Posted September 23, 2019

1:00 pm - 4:00 pm Lockett 232Comprehensive/PhD Qualifying Exam in Topology

This exam is part of the PhD Qualifying Examination in Mathematics. Use this link for the registration form: Comprehensive Exam Registration

Posted September 23, 2019

1:00 pm - 4:00 pm Lockett 232Comprehensive/PhD Qualifying Exam in Analysis

This exam is part of the PhD Qualifying Examination in Mathematics. Use this link for the registration form: Comprehensive Exam Registration

Posted September 23, 2019

1:00 pm - 4:00 pm Lockett 232Comprehensive/PhD Qualifying Exam in Algebra

This exam is part of the PhD Qualifying Examination in Mathematics. Use this link for the registration form: Comprehensive Exam Registration

Colloquium
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Posted December 19, 2019

Last modified January 8, 2020

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.

Applied Analysis Seminar
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Posted January 11, 2020

3:30 pm - 4:20 pm 232 Lockett Hall
Quoc-Hung Nguyen, ShanghaiTech University

Quantitative estimates for Lagrangian flows associated to non-Lipschitz vector fields

Since the work by DiPerna and Lions (89) the continuity and transport equation under mild regularity assumptions on the vector field have been extensively studied, becoming a florid research field. In this talk, we give an overview of this theory presenting classical results and new quantitative estimates. One important tool in our investigation is a Kakeya type singular operator. We establish the weak type (1,1) bound for this operator and we exploit it to prove well-posedness and stability results for the continuity and transport equation associated to vector fields represented as singular integrals of BV functions. We also discuss the optimality of this result. Finally, we present sharp regularity estimates for solutions of the continuity equation under various assumptions on the velocity fields.

Harmonic Analysis Seminar
Abstract and additional information

Posted January 21, 2020

2:30 pm - 3:20 pm Lockett 235
Sergio Carrillo, Universidad Sergio Arboleda, Bogata, Columbia

Gevrey power series solutions in analytic functions of first order holomorphic PDEs

Abstract: The goal of this talk is to explain a new Gevrey type -in an analytic function P- for formal power series solutions of some families of singular first order holomorphic PDEs. We will show that under a suitable geometric condition, if P generates the singular locus of the equation, then P is the generic source of divergence of the formal solution. In fact, our result recovers sistematically many well-known cases of singularly perturbed holomorhpic ODEs. The key estimates we use are based on Nagumo norms and their compatibilty with a Weierstrass division theorem. This work is a first step into the study of a Borel-type summability for these series as we shall describe by examples for the case P equal to a monomial.

Colloquium
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Posted January 19, 2020

Last modified January 21, 2020

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.

Colloquium
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Posted January 20, 2020

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.

Colloquium
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Posted January 21, 2020

Last modified January 24, 2020

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 February 13, 2020

Keiser Math Lounge (Lockett 321)ASA Club Meeting

Taylor Daigle, who is an actuarial analyst at Pinnacle Actuarial Resources and a 2018 LSU graduate will visit. Pizza will be served.

Colloquium
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Posted January 21, 2020

Last modified January 26, 2020

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.

Colloquium
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Posted January 21, 2020

Last modified January 23, 2020

Li Chen, University of Connecticut

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.

Applied Analysis Seminar
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Posted January 30, 2020

Last modified February 2, 2020

Khai Nguyen, NCSU

The metric entropy for nonlinear PDEs

Inspired by a question posed by Lax in 2002, in recent years it has received an increasing attention the study on the metric entropy (epsilon entropy) for nonlinear PDEs. In this talk, I will present recent results on sharp estimates in terms of epsilon entropy for hyperbolic conservation laws and Hamilton-Jacobi equations. Estimates of this type play a central role in various ares of information theory and statistics as well as of ergodic and learning theory. In the present setting, this concept could provide a measure of the order of "resolution" of a numerical method for the corresponding equations.

Colloquium
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Posted January 22, 2020

Last modified January 29, 2020

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.

Faculty Meeting
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Posted January 27, 2020

Last modified January 28, 2020

Faculty Meeting

Colloquium
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Posted February 3, 2020

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.

Colloquium
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Posted January 10, 2020

Last modified February 2, 2020

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 February 4, 2020

8:30 am - 9:30 am Keisler Lounge, Lockett 321A Conversation with Prof. Lili Ju, University of South Carolina

Posted August 31, 2019

12:00 pm - 4:00 pm Saturday, February 8, 2020 Digital Media Center TheatreScientific Computing Around Louisiana (SCALA 2020)

Faculty Meeting
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Posted February 6, 2020

2:00 pm - 3:30 pm Lockett 10Faculty Meeting

Applied Analysis Seminar
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Posted December 2, 2019

Last modified February 2, 2020

Junshan Lin, Auburn University

Scattering Resonances Through Subwavelength Holes and Their Applications in Imaging and Sensing

The so-called extraordinary optical transmission (EOT) through metallic nanoholes has triggered extensive research in modern plasmonics, due to its significant applications in bio-sensing, imaging, etc. The mechanisms contributing to the EOT phenomenon can be complicated due to the multiscale nature of the underlying structure. In this talk, I will focus on mechanisms induced by scattering resonances.

In the first part of the talk, based upon the layer potential technique, asymptotic analysis and the homogenization theory, I will present rigorous mathematical analysis to investigate the scattering resonances for several typical two-dimensional structures, these include Fabry-Perot resonance, Fano resonance, spoof surface plasmon, etc. In the second part of the talk, preliminary mathematical studies for their applications in sensing and super-resolution imaging will be given. I will focus on the resonance frequency sensitivity analysis and how one can achieve super-resolution by using subwavelength structures.

Algebra and Number Theory Seminar
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Posted October 11, 2019

Last modified January 31, 2020

Kent Vashaw, Louisiana State University

Noncommutative tensor triangular geometry

We describe a general theory of the prime spectrum of non-braided monoidal triangulated categories. These notions are a noncommutative analogue to Paul Balmer's prime spectra of symmetric tensor-triangulated categories. Noncommutative monoidal triangulated categories appear naturally as stable module categories for non-quasitriangular Hopf algebras and as derived categories of bimodules of noncommutative algebras. In stable module categories of Hopf algebras, the support theory of the category, as described by Benson-Iyengar-Krause, is linked to the Balmer spectrum, which is shown to be the final support datum. We will describe how this connection can be used to compute Balmer spectra in general, and we will compute the Balmer spectra for stable module categories of the small quantum group of a Borel subalgebra at a root of unity, and the stable module categories for smash coproduct Hopf algebras of group algebras with coordinate rings of groups. This is joint work with Daniel Nakano and Milen Yakimov.

Colloquium
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Posted January 22, 2020

Last modified February 9, 2020

Jeffrey Danciger, UT Austin

Affine geometry and the Auslander Conjecture

Abstract: 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 Teichmuller theory will enter the picture.

Algebra and Number Theory Seminar
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Posted January 15, 2020

Last modified February 7, 2020

Po-Han Hsu, Louisiana State University

Erd ̋os-Kac theorem and its deviation principles

Computational Mathematics Seminar

Posted January 7, 2020

3:30 pm - 4:20 pm Digital Media Center 1034
Giordano Tierra-Chica, University of North Texas

Numerical Schemes for Mixtures of Isotropic and Nematic Flows Taking into Account Anchoring and Stretching Effects

The study of interfacial dynamics between two different components has become the key role to understand the behavior of many interesting systems. Indeed, two-phase flows composed of fluids exhibiting different microscopic structures are an important class of engineering materials. The dynamics of these flows are determined by the coupling among three different length scales: microscopic inside each component, mesoscopic interfacial morphology and macroscopic hydrodynamics. Moreover, in the case of complex fluids composed by the mixture between isotropic (newtonian fluid) and nematic (liquid crystal) flows, its interfaces exhibit novel dynamics due to anchoring effects of the liquid crystal molecules on the interface.

In this talk I will introduce a PDE system to model mixtures composed by isotropic fluids and nematic liquid crystals, taking into account viscous, mixing, nematic, stretching and anchoring effects and reformulating the corresponding stress tensors in order to derive a dissipative energy law. Then, I will present new linear unconditionally energy-stable splitting schemes that allows us to split the computation of the three pairs of unknowns (velocity-pressure, phase field-chemical potential and director vector-equilibrium) in three different steps. The fact of being able to decouple the computations in different linear sub-steps maintaining the discrete energy law is crucial to carry out relevant numerical experiments under a feasible computational cost and assuring the accuracy of the computed results.

Finally, I will present several numerical simulations in order to show the efficiency of the proposed numerical schemes, the influence of the shape of the nematic molecules (stretching effects) in the dynamics and the importance of the interfacial interactions (anchoring effects) in the equilibrium configurations achieved by the system.

(Refreshments at 3:00PM in the Computational Math Area of LDMC)

Harmonic Analysis Seminar
Abstract and additional information

Posted February 7, 2020

Last modified February 16, 2020

Gestur Olafsson, Mathematics Department, LSU

Toeplitz operators and representation theory I

We will discuss the basic ideas how representation theory can be used in the Theory of Toeplitz operators. We start with a general set up of a group acting on a complex manifold with a quasi-invariant measure such that there are non-trivial holomorphic L^2-functions and discuss how that leads to Toeplitz operators. We then introduce several examples. We then connect this to representation theory and explain how representation theory can be used to obtain commutative C^*-algebras of Toeplitz operators. Finally we describe how those ideas can be used to determine the spectrum of the so obtained C^*-algebra by constructing an isomorphism into a L^2-space which is easier to understand. The talks should be accessible to graduate students. We will at least use two seminar talks for the material.

Most of this material is a joint work with M. Dawson and R. Quiroga.

Geometry and Topology Seminar
Seminar website

Posted February 7, 2020

Last modified February 17, 2020

Mike Wong, Louisiana State University

Ribbon Homology Cobordisms

Abstract: A cobordism between 3-manifolds is ribbon if it has no 3-handles. Such cobordisms arise naturally from several different topological and geometric contexts. In this talk, we describe a few obstructions to their existence, from Thurston geometries, character varieties, and instanton and Heegaard Floer homologies, and some applications. This is joint work with Aliakbar Daemi, Tye Lidman, and Shea Vela-Vick.

Algebra and Number Theory Seminar
Questions or comments?

Posted January 14, 2020

Mardi Gras Holiday

Posted February 24, 2020

6:00 pm 321 Lockett in the Keisler LoungeActuarial club meeting

Matthew Arnold from Blue Cross Blue Shield of Louisiana will be our guest speaker. He is one of their associate actuaries and one of LSU''s advising actuaries. His topic will be regarding Medicare Supplement/ Medicare Advantage. The meeting will take place at 6pm. As always pizza will be served at the meeting.

Algebra and Number Theory Seminar
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Posted February 6, 2020

Last modified February 27, 2020

John Doyle, Louisiana Tech University

Dynamical modular curves and uniform boundedness of preperiodic points

In the mid-1990's, Merel proved the strong uniform boundedness conjecture for torsion points on elliptic curves over number fields. Around the same time, work of Nguyen-Saito and Abramovich established the function field analogue by showing that the gonalities of the modular curves X_1(n) tend to infinity. By studying the geometry of dynamical modular curves, one can prove uniform boundedness for preperiodic points for certain interesting families of polynomial maps over function fields. I will discuss this result as well as a consequence for the dynamical uniform boundedness conjecture over number fields, originally posed by Morton and Silverman. This is joint work with Bjorn Poonen.

Computational Mathematics Seminar

Posted February 15, 2020

3:30 pm - 4:30 pm Digital Media Center 1034
Li Wang, University of Texas at Arlington

Probabilistic Semi-supervised Learning via Sparse Graph Structure Learning

Abstract: We present a probabilistic semi-supervised learning (SSL) framework based on sparse graph structure learning. Different from existing SSL methods with either a predefined weighted graph heuristically constructed from the input data or a learned graph based on the locally linear embedding assumption, the proposed SSL model is capable of learning a sparse weighted graph from the unlabeled high-dimensional data and a small amount of labeled data, as well as dealing with the noise of the input data. Our representation of the weighted graph is indirectly derived from a unified model of density estimation and pairwise distance preservation in terms of various distance measurements, where latent embeddings are assumed to be random variables following an unknown density function to be learned and pairwise distances are then calculated as the expectations over the density for the model robustness to the data noise. Moreover, the labeled data based on the same distance representations is leveraged to guide the estimated density for better class separation and sparse graph structure learning. A simple inference approach for the embeddings of unlabeled data based on point estimation and kernel representation is presented. Extensive experiments on various data sets show the promising results in the setting of SSL compared with many existing methods, and significant improvements on small amounts of labeled data. div

Harmonic Analysis Seminar
Abstract and additional information

Posted February 7, 2020

Last modified February 16, 2020

Gestur Olafsson, Mathematics Department, LSU

Toeplitz operators and representation theory II

We will discuss the basic ideas how representation theory can be used in the Theory of Toeplitz operators. We start with a general set up of a group acting on a complex manifold with a quasi-invariant measure such that there are non-trivial holomorphic L^2-functions and discuss how that leads to Toeplitz operators. We then introduce several examples. We then connect this to representation theory and explain how representation theory can be used to obtain commutative C^*-algebras of Toeplitz operators. Finally we describe how those ideas can be used to determine the spectrum of the so obtained C^*-algebra by constructing an isomorphism into a L^2-space which is easier to understand. The talks should be accessible to graduate students. We will at least use two seminar talks for the material.

Most of this material is a joint work with M. Dawson and R. Quiroga.

Geometry and Topology Seminar
Seminar website

Posted January 30, 2020

Last modified March 3, 2020

Eva Elduque, University of Michigan

Mixed Hodge structures on Alexander modules

Abstract: Given an epimorphism from the fundamental group of a smooth complex algebraic variety U onto the integers Z, one naturally obtains an infinite cyclic cover of the variety. In analogy with knot theory, the homology groups of this infinite cyclic cover, which are endowed with Z-actions by deck transformations, determine the family of Alexander modules associated to the epimorphism. In this talk, we will talk about how to equip the torsion part of the Alexander modules (with respect to the Z-actions) with canonical mixed Hodge structures in the case when the epimorphism is the induced map on fundamental groups of an algebraic map f from U into the punctured complex plane. Furthermore, we will compare the resulting mixed Hodge structure to other well studied mixed Hodge structures in the literature, including the limit mixed Hodge structure on the generic fiber of f. The relevant concepts will be introduced during the talk. Joint work with C. Geske, L. Maxim, and B. Wang.

Colloquium
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Posted February 9, 2020

Last modified March 2, 2020

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.

Applied Analysis Seminar
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Posted December 2, 2019

Last modified March 8, 2020

Rudi Weikard, University of Alabama at Birmingham

Topics in inverse problems of differential equations

Rudi Weidard's research interests are currently in Inverse Problems. He also investigates differential equations in the complex domain and in abelian functions.

Faculty Meeting
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Posted March 7, 2020

3:30 pm - 4:20 pm tbaFaculty Meeting

Geometry and Topology Seminar
Seminar website

Posted November 11, 2019

Last modified March 8, 2020

Zhenkun Li, MIT

Decomposing sutured Instanton Floer homology

Abstract: Sutured Instanton Floer homology was introduced by Kronheimer and Mrowka. In this talk I will explain how to decompose sutured Instanton Floer homology with respect to a properly embedded surfaces inside the sutured manifold, and explain how this decomposition could be used to study the topological complexities of sutured manifolds and taut foliations. This work is partially joint with Sudipta Ghosh.

Posted March 9, 2020

3:30 pm - 4:30 am Lockett 235
Gestur Olafsson, Mathematics Department, LSU

Toeplitz operators and representation theory III

Abstract: We will discuss the basic ideas how representation theory can be used in the Theory of Toeplitz operators. We start with a general set up of a group acting on a complex manifold with a quasi-invariant measure such that there are non-trivial holomorphic L^2-functions and discuss how that leads to Toeplitz operators. We then introduce several examples. We then connect this to representation theory and explain how representation theory can be used to obtain commutative C^*-algebras of Toeplitz operators. Finally we describe how those ideas can be used to determine the spectrum of the so obtained C^*-algebra by constructing an isomorphism into a L^2-space which is easier to understand. The talks should be accessible to graduate students. We will at least use two seminar talks for the material. Most of this material is a joint work with M. Dawson and R. Quiroga.

Faculty Meeting
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Posted March 10, 2020

Last modified March 11, 2020

Faculty Meeting

Algebra and Number Theory Seminar
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Posted January 14, 2020

Spring Break

Algebra and Number Theory Seminar
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Posted March 30, 2020

Last modified April 5, 2020

Richard Gottesman, Queen's University

Vector-Valued Modular Forms

The collection of vector-valued modular forms form a graded module over the ring of modular forms. I will explain how understanding the structure of this module allows one to show that the component functions of vector-valued modular forms satisfy an ordinary differential equation whose coefficients are modular forms. This enables one to give explicit formulas for vector-valued modular forms in terms of hypergeometric series. In certain cases, one can use such formulas to prove the unbounded denominator conjecture.

Algebra and Number Theory Seminar
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Posted January 14, 2020

Last modified April 9, 2020

Alyson Deines, Center for Communications Research La Jolla (CCR-L)

Elliptic Curves of Prime Conductor

The torsion order elliptic curves over $Q$ with prime conductor have been well studied. In particular, we know that for an elliptic curve $E/Q$ with conductor $p$ a prime, if $p > 37$, then E has either no torsion, or is a Neumann-Setzer curve and has torsion order 2. In this talk we examine similar behavior for elliptic curves of prime conductor defined over totally real number fields.

Faculty Meeting
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Posted April 20, 2020

3:30 pm - 4:20 pm CloudMeeting of Tenured Faculty