017 Posted March 1, 2017 2:30 pm - 3:20 pm Lockett 136 Kenny De Commer, Vrije Universiteit Brussel Central approximation properties for quantum groups Abstract: Several approximation properties for discrete groups (Haagerup property, weak amenability, property (T), ...) can be formulated also for discrete quantum groups, which are Hopf algebras with an involution and integral, to be seen as the group algebra of the discrete quantum group. In this talk, I will explain how one can formulate an extra condition on the approximation properties called centrality, which is automatically satisfied in the discrete group case. We will then show how these central approximation properties have good permanence properties for discrete quantum groups, and will illustrate the theory by showing that the free orthogonal quantum groups of Wang and Van Daele have the Haagerup property and are weakly amenable. If time permits, we will also comment on recent results by Y. Arano and by S. Popa and S. Vaes. This is joint work with A. Freslon and M. Yamashita. Wednesday, March 22, 2017 Posted March 13, 2017 Last modified March 14, 2017 3:29 pm - 4:30 pm Lockett 276 Stefan Kolb, Newcastle University Radial part calculations for affine sl2. Abstract: In their seminal work in the 70s Olshanetsky and Perelomov used radial part calculations for symmetric spaces to prove integrability of the Calogero-Moser Hamiltonian for special parameters. In this talk I will explain these notions. Then, restricting to affine sl2, I will try to explain what happens if one extends their argument to Kac-Moody algebras. One obtains a blend of the KZB-heat equation with Inozemtsev's extension of the elliptic Calogero-Moser Hamiltonian. Wednesday, April 26, 2017 Posted April 24, 2017 3:30 pm - 4:20 pm Lockett 285 Hongyu He, Department of Mathematics, LSU Interlacing relations in Representation theory Given an irreducible representation of U(n) with highest weight$\lambda$, its restriction to U(n-1) decomposes into a direct sum of irreducible representations of U(n-1) with highest weights$\mu$. It is well-known that$\lambda$and$\mu$must satisfy the Cauchy interlacing relations $$\lambda_1 \geq \mu_1 \geq \lambda_2 \geq \mu_2...$$ and vice versa. In this talk, I shall discuss the noncompact analogue for the discrete series of$U(p,q)$as conjectured by Gan, Gross and Prasad. I will introduce the Gan-Gross-Prasad interlacing relations and discuss some recent progress. Wednesday, October 25, 2017 Posted September 27, 2017 Last modified October 16, 2017 3:30 pm - 4:30 pm Lockett 285 Originally scheduled for 3:30 pmWednesday, October 18, 2017 Reflection Positivity; Representation Theory meets CQFT Moved by one week: We will give an overview over our work with K-H. Neeb on reflection positivity. We start with recalling the Osterwalder-Schrader Axioms for Constructive Quantum Field Theory and the Osterwalder-Schrader (OS) quantization. We then point out the natural generalization and discuss some examples. We then discuss reflection positive representations, in particular reflection positive 1-parameter subgroups. In the second part we discuss OS quantization related to the sphere. Wednesday, November 1, 2017 Posted September 27, 2017 Last modified October 16, 2017 3:30 pm - 4:30 pm Locett 285 Originally scheduled for 3:30 pmWednesday, October 25, 2017 Reflection Positivity; Representation Theory meets CQFT, part II This is the second part of the series on Reflection positivity. Both talks are accessible for graduate students. Wednesday, November 8, 2017 Posted October 10, 2017 3:30 pm - 4:30 pm Lockett 285 Boris Rubin, Lousiana State University Weighted Norm Estimates for Radon Transforms and Geometric Inequalities We obtain sharp inequalities for the Euclidean k-plane transforms and the " j-plane to k-plane'' transforms acting in$L^p$spaces on$R^n$with a radial power weight. The corresponding operator norms are explicitly evaluated. The results extend to Funk-type transforms on the sphere and Grassmann manifolds. As a consequence, we obtain new weighted estimates of measures of planar sections for measurable subsets of$R^n$. The corresponding unweighted$L^p -L^q\$ estimates and related open problems will be discussed.

Wednesday, December 6, 2017

Posted November 14, 2017

3:30 pm - 4:20 pm Lockett 285

Anton Zeitlin, LSU
Enumerative geometry and quantum integrable systems

Abstract: The miraculous correspondence between 3-dimensional Gauge theory and integrable models based on quantum groups was observed by Nekrasov and Shatashvili in 2009. That discovery led to a lot of interesting developments in mathematics, in particular in enumerative geometry, bringing a new life to older ideas of Givental, and enriching it with flavors of geometric representation theory via the results of Braverman, Maulik, Okounkov and many others. In this talk I will focus on recent breakthroughs, originating from the work of Okounkov on the subject, leading to proper mathematical understanding of Nekrasov-Shatashvili original papers.

Posted November 14, 2017

3:30 pm - 4:20 pm Lockett 285

Anton Zeitlin, LSU
Enumerative geometry and quantum integrable systems

Abstract: The miraculous correspondence between 3-dimensional Gauge theory and integrable models based on quantum groups was observed by Nekrasov and Shatashvili in 2009. That discovery led to a lot of interesting developments in mathematics, in particular in enumerative geometry, bringing a new life to older ideas of Givental, and enriching it with flavors of geometric representation theory via the results of Braverman, Maulik, Okounkov and many others. In this talk I will focus on recent breakthroughs, originating from the work of Okounkov on the subject, leading to proper mathematical understanding of Nekrasov-Shatashvili original papers.

Wednesday, December 13, 2017

Posted December 7, 2017

3:30 pm - 4:20 pm Lockett 285

Kenny De Commer, Vrije Universiteit Brussel
Three categorical pictures for quantum symmetric spaces

Abstract: Using Tannaka-Krein methods, a duality can be constructed between actions of a compact quantum group on the one hand, and module C*-categories over its representation category on the other. In this talk, we will construct three module C*-categories for the q-deformed representation category of a compact semisimple Lie group G, starting from a compact symmetric space G/K for G. The first construction is based on the theory of cyclotomic KZ-equations developed by B. Enriquez. The second construction uses the notion of quantum symmetric pair as developed by G. Letzter. The third construction uses the notion of twistedHeisenberg algebra. In all cases, we show that the module C*-category is twist-braided - this is due to B. Enriquez in the first case, S. Kolb in the second case, and closely related to work of J. Donin, P. Kulish and A. Mudrov in the third case. We formulate a conjecture concerning equivalence of these twist-braided module C*-categories, and prove the equivalence in the simplest case of quantum SU(2). This is joint work with S. Neshveyev, L. Tuset and M. Yamashita.

Wednesday, February 7, 2018

Posted February 5, 2018

3:30 pm - 4:30 pm Lockett 243

Raul Quiroga, Centro de Investigacion en Matematicas (cimat)
K-invariant Toeplitz operators on bounded symmetric domains