Posted September 2, 2003

12:40 pm - 1:30 pm Lockett 381
Gestur Olafsson, Mathematics Department, LSU

Square Integrable Representations and Frames

Posted September 5, 2003

Last modified September 8, 2003

Gestur Olafsson, Mathematics Department, LSU

Square Integrable Representations and Frames II

Posted September 8, 2003

12:40 pm - 1:30 pm Lockett 381
Gestur Olafsson, Mathematics Department, LSU

Square Integrable Representations and Frames III

Posted September 18, 2003

Last modified January 27, 2004

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

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)

Posted October 17, 2003

12:40 pm - 1:30 pm Lockett 381
Shijun Zheng, LSU

The wavelet decomposition for operator multiplication

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

Posted October 30, 2003

12:40 pm - 1:30 pm Lockett 381
Mark Davidson, Mathematics Department, LSU

Generating Functions and Representation Theory

Posted November 7, 2003

12:40 pm - 1:30 pm
Mark Davidson, Mathematics Department, LSU

Generating Functions and Representation Theory

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

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

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

Posted March 7, 2004

3:00 pm - 3:50 am Lockett 381
Gestur Olafsson, Mathematics Department, LSU

The Fuglede conjecture and related problems.

Posted March 8, 2004

3:03 pm - 3:52 pm Lockett 381
Simon Gindikin, Rutgers University

Some explicit formulas in integral geometry

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

Posted April 1, 2004

Last modified March 2, 2021

Shijun Zheng, LSU

Operator representation in wavelet bases and Application in PDEs, Part 2

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.

Posted April 20, 2004

3:00 pm - 3:50 pm Lockett 381
Boris Rubin, Louisiana State University

Zeta integrals and Radon transforms on the space of rectangular matrices

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

Posted April 27, 2004

3:00 pm - 3:50 pm Lockett 381
Dave Larson, Texas A&M

Wavelet sets and Frames

Posted September 9, 2004

2:40 pm - 3:30 pm Lockett 381
Jens Christensen, Mathematics Department, LSU

Uncertainty principles generated by Lie-groups

Posted September 17, 2004

2:40 pm - 3:30 pm Lockett 381
Jens Christensen, Mathematics Department, LSU

Uncertainty principles generated by Lie-groups

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.

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.

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

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

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

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

Posted January 25, 2005

10:40 am - 11:30 am Lockett 282
Jimmie Lawson, Mathematics Department, LSU

Symmetric Spaces with Seminegative Curvature

Posted January 25, 2005

10:40 am - 11:30 am Lockett 282
Jimmie Lawson, Mathematics Department, LSU

Symmetric Spaces of Seminegative Curvature

Posted February 4, 2005

10:40 am - 11:30 am Lockett 282
Jimmie Lawson, Mathematics Department, LSU

Symmetric Spaces of seminegative curvature

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)

Posted February 22, 2005

10:40 am - 11:30 am Lockett 282
Boris Rubin, Louisiana State University

The Composite Cosine Transform on the Stiefel Manifold

Posted March 3, 2005

10:40 am - 11:30 am Lockett 282
Boris Rubin, Louisiana State University

The Composite Cosine Transform on the Stiefel Manifold II

Posted March 31, 2005

10:40 am - 11:30 am Lockett 282
Daniel Sage, Mathematics Department, LSU

Racah Coefficients and Subrepresentation Semirings

Posted April 11, 2005

10:40 am - 11:30 am Lockett 282
Daniel Sage, Mathematics Department, LSU

Racah Coefficients and Subrepresentation semigroups. II

Posted April 19, 2005

10:40 am - 11:30 am Lockett 282
Daniel Sage, Mathematics Department, LSU

Racah Coefficients and Subrepresentation Semirings III

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

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

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

Posted September 21, 2005

2:40 pm - 3:30 pm Lockett 381
Gestur Olafsson, Mathematics Department, LSU

The Image of the Segal-Bargman Transform

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

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

Posted October 13, 2005

Last modified March 3, 2021

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 problem of Godement. This talk will be accessible to graduate students.

Posted October 20, 2005

2:40 pm - 1:30 pm Lockett 381
Boris Rubin, Louisiana 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.

Posted October 27, 2005

Last modified October 28, 2005

Boris Rubin, Louisiana 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.

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.

Posted October 28, 2005

2:40 pm - 3:30 pm 381 Lockett Hall
Ricardo Estrada, Mathematics Department, LSU

Average local behavior of functions and Fourier Series

Posted January 24, 2006

3:40 pm - 4:30 pm Lockett 381
Gestur Olafsson, Mathematics Department, LSU

Determining Intertwining Operators

Posted February 8, 2006

3:40 pm - 4:30 pm Lockett 381
Gestur Olafsson, Mathematics Department, LSU

Determining Intertwining Operators II

Posted February 2, 2006

Last modified February 22, 2006

See program announcement

Workshop in Harmonic Analysis and Fractal Geometry

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

Posted March 14, 2006

3:40 pm - 4:30 pm Lockett 381
Gestur Olafsson, Mathematics Department, LSU

Determining Intertwining Operators III

Posted March 24, 2006

3:40 pm - 4:30 pm Lockett 381
Boris Rubin, Louisiana 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.

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.

Posted April 25, 2006

Last modified March 2, 2021

Tomasz Przebinda, University of Oklahoma

Orbital Integrals and Howe's Correspondence

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.

Posted September 11, 2006

Last modified March 3, 2021

Hongyu He, Department of Mathematics, LSU

Complementary series of the Universal Covering of the Symplectic Group

Complementary series arise as perturbation 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.

Posted September 14, 2006

Last modified March 3, 2021

Hongyu He, Mathematics Department, LSU

Complementary series of the Universal Covering of the Symplectic Group II

Complementary series arise as perturbation 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.

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.

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.

Posted November 13, 2006

Last modified November 27, 2006

Suat Namli, Louisiana State University
Graduate Student

A white noise analysis idea applied to orthogonal polynomials

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.

Posted March 14, 2007

Last modified March 15, 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.

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.

Posted April 24, 2007

Last modified September 17, 2021

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.

Posted May 4, 2007

Last modified March 3, 2021

Raul Quiroga, Centro de Investigacion en Matematicas (CIMAT)

Rigidity results for pseudo-Riemannian manifolds

We will continue our discussion of compact pseudo-Riemannian manifolds with a noncompact simple Lie group of isometries. It will be seen that such pseudo-Riemannian 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 pseudo-Riemannian manifolds considered. As a consequence, we will prove that if $M$ is an irreducible pseudo-Riemannian 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.

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.

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.

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.

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.

Posted September 19, 2007

Last modified March 3, 2021

Alex Iosevich , University of Missouri–Columbia

Bounds for discrete Radon transforms and application to problems in geometric combinatorics and additive number theory

Posted October 5, 2007

Last modified March 3, 2021

Martin Laubinger, LSU
Graduate Student

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

Posted November 12, 2007

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

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

Posted February 15, 2008

Last modified March 3, 2021

Gestur Olafsson, Mathematics Department, LSU

Seminar on Symmetric Spaces

Posted February 18, 2008

Last modified February 26, 2008

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.

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.

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.

Posted March 5, 2008

Last modified March 3, 2021

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.

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.

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.

Posted April 11, 2008

Last modified April 23, 2008

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).

Posted April 15, 2008

Last modified March 3, 2021

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-Véron in the case $\sigma\equiv 1$, and extend earlier results due to Kalton and Verbitsky, Brezis and Cabré for general $\sigma$ to nonlinear operators. This talk is based on joint work with Igor E. Verbitsky.

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.

Posted January 29, 2009

Last modified March 3, 2021

Dimitar Grantcharov, University of Texas, Arlington

Weight Modules of Affine Lie Algebras

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.

Posted February 19, 2009

Last modified March 3, 2021

Martin Laubinger, University of Münster

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 $SL(2,K)$.

Posted March 20, 2009

Last modified March 3, 2021

Daniel Sage, Mathematics Department, LSU

An explicit basis of lowering operators for irreducible representations of unitary groups

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.

Posted April 20, 2009

Last modified March 3, 2021

Boris Rubin, Louisiana 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.

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.

Posted October 29, 2009

Last modified March 3, 2021

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.

Posted November 13, 2009

3:40 pm - 4:30 pm Lockett 381
Boris Rubin, Louisiana 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.

Posted March 4, 2010

Last modified March 3, 2021

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 are satisfied, then $Z(F)$ is a characteristic normal abelian open subgroup of $G$ and $G$ is abelian by finite.

Posted April 14, 2010

Last modified March 3, 2021

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.

Posted August 30, 2010

Last modified March 2, 2021

Ivan Dimitrov, Queen's University (Canada)

Borel subalgebras of $gl_\infty$

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.

Posted September 14, 2010

Last modified March 3, 2021

Karl Heinrich Hofmann, Darmstadt University, Germany
Professor Emeritus

On Spaces Whose Homeomorphism Group is Compact

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 profinite 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 $h\rightarrow H/K$ 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 be explained in detail.

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.

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.

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.

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.

Posted November 12, 2010

Last modified March 3, 2021

Rodolfo Torres, University of Kansas

Weighted estimates for multilinear singular integrals, commutators, and maximal functions.

We will recall a theory of weights developed for multilinear Calderón-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.

Posted January 27, 2011

Last modified March 3, 2021

Benjamin Harris, MIT

Fourier Transforms of Nilpotent, Coadjoint Orbits and Leading Terms of Tempered Characters.

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.

Posted January 20, 2011

Last modified March 2, 2021

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.

Posted September 9, 2011

Last modified March 3, 2021

Daniel Maier, University of Tübingen

Compact Monothetic Groups in Dynamical Systems

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.

Posted September 12, 2011

Last modified March 3, 2021

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, Università degli studi, Palermo.

Posted September 26, 2011

Last modified October 17, 2011

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.

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.

Posted October 12, 2011

Last modified March 2, 2021

Boris Rubin, Louisiana State University

Philomena Mader's Inversion Formulas for Radon Transforms

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.

Posted September 26, 2011

Last modified October 28, 2011

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 $\varphi_n (x) = \langle e_n, \pi_n (x)e_n\rangle$ where $e_n$ is a $K_n$--fixed unit vector for $\pi_n$. The main result is that the limit $\lim_{n\to \infty} \varphi_n(x)$ defines a spherical function on $G_\infty /K_\infty$ if and only if the rank of $G_n/K_n$ is bounded.

Posted January 30, 2012

Last modified February 1, 2012

Ricardo Estrada, Mathematics Department, LSU

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

Posted February 6, 2012

Last modified February 15, 2012

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.

Posted January 25, 2012

Last modified March 3, 2021

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
$$[\sigma(X), \square] = C(X) \square$$
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 homomorphisms 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.

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

Posted January 20, 2012

Last modified April 19, 2012

Andreas Seeger, University of Wisconsin, Madison

Singular Integrals and a Problem on Mixing

Posted August 8, 2012

Last modified March 3, 2021

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 embedded 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 hyperboloid 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.

Posted August 21, 2012

Last modified March 3, 2021

Boris Rubin, Louisiana State University

Weighted norm inequalities for $k$-plane transforms

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.

Posted September 14, 2012

Last modified February 6, 2021

Raul Quiroga, Centro de Investigacion en Matematicas (CIMAT)

Commutative C*-algebras of Toeplitz operators and their geometric aspects, Part 1

Our main object of interest are the Toeplitz operators on weighted Bergman spaces over complex bounded domains. Such operators are given by a multiplication 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.

Posted September 14, 2012

Last modified March 3, 2021

Raul Quiroga, Centro de Investigacion en Matematicas (CIMAT)

Commutative C*-algebras of Toeplitz operators and their geometric aspects, Part 2

Our main object of interest are the Toeplitz operators on weighted Bergman spaces over complex bounded domains. Such operators are given by a multiplication 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.

Posted September 26, 2012

Last modified March 3, 2021

Moritz Egert, University of Darmstadt

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 $L^p$.

Posted September 10, 2012

Last modified March 3, 2021

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

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.

Posted October 9, 2012

Last modified October 12, 2012

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.

Posted March 19, 2013

Last modified March 3, 2021

Karl Heinrich Hofmann, Darmstadt University, Germany
Professor Emeritus

Transitive actions of a compact group on a locally contractible space: a Theorem by Janos Szenthe revisited and recast

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 Münster (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 different from ours. A preprint of A. A George Michael surfaced in November of last year with another proof similar to theirs.

Posted January 24, 2013

Last modified March 2, 2021

Matthew Dawson, Centro de Investigacion en Matematicas

TBA

Posted January 24, 2013

Last modified April 10, 2013

Joachim Hilgert, Paderborn University

Fock spaces and small representations

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.

Posted September 10, 2013

Last modified September 11, 2013

Matthew Dawson, Centro de Investigacion en Matematicas

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.

Posted November 19, 2013

3:30 pm - 4:20 pm Lockett 235
Boris Rubin, Louisiana 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.

Posted January 9, 2014

Last modified January 20, 2014

Palle Jorgensen, University of Iowa

Cross roads of stochastic processes, representations of Lie groups, and their applications in physics

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 $D$.

Posted October 22, 2014

Last modified October 23, 2014

Boris Rubin, Louisiana 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.

Posted February 27, 2015

Last modified March 3, 2021

Eli V Roblero-Mendez, LSU

Rigidity of actions of simple Lie groups, I

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.

Posted February 27, 2015

Last modified March 3, 2021

Eli V Roblero-Mendez, LSU

Rigidity of actions of simple Lie group, II

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.

Posted November 5, 2015

Last modified February 6, 2021

Amer Darweesh, LSU

Wavelets, Coorbit Theory, and Projective Representation

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.

Posted March 13, 2017

Last modified March 14, 2017

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.

Posted April 24, 2017

Last modified March 3, 2021

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.

Posted September 27, 2017

Last modified October 16, 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.

Posted September 27, 2017

Last modified March 3, 2021

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.

Posted October 10, 2017

3:30 pm - 4:30 pm Lockett 285
Boris Rubin, Louisiana 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.

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.

Posted December 7, 2017

Last modified March 3, 2021

Kenny De Commer, Vrije Universiteit Brussel

Three categorical pictures for quantum symmetric spaces

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 twisted Heisenberg 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.

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

Posted March 21, 2018

3:30 pm - 4:20 pm Lockett 243
Joseph Grenier, Louisiana State University

Constructive and Topological Reflection Positivities

In the 1970''s, Osterwalder and Schrader introduced an axiom for Constructive Quantum Field Theories called Reflection Positivity. The uses and consequence of Reflection Positivity have been explored by Jaffe, Neeb, Olafsson, and more with numerous interesting results. It was only recently, in 2016, that Reflection Positivity was adapted to Topological Quantum Field Theory through the use of bordisms and categories. This talk will briefly introduce the constructive form of Reflection Positivity before discussing the categorical approach taken by Freed and Hopkins. The talk will be suitable for graduate students in algebra, analysis and topology.

Posted September 12, 2018

Last modified February 2, 2022

Andreas Debrouwere, LSU

Factorization of quasianalytic vectors

In 1978 [1], Dixmier and Malliavin addressed the following problem: Let
$E$ be a Banach space and let $(π, E)$ be a representation of a Lie group $G$ on
$E$. This representation induces a continuous action $Π$ of the algebra $C_c^∞(G)$ on $E$ given by
$$Π(f)e = \int_G f(g)π(g)e\mathrm{d}g,\quad f \in \mathcal{D}(G), e \in E,$$
and it restricts to a continuous action on the space of smooth vectors $E^∞$.
Dixmier and Malliavin proved the following beautiful factorization result
$$E^∞ = \text{span}(Π(\mathcal{D}(G))E^∞).$$
Recently, a similar factorization result was shown for analytic vectors [2, 3].

In this talk we will generalize these results for the case $G = (\mathbb{R}^d, +)$ in the following way: We consider a representation $(π, E)$ of $(\mathbb{R}^d, +)$ on a quasicomplete locally convex space $E$, introduce the notion of a *quasianalytic* vector (w.r.t. a general Denjoy-Carleman class) and show a Dixmier-Malliavin type result for the space of quasianalytic vectors. As an application, we present factorization results for various weighted convolution algebras of
quasianalytic functions.

This talk is based on collaborative work with Bojan Prangoski and Jasson
Vindas.

References

[1] J. Dixmier, P. Malliavin, *Factorisations de fonctions et de vecteurs indéfiniment différentiables*, Bull. Sci. Math. **102** (1978), 307–330.

[2] H. Gimperlein, B. Krötz, C. Lienau, *Analytic factorization of Lie group representations*, J. Funct. Anal. **262** (2012), 667–681.

[3] C. Lienau, *Analytic representation theory of $(\mathbb{R}; +)$*, J. Funct. Anal. **257** (2009), 3293–3308.

Posted September 12, 2018

Last modified September 18, 2018

Jiuyi Zhu, LSU

Quantitative unique continuation of partial differential equations

Abstract: Motivated by the study of eigenfunctions, we consider the quantitative unique continuation (or quantitative uniqueness) of partial differential equations. The quantitative unique continuation is characterized by the order of vanishing of solutions, which describes quantitative behavior of strong unique continuation property. Strong unique continuation property states that if a solution that vanishes of infinite order at a point vanishes identically. It is interesting to know how the norms of the potential functions and gradient potentials control the order of vanishing. We will report some recent progresses about quantitative unique continuation in different Lebesgue spaces for semilinear elliptic equations, parabolic equations and higher order elliptic equations.

Posted September 28, 2018

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

Mixed norm spaces of analytic functions as spaces of generalized fractional derivatives of functions in Hardy type spaces

We present a new general approach to the definition of a class of mixed norm spaces of analytic functions on the unit disc in complex plane. We study a problem of boundedness of Bergman projection in this general setting. We apply this general approach for the new concrete cases when the norms of variable exponent Lebesgue space, Orlicz space or generalized Morrey space are used. In general, such introduced spaces are the spaces of functions which are in a sense the generalized Hadamard type derivatives of analytic functions having lq summable Taylor coefficients.

Posted September 12, 2018

Last modified March 3, 2021

Anton Zeitlin, LSU

Thoughts on Supermoduli

I will talk about various approaches to supermoduli spaces of punctured surfaces. After short introduction, I will briefly describe earlier results, regarding the construction of Penner-like coordinates on super-Teichmüller spaces for punctured surfaces. Then I will describe a "parallel" construction explicitly describing deformations of superconformal structures via data on fatgraphs and the associated Strebel differentials.

Posted February 3, 2019

3:30 pm - 4:20 pm Lockett 276
Boris Rubin, Louisiana State University

Radon Transforms for Mutually Orthogonal Affine Planes

Abstract: We study a Radon-like transform that takes functions on the Grassmannian of j-dimensional affine planes in $R^n$ to functions on a similar manifold of k-dimensional planes by integration over the set of all j-planes that meet a given k-plane at a right angle. This transform has a mixed structure, combining the k-plane transform and the dual j-plane transform. The main results include action of such transforms on rotation invariant functions, sharp existence conditions, intertwining properties, connection with Riesz potentials and inversion formulas. The case j+k=n-1 for n odd was considered by S. Helgason and F. Gonzalez. This is a joint work with Yingzhan Wang (Guangzhou University).

Posted February 11, 2019

3:30 pm - 4:20 pm Lockett 276
Andreas Debrouwere, LSU

The solution to the first Cousin problem for classes of quasianalytic functions

The solution to the (first) Cousin problem on Stein open sets is a classical result in the theory of functions of several complex variables. As a consequence, the Cousin problem is solvable for the class of real analytic functions on an arbitrary open set of \R^d. The aim of this talk is to explain this problem and show that it is solvable for general classes of quasianalytic functions. Our solution makes use of the so-called Mittag-Leffler procedure, which will also be discussed during the talk.

Posted March 7, 2019

3:30 pm - 4:20 pm Lockett 276
Jens Christensen, Colgate University

Mean Value Operators

Abstract: Mean value operators have many uses in mathematics. They can be used to characterize harmonic functions, and several generalization of mean value operators have been used in the theory of PDEs. The operators are a special type of Radon transforms, and they show up in thermo and photo acoustic tomography. In this talk we will study mapping properties of spherical mean value operators. Our main result is that any smooth function (on a non-compact symmetric space of rank one) can be written as the spherical mean value of another smooth function. The work builds on results by Ehrenpreis regarding division in Paley-Wiener space.

Posted March 23, 2019

Last modified March 15, 2021

Joachim Hilgert, Paderborn University

Helgason's conjecture revisited

Helgason's conjecture from 1976 said that for generic spectral parameters of a Riemannian symmetric space of non-compact type the corresponding Poisson transform is a bijection between hyperfunction sections of the corresponding principal series (realized on the Furstenberg boundary) and the joint eigenfunctions on the symmetric space. The conjecture was settled in a famous paper by Kashiwara-Kowata-Kimura-Okamoto-Oshima-Tanaka in 1978 using heavy machinery from the so-called algebraic analysis started by Mikio Sato. In the 1980s Oshima sketched an alternative approach using less machinery. The key issue is the construction of boundary values of joint eigenfunctions. In this talk I will explain a construction of boundary values in the spirit of Oshima as worked out in recent joint work with Soenke Hansen and Aprameyan Parthasarathy.

Posted April 9, 2019

3:30 pm - 4:20 pm Lockett 276Some Applications of Harmonic Analysis to Toeplitz Operators for Bounded Symmetric Domains

Abstract: In this talk, we will review some recent advances in the theory of Toeplitz operators defined on Bergman spaces of holomorphic functions on complex bounded symmetric domains. These results were made possible by applying the powerful machinery of harmonic analysis and representation theory. In particular, we will see how the problems of finding large commuting families of Toeplitz operators and of calculating the spectra of Toeplitz operators in such families is closely related to the structure of the scalar-type holomorphic discrete series representations of Hermitian Lie groups.

Posted January 21, 2020

Last modified March 3, 2021

Sergio Carrillo, Universidad Sergio Arboleda, Bogata, Columbia

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

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 systematically many well-known cases of singularly perturbed holomorphic ODEs. The key estimates we use are based on Nagumo norms and their compatibility 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.

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.

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.

Posted June 3, 2020

Last modified June 8, 2020

Gestur Olafsson, Mathematics Department, LSU

Toeplitz operators and representation theory III

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.

Posted June 15, 2020

3:30 pm - 4:30 pm zoom link: https://lsu.zoom.us/j/99528757270?pwd=SUprWlJyczd3VUhEZ3Z3MTJjdjlwdz09
Gestur Olafsson, Mathematics Department, LSU

Toeplitz operators and representation theory IV

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

Posted June 15, 2020

3:30 pm - 4:30 pm zoom link: https://lsu.zoom.us/j/99528757270?pwd=SUprWlJyczd3VUhEZ3Z3MTJjdjlwdz09
Gestur Olafsson, Mathematics Department, LSU

Toeplitz operators and representation theory V

Posted June 15, 2020

3:30 pm - 4:30 pm zoom link: https://lsu.zoom.us/j/99528757270?pwd=SUprWlJyczd3VUhEZ3Z3MTJjdjlwdz09
Jens Christensen, Colgate University

Atomic decompositions of Bergman spaces

In the 1980''s Coifman and Rochberg provided atomic decompositions for Bergman spaces on (the unbounded realization of) bounded symmetric domains as well as on the unit ball. Their atoms were point evaluations of the Bergman kernel. Also, their results did not readily transfer to the bounded realization of the domain except in the case of the unit ball. By applying representation/coorbit theory we obtain a large family of new atoms (including the classical ones) for Bergman spaces on bounded symmetric domains. Our approach also allows us to describe the relation between atoms for the bounded and unbounded realizations of the domain thus solving one of the issues raised by Coifman and Rochberg. We finally list a few open questions for domains of rank higher than one. This is joint work with Gestur Olafsson.

Posted July 6, 2020

3:30 pm - 4:30 pm zoom link: https://lsu.zoom.us/j/99528757270?pwd=SUprWlJyczd3VUhEZ3Z3MTJjdjlwdz09
Jens Christensen, Colgate University

Atomic decompositions of Bergman spaces II

In the 1980''''s Coifman and Rochberg provided atomic decompositions for Bergman spaces on (the unbounded realization of) bounded symmetric domains as well as on the unit ball. Their atoms were point evaluations of the Bergman kernel. Also, their results did not readily transfer to the bounded realization of the domain except in the case of the unit ball. By applying representation/coorbit theory we obtain a large family of new atoms (including the classical ones) for Bergman spaces on bounded symmetric domains. Our approach also allows us to describe the relation between atoms for the bounded and unbounded realizations of the domain thus solving one of the issues raised by Coifman and Rochberg. We finally list a few open questions for domains of rank higher than one. This is joint work with Gestur Olafsson.

Posted July 20, 2020

Last modified July 28, 2020

Matthew Dawson, Centro de Investigacion en Matematicas

Infinite-dimensional groups and virtual root systems

The study of infinite-dimensional Lie groups and Lie algebras is a growing area of mathematics with interesting connections to mathematical physics and harmonic analysis. In this talk we will focus mainly on such groups and Lie algebras that are constructed from finite-dimensional Lie groups and Lie algebras via direct limits. This allows one to construct infinite-dimensional analogues of the classical groups that are nonetheless of countable dimension (so that they are, in some sense, the "smallest" infinite-dimensional Lie groups that can be constructed). They inherit many properties of their finite-dimensional counterparts, but present new phenomena only seen in infinite-dimensional groups. We will finish with a discussion of ongoing joint work with Johanna Hennig on the structure of certain direct limits of semisimple Lie algebras that are known not to possess root-space decompositions in the traditional sense. Nonetheless, we construct a "virtual root-space decomposition" by way of direct integrals, a tool from harmonic analysis.

Posted July 20, 2020

Last modified March 2, 2021

Matthew Dawson, Centro de Investigacion en Matematicas

Infinite-dimensional groups and virtual root systems

The study of infinite-dimensional Lie groups and Lie algebras is a growing area of mathematics with interesting connections to mathematical physics and harmonic analysis. In this talk we will focus mainly on such groups and Lie algebras that are constructed from finite-dimensional Lie groups and Lie algebras via direct limits. This allows one to construct infinite-dimensional analogues of the classical groups that are nonetheless of countable dimension (so that they are, in some sense, the "smallest" infinite-dimensional Lie groups that can be constructed). They inherit many properties of their finite-dimensional counterparts, but present new phenomena only seen in infinite-dimensional groups. We will finish with a discussion of ongoing joint work with Johanna Hennig on the structure of certain direct limits of semisimple Lie algebras that are known not to possess root-space decompositions in the traditional sense. Nonetheless, we construct a "virtual root-space decomposition" by way of direct integrals, a tool from harmonic analysis.

Posted July 20, 2020

3:30 pm - 4:30 pm zoom link: https://lsu.zoom.us/j/99528757270?pwd=SUprWlJyczd3VUhEZ3Z3MTJjdjlwdz09
Vignon Oussa, Bridgewater State University

HRT conjecture and linear independence of translates on the Heisenberg group.

In this talk, we will establish the relationship between the HRT Conjecture and linear independence of translation systems on the Heisenberg group. We will show that the HRT Conjecture is equivalent to the conjecture that co-central translates of square-integrable functions on the Heisenberg group are linearly independent. This result affirmatively answers a question asked at the HRT workshop in Saint Louis University in 2016

Posted August 22, 2020

Last modified September 14, 2020

Stephen Shipman, Mathematics Department, LSU

Introduction to Fourier analysis for Z^d and applications

The aim of my talks is to develop the Fourier analysis underlying the study of periodic operators; to show how the theory is applied to phenomena of crystal-type structures in solid-state physics; and to indicate the kinds of problems of interest today. The first talk will concentrate on the Fourier analysis for actions of Z^d and its finite-index extensions. The second talk will delve into aspects of the spectrum of periodic operators and their perturbations, including continuous pre-fractal and fractal continuous spectrum; and eigenvalues embedded in the continuum. The third talk will continue with topics of current interest, such as Dirac cones in graphene, multi-layer structures, Berry phase, twisted bi-layer graphene, etc.

Posted August 22, 2020

Last modified September 14, 2020

Stephen Shipman, Mathematics Department, LSU

Introduction to Fourier analysis for Z^d and applications

The aim of my talks is to develop the Fourier analysis underlying the study of periodic operators; to show how the theory is applied to phenomena of crystal-type structures in solid-state physics; and to indicate the kinds of problems of interest today. The first talk will concentrate on the Fourier analysis for actions of Z^d and its finite-index extensions. The second talk will delve into aspects of the spectrum of periodic operators and their perturbations, including continuous pre-fractal and fractal continuous spectrum; and eigenvalues embedded in the continuum. The third talk will continue with topics of current interest, such as Dirac cones in graphene, multi-layer structures, Berry phase, twisted bi-layer graphene, etc.

Posted August 22, 2020

Last modified September 14, 2020

Stephen Shipman, Mathematics Department, LSU

Introduction to Fourier analysis for Z^d and applications.

The aim of my talks is to develop the Fourier analysis underlying the study of periodic operators; to show how the theory is applied to phenomena of crystal-type structures in solid-state physics; and to indicate the kinds of problems of interest today. The first talk will concentrate on the Fourier analysis for actions of Z^d and its finite-index extensions. The second talk will delve into aspects of the spectrum of periodic operators and their perturbations, including continuous pre-fractal and fractal continuous spectrum; and eigenvalues embedded in the continuum. The third talk will continue with topics of current interest, such as Dirac cones in graphene, multi-layer structures, Berry phase, twisted bi-layer graphene, etc.

Posted September 17, 2020

Last modified October 8, 2020

Raul Quiroga, Centro de Investigacion en Matematicas (CIMAT)

Moment maps on the unit ball and commuting Toeplitz operators I.

Weighted Bergman spaces on the unit ball are reproducing kernel Hilbert spaces that provide an interesting object to study in functional analysis. These spaces come with the so-called Toeplitz operators, defined as multiplication operators by essentially bounded symbols followed by the orthogonal (Bergman) projection. The C*-algebra generated by all Toeplitz operators is highly non-commutative. However, it was discovered the existence of many large commutative C*-subalgebras contained in the C*-algebra generated by Toeplitz operators. Some of these commutative C*-algebras turned out to be associated with maximal Abelian subgroups (MASG) of the automorphism group of the unit ball. More precisely, the C*-algebra generated by Toeplitz operators with G-invariant symbols is commutative for G a MASG. This is no longer true for an arbitrary connected Abelian group.

We will present a geometric construction that associates to any connected Abelian subgroup H of automorphisms of the unit ball a set of symbols whose Toeplitz operators generate a commutative C*-algebra, regardless of whether H is maximal or not. Such construction is based on the moment map associated with the H-action on the unit ball which uses the symplectic structure involved. The families of symbols so obtained include all the special families of symbols currently found in the literature whose Toeplitz operators generate commutative C*-algebras. Furthermore, our construction provides new families of symbols not previously considered. Also, for all of our special symbols we can obtain spectral integral formulas for the corresponding Toeplitz operators.

Posted September 17, 2020

Last modified October 8, 2020

Raul Quiroga, Centro de Investigacion en Matematicas (CIMAT)

Moment maps on the unit ball and commuting Toeplitz operators II.

Weighted Bergman spaces on the unit ball are reproducing kernel Hilbert spaces that provide an interesting object to study in functional analysis. These spaces come with the so-called Toeplitz operators, defined as multiplication operators by essentially bounded symbols followed by the orthogonal (Bergman) projection. The C*-algebra generated by all Toeplitz operators is highly non-commutative. However, it was discovered the existence of many large commutative C*-subalgebras contained in the C*-algebra generated by Toeplitz operators. Some of these commutative C*-algebras turned out to be associated with maximal Abelian subgroups (MASG) of the automorphism group of the unit ball. More precisely, the C*-algebra generated by Toeplitz operators with G-invariant symbols is commutative for G a MASG. This is no longer true for an arbitrary connected Abelian group.

We will present a geometric construction that associates to any connected Abelian subgroup H of automorphisms of the unit ball a set of symbols whose Toeplitz operators generate a commutative C*-algebra, regardless of whether H is maximal or not. Such construction is based on the moment map associated with the H-action on the unit ball which uses the symplectic structure involved. The families of symbols so obtained include all the special families of symbols currently found in the literature whose Toeplitz operators generate commutative C*-algebras. Furthermore, our construction provides new families of symbols not previously considered. Also, for all of our special symbols we can obtain spectral integral formulas for the corresponding Toeplitz operators.

Posted October 8, 2020

Last modified October 26, 2020

Nicholas J Christoffersen, LSU

Representations of Cuntz algebras associated to random walks on graphs

We introduce a class of representations of the Cuntz algebra associated to random walks on graphs. The representations are constructed using the dilation theory of row coisometries. We study these representations, their commutant and the intertwining operators. This is a joint work with Dr. Dorin Dutkay at the University of Central Florida.

Posted February 3, 2021

Last modified February 12, 2021

Nikolai Vasilevski, CINVESTAV

Toeplitz operators on the Bergman space

The talk is intended for a wide audience, not necessarily consisting of experts in the theory of Toeplitz operators, and is a review of the results on the description of algebras generated by Toeplitz operators, with symbols from various special classes, and acting in standard weighted Bergman spaces on the unit ball. We begin with a somewhat surprising and unpredictable result on the existence of a large class of non-isomorphic commutative C*-algebras generated by Toeplitz operators. As it turned out, their symbols must be invariant under the action of maximal Abelian subgroups of the biholomorphisms of the unit ball. The next surprise was the discovery of a large number of Banach (not C*) algebras, which turned out to be, as a rule, not semisimple. The problem here is to find a compact set of maximal ideals and to describe the radical.
Finally we consider in more detail non-commutative C*-algebras generated by Toeplitz operators whose symbols are invariant under the action of a subgroup of some maximal Abelian group of biholomorphisms. As it turned out, different types of the action of the same subgroup lead to completely different properties of the corresponding algebras.

Posted February 19, 2021

3:30 pm Zoom Link https://lsu.zoom.us/j/99528757270?pwd=SUprWlJyczd3VUhEZ3Z3MTJjdjlwdz09
Nikolai Vasilevski, CINVESTAV

Toeplitz operators on the Bergman space

The talk is intended for a wide audience, not necessarily consisting of experts in the theory of Toeplitz operators, and is a review of the results on the description of algebras generated by Toeplitz operators, with symbols from various special classes, and acting in standard weighted Bergman spaces on the unit ball. We begin with a somewhat surprising and unpredictable result on the existence of a large class of non-isomorphic commutative C*-algebras generated by Toeplitz operators. As it turned out, their symbols must be invariant under the action of maximal Abelian subgroups of the biholomorphisms of the unit ball. The next surprise was the discovery of a large number of Banach (not C*) algebras, which turned out to be, as a rule, not semisimple. The problem here is to find a compact set of maximal ideals and to describe the radical. Finally we consider in more detail non-commutative C*-algebras generated by Toeplitz operators whose symbols are invariant under the action of a subgroup of some maximal Abelian group of biholomorphisms. As it turned out, different types of the action of the same subgroup lead to completely different properties of the corresponding algebras.

Posted March 6, 2021

Last modified March 15, 2021

Pierre Clare, William & Mary

Group C*-algebras, Constructions and representations

The purpose of these two talks is to describe some of the ways in which C*-algebras associated with topological groups allow to use tools from operator algebras and noncommutative geometry to study unitary representations. The first talk will be devoted to describing the unitary dual in the language of C*-algebras. In the second talk, we will focus on induced representations and aspects of the Mackey machine expressed in that language.

Quick review of abstract and concrete C*-algebras. The commutative case (Gelfand transform). Convolution algebras, group C*-algebras. Integration of representations. The abelian case (Fourier transform). Fell topology. The unitary dual as a noncommutative space.

Posted March 6, 2021

Last modified March 23, 2021

Irfan Alam, LSU

Generalizing de Finetti's theorem using nonstandard methods

In its classical form, de Finetti's theorem provides a representation of any exchangeable sequence of Bernoulli random variables as a mixture of sequences of iid random variables. Following the work of Hewitt and Savage, such a representation was known for exchangeable random variables taking values in any Polish space. In a recent work, the author has used nonstandard analysis to show that such a representation holds for a sequence of exchangeable random variables taking values in any Hausdorff space as long as their underlying distribution is Radon (in fact, tightness and outer regularity on compact sets are also sufficient). An overview of this work will be presented. The arguments have topological measure theoretic and combinatorial flavors, with nonstandard analysis serving as a bridge between these themes. A main component of this work involves generalizing a result of Paul Ressel that was previously obtained using abstract harmonic analysis.

Posted March 17, 2021

3:30 pm - 4:30 pm https://lsu.zoom.us/j/99528757270?pwd=SUprWlJyczd3VUhEZ3Z3MTJjdjlwdz09
Phuc Nguyen, Department of Mathematics, Louisiana State University

The Hardy-Littlewood maximal function on Choquet spaces

I will present my recent joint work with Keng Hao Ooi in which we obtain the boundedness of the Hardy-Littlewood maximal function on $L^q$ type spaces defined via Choquet integrals associate to Sobolev capacities. The bounds are obtained in full range of exponents which also include a weak type end-point bound.

Posted March 31, 2021

Last modified April 5, 2021

Pierre Clare, William & Mary

Group C*-algebras II

The purpose of these two talks is to describe some of the ways in which C*-algebras associated with topological groups allow to use tools from operator algebras and noncommutative geometry to study unitary representations. In the this talk, we will focus on induced representations and aspects of the Mackey machine expressed in that language.

Hilbert modules, crossed products and the Mackey Machine.

Induced representations a la Mackey. Hilbert modules. Induced representations a la Rieffel. Crossed products C*-algebras. Imprimitivity

Posted April 7, 2021

Last modified May 1, 2021

Egor Maximenko, National Polytechnic Institute, Mexico

Radial Toeplitz operators on the Fock space and square-root-slowly oscillating sequences

In this talk, based on a joint article with Kevin Esmeral (https://doi.org/10.1007/s11785-016-0557-0), we describe the C*-algebra generated by radial Toeplitz operators with bounded symbols acting on the Fock space. We prove that this C*-algebra is isometrically isomorphic to the C*-algebra of bounded sequences uniformly continuous with respect to the square-root-metric $\rho(j,k)=|\sqrt(j)-\sqrt(k)|$. More precisely, we show that the spectral sequences (i.e., the sequences of the eigenvalues) of radial Toeplitz operators form a dense subset of the latter C*-algebra of sequences. The main idea is to approximate the spectral sequences by convolutions and apply an appropriate version of Wiener's density theorem.

Posted April 18, 2021

3:30 pm - 4:30 pm https://lsu.zoom.us/j/99528757270?pwd=SUprWlJyczd3VUhEZ3Z3MTJjdjlwdz09
Gerardo Ramos-Vazquez

Homogeneous polyanalytic kernels in the Bergman space

The aim of this talk is to compute the reproducing kernels (RK) of the Bergman spaces of polyanalytic functions over the unit ball and the Siegel domain. First, we show a mean-value property for polyanalytic functions on the unit ball using the reproducing property of Jacobi polynomials. The RK of the Bergman space of such functions in the unit ball is a simple consequence. Secondly, we build a unitary "jump" operator from the Bergman space over the unit ball to the Bergman space over the Siegel domain. With the help of this operator, we compute the RK of the Bergman space over the Siegel domain from the previous calculations.

Posted April 26, 2021

Last modified July 25, 2021

Vishwa Dewage, Louisiana State University

C*-algebra generated by Toeplitz operators with quasi-radial symbols

The n-dimensional Fock space is defined to be the space of holomorphic functions that are square integrable with respect to the Gaussian measure.

Using representation theory, we diagonalize the Toeplitz operators on Fock space with essentially bounded quasi-radial symbols. Then we show that the commutative C*-algebra generated by these Toeplitz operators is isometrically isomorphic to a space of bounded functions that are uniformly continuous with respect to the square root metric.

This is a joint work with my advisor, Prof. Olafsson.

Posted April 30, 2021

Last modified July 25, 2021

Christian Jakel, University of Sao Paulo

Free massless fields and interacting massive fields in 1 + 1 de Sitter space

We report on the current status of our program physics without assumptions. The basic idea is to analyze the physics determined by the space-time itself. In particular, we describe massless and massive scalar particles on the two-dimensional de Sitter space dS. The quantum theory arises, without any further assumption, from the representation theory of the isometry group SO_0(1,2) of dS. Tomita-Takesaki modular theory provides a group theoretic localization of the observables on dS. In the massless case, it gives rise to an infinite number of one-parameter groups (the modular groups) which implement the geometric action of the conformal transformations on the observables. Their generators (the modular Hamiltonians) satisfy the Virasoro algebra commutation relations. In the massive case, modular theory can be used add polynomial interactions. This is a joint work with Urs Achim Wiedemann and Jens Mund.

Posted April 7, 2021

Last modified December 6, 2021

Egor Maximenko, National Polytechnic Institute, Mexico

Radial operators on polyanalytic weighted Bergman spaces

In this talk, based on a recent paper with Roberto Moisés Barrera-Castelán and Gerardo Ramos-Vazquez, we describe the von Neumann algebra $\mathcal{R}_n$ of radial operators acting on the $n-$ analytic weighted Bergman space $\mathcal{A}_n^2$ on the unit disk. First, extending the results of Ramazanov (1999, 2002), we explain that disk polynomials (studied by Koornwinder in 1975 and Wunsche in 2005) form an orthonormal basis of $\mathcal{A}_n^2$. Using this basis, we provide the Fourier decomposition of $\mathcal{A}_n^2$ into the orthogonal sum of the subspaces associated with different frequencies. This leads to the decomposition of the von Neumann algebra $\mathcal{R}_n$ into the direct sum of some matrix algebras. In other words, all radial operators are represented as matrix sequences. In particular, we represent in this form the Toeplitz operators with bounded radial symbols, acting in $\mathcal{A}_n^2$. Moreover, using ideas by Englis (1996), we show that the set of all Toeplitz operators with bounded generating symbols is not weakly dense in the algebra of all bounded linear operators acting in $\mathcal{A}_n^2)$

Posted May 24, 2021

Last modified May 28, 2021

Siddhartha Sahi, Rutgers University

Some properties of the Macdonald kernel and associated integral transforms

Abstract: Jack polynomials are an important family of symmetric polynomials that depend on a parameter $\alpha$. For certain values of $\alpha$ they specialize to radial parts of spherical functions on symmetric cones; in particular the values $\alpha=2/d,\ d=1,2,4$ correspond to positive definite Hermitian matrices over $\R,\C,\H$, respectively. I.G. Macdonald has introduced a certain kernel function $e(x,y)$, which is defined as a multivariate power series involving Jack polynomials in two sets of variables $ x,y\in R^n$. In this paper we establish three key properties of the Macdonald kernel and associated integral transforms. As anticipated by Macdonald, these results allow one to develop a reasonable theory of Fourier and Laplace transforms, and hypergeometric functions, for arbitrary $\alpha>0$; thereby generalizing classic results of Bochner, Herz, and many others, for symmetric cones. This is joint work with Gestur Olafsson. $\\$ The talk will start at 3:30-4:30 PM Central time ( 4:30- 5:30 PM Eastern time)

Posted September 21, 2021

Last modified October 4, 2021

Franz Luef, Norwegian University of Science and Technology

Wiener’s Tauberian Theorem in Quantum Harmonic Analysis

Abstract: We present variants of Wiener’s Tauberian Theorems for operators as well as of the Wiener-Pitt theorem, which are based on operator convolutions introduced by R.F. Werner in his seminal work on quantum harmonic analysis on phase space. These results have applications in time-frequency analysis and for quantization schemes. This talk is based on joint work with Eirik Skrettingland.

Posted October 8, 2021

Last modified October 10, 2021

Allan Merino, University of Ottawa

Howe duality and characters (Part I)

Abstract: For every irreducible reductive dual pair (G, G’) in Sp(W), Roger Howe proved the existence of an isomorphism between the spaces R(G) and R(G’), where R(G) is the set of infinitesimal equivalence classes of irreducible admissible representations of $\tilde{G}$ (preimage of G in the metaplectic group) which can be realized as a quotient of the metaplectic representation. All the representations appearing in the correspondence have a distribution character, and characters are analytic objects completely identifying the irreducible representations. In particular, one natural question is to understand the transfer of characters in the theta correspondence (or Howe’s duality). The goal of my first talk is to define carefully the notions I mentioned previously: Metaplectic representation, character of an infinite dimensional representation and Howe’s duality theorem with some well-known results and applications. In the last minutes of my talk, I will explain how this transfer of characters work when G is compact, and give an explicit way to compute the character of the corresponding representation $ \pi’$ by using the Howe oscillator semigroup.

Posted October 10, 2021

3:30 pm - 4:30 pm https://lsu.zoom.us/j/94413235134
Allan Merino, University of Ottawa

Howe duality and characters (Part II)

In 2000, T. Przebinda introduced the Cauchy-Harish-Chandra integral and conjectured that the transfer of characters should be obtained via this map. This conjecture is known to be true if G is compact and was proven by Przebinda for unitary representations in the stable range case. In general, it has been established that the Cauchy-Harish-Chandra integral sends the $\tilde{G}$-invariant eigendistributions on $\tilde{G}$ into the $\tilde{G’}$-invariant eigendistributions on $\tilde{G’} $as long as $rk(G) \leq rk(G’)$. After recalling carefully the construction of the Cauchy-Harish-Chandra integral and stating Przebinda’s conjecture, I am going to explain, by using results of A. Paul, how to prove this conjecture for the pair (G,G’) = (U(p, q), U(r, s)), with p+q = r+s, starting with a discrete series representation of $\tilde{G}$. At the end of my talk, I will discuss some auxiliary results I got in my paper and an ongoing project on transfer of characters for (G, G’), with $rk(G) \leq rk(G’)$, starting from a discrete series representation. At the end of my talk, I am going to discuss an ongoing project on the proof of the previous conjecture for an arbitrary dual pair (G, G’) starting with a discrete series representation of \tilde{G}.

Posted October 24, 2021

3:30 pm - 4:30 pm Zoom https://lsu.zoom.us/j/94413235134
Cody Stockdale, Clemson University

Weighted theory of Toeplitz operators on the Bergman space

Abstract: We discuss the weighted compactness and boundedness properties of Toeplitz operators, $T_u$, on the Bergman space with respect to B\'ekoll\`e-Bonami type weights. We give sufficient conditions on $u$ that imply the compactness of $T_u$ on $L^p_{\sigma}$ for $p \in (1,\infty)$ and all weights $\sigma \in B_p$, and from $L^1_{\sigma}$ to $L_{\sigma}^{1,\infty}$ for all $\sigma \in B_1$. Additionally, using a new extrapolation result, we characterize the compact Toeplitz operators on the weighted Bergman space $\mathcal{A}^p_\sigma$ for all $\sigma$ belonging to a nontrivial subclass of $B_p$. Concerning boundedness, we show that $T_u$ extends boundedly on $L^p_{\sigma}$ for $p \in (1,\infty)$ and weights $\sigma$ in a $u$-adapted class of weights containing $B_p$. Finally, we establish an analogous weighted endpoint weak-type $(1,1)$ bound for weights beyond $B_1$. This is joint work with Nathan Wagner (Washington University in St. Louis).

Posted November 2, 2021

3:30 pm - 4:30 pm https://lsu.zoom.us/j/94413235134
Cody Stockdale, Clemson University

Weighted theory of Toeplitz operators on the Bergman space II

Abstract: We discuss the weighted compactness and boundedness properties of Toeplitz operators, $T_u$, on the Bergman space with respect to B\'ekoll\`e-Bonami type weights. We give sufficient conditions on $u$ that imply the compactness of $T_u$ on $L^p_{\sigma}$ for $p \in (1,\infty)$ and all weights $\sigma \in B_p$, and from $L^1_{\sigma}$ to $L_{\sigma}^{1,\infty}$ for all $\sigma \in B_1$. Additionally, using a new extrapolation result, we characterize the compact Toeplitz operators on the weighted Bergman space $\mathcal{A}^p_\sigma$ for all $\sigma$ belonging to a nontrivial subclass of $B_p$. Concerning boundedness, we show that $T_u$ extends boundedly on $L^p_{\sigma}$ for $p \in (1,\infty)$ and weights $\sigma$ in a $u$-adapted class of weights containing $B_p$. Finally, we establish an analogous weighted endpoint weak-type $(1,1)$ bound for weights beyond $B_1$. This is joint work with Nathan Wagner (Washington University in St. Louis).

Posted November 4, 2021

3:30 pm - 4:30 pm https://lsu.zoom.us/j/94413235134
Cody Stockdale, Clemson University

Weighted theory of Toeplitz operators on the Bergman space II

Abstract: We discuss the weighted compactness and boundedness properties of Toeplitz operators, $T_u$, on the Bergman space with respect to B\'ekoll\`e-Bonami type weights. We give sufficient conditions on $u$ that imply the compactness of $T_u$ on $L^p_{\sigma}$ for $p \in (1,\infty)$ and all weights $\sigma \in B_p$, and from $L^1_{\sigma}$ to $L_{\sigma}^{1,\infty}$ for all $\sigma \in B_1$. Additionally, using a new extrapolation result, we characterize the compact Toeplitz operators on the weighted Bergman space $\mathcal{A}^p_\sigma$ for all $\sigma$ belonging to a nontrivial subclass of $B_p$. Concerning boundedness, we show that $T_u$ extends boundedly on $L^p_{\sigma}$ for $p \in (1,\infty)$ and weights $\sigma$ in a $u$-adapted class of weights containing $B_p$. Finally, we establish an analogous weighted endpoint weak-type $(1,1)$ bound for weights beyond $B_1$. This is joint work with Nathan Wagner (Washington University in St. Louis).

Posted November 4, 2021

3:30 pm - 4:30 pm Zoom link: https://lsu.zoom.us/j/94413235134
Gerardo Vazquez, National Polytechnic Institute, Mexico

Translation-invariant operators in reproducing kernel Hilbert spaces

Abstract.: Let $G$ be a locally compact abelian group with a Haar measure, and $Y$ be a measure space. Suppose that H is a reproducing kernel Hilbert space of functions on $G\times Y$, such that $H$ is naturally embedded into $L^2(G\times Y)$ and is invariant under the translations associated with the elements of $G$. Under some additional technical assumptions, we study the W*-algebra $\mathcal{V}$ of translation-invariant bounded linear operators acting on $H$. We obtain an explicit and constructive description of $\mathcal{V}$ in terms of the Fourier transform of the reproducing kernel. The talk is based on a joint work with Crispin Herrera Yañez and Egor Maximenko. We use many ideas of Nikolai Vasilevski and his colleagues.

Posted November 4, 2021

3:30 pm - 4:30 pm Zoom link: https://lsu.zoom.us/j/94413235134
Egor Maximenko, National Polytechnic Institute, Mexico

Examples of reproducing kernel Hilbert spaces with translation-invariant operators

This is a continuation of the talk given by Gerardo Ramos Vazquez. It is based on our joint paper with Crispin Herrera Yañez. Here we apply our scheme to a series of examples: vertical and angular operators in the analytic Bergman space on the upper half-plane, vertical and angular operators in the harmonic Bergman space on the upper half-plane, vertical operators in the poly-analytic and true-poly-analytic Bergman spaces on the upper half-plane, vertical operators in the wavelet spaces associated to the positive affine group, radial operators in analytic and harmonic Bergman spaces on the unit disk, vertical operators in the RKHS associated to the Gauss-Weierstrass kernel on $\mathbb{C}^n$. In most of the examples, the Toeplitz operators with group-invariant symbols were studied before, but we deal with the whole W*-algebra of group-invariant operators.

Posted November 16, 2021

Last modified November 23, 2021

Vignon Oussa, Bridgewater State University

Phase retrieval for nilpotent groups

This talk presents new results on phase retrieval for group representations. Precisely, we show that the (unitary) irreducible representations of nilpotent groups (finite and infinite) allow phase retrieval. This is joint work with Hartmut Fuehr.

Posted April 5, 2022

Last modified April 7, 2022

Wavelet Spaces as Reproducing Kernel Hilbert Spaces

In this talk, I will explore wavelet spaces arising from time-frequency analysis and wavelet analysis. These spaces have an intriguing reproducing kernel Hilbert space structure. Additionally, the spaces can be investigated from an abstract harmonic analysis perspective through representation theory. I will demonstrate a non-trivial distinctness result for wavelet spaces that heavily utilize the representation-theoretic viewpoint. Finally, if time permits, I will relate the reproducing kernel Hilbert space structure of these spaces to the HRT-conjecture in time-frequency analysis.

Posted April 5, 2022

3:30 pm - 4:30 pm zoomTBA

Posted April 23, 2022

3:30 pm - 4:30 pm Lockett, 232
Rui Han, LSU

Matthew McCoy

Dylan Spedale

Fan Yang, LSU

Lp improving estimates for averages in R^d, F_q^d and Z^d

We will survey the L^p improving estimates for spherical averages in the Euclidean space R^d, and talk about some recent sharp results for spherical averages in the finite field F_q^d and polynomial and prime number averages in the integer setting.

Posted April 5, 2022

3:30 pm - 4:30 pm zoomTBA

Posted April 23, 2022

3:30 pm - 4:30 pm Lockett, 232
Rui Han, LSU

Matthew McCoy

Dylan Spedale

Fan Yang, LSU

Lp improving estimates for averages in R^d, F_q^d and Z^d

We will survey the L^p improving estimates for spherical averages in the Euclidean space R^d, and talk about some recent sharp results for spherical averages in the finite field F_q^d and polynomial and prime number averages in the integer setting.

Posted July 11, 2022

2:30 pm - 3:30 pm Lockett Hall 232, $\\$ Zoom Link: https://lsu.zoom.us/j/94413235134
Jens Christensen, Colgate University

The Uncertainty Principle and Representation Theory

We discuss an uncertainty principle for self-adjoint operators generated by unitary representations of Lie groups. We then apply this uncertainty principle to differential operators on Bergman spaces on the unit disc. In the process we classify the self-adjoint differential operators on these spaces, and hint at extensions to higher dimensional domains.

Posted July 28, 2022

Last modified August 30, 2022

Howard Cohl, National Institute of Standards and Technology

Representations and special values for nonsymmetric and symmetric Poisson kernels of the Askey-Wilson polynomials

We discuss the literature on symmetric and nonsymmetric representations for Poisson kernels of Askey-Wilson polynomials. This goes back to the works of Rahman, Verma, Askey, Gasper, Suslov during the 80s and 90s. The symmetric Poisson kernel for Askey-Wilson polynomials was treated in a series of papers by Rahman, Verma and Suslov, and Askey, Rahman and Suslov treated a special form of the nonsymmetric Poisson kernel for Askey-Wilson polynomials in 1996. Even though in all these papers, the analysis was correct, unfortunately, this and all previous symmetric and nonsymmetric Poisson kernels contained typographical errors. We have corrected all symmetric and nonsymmetric representations for these Poisson kernels and as well have computed a new representation of the Askey-Rahman-Suslov nonsymmetric Poisson kernel for Askey-Wilson polynomials using the method of integral representations treated in a recent paper by Cohl and Costas-Santos. We will discuss the form of these representations and as well discuss generating function and arbitrary argument transformation formulas which arise by taking special values of the symmetric representations of these nonsymmetric Poisson kernels. We will also discuss future versions of these results through extension of parameter nonsymmetry.

Posted July 28, 2022

Last modified September 26, 2022

Camilo Montoya, National Institute of Standards and Technology

Laplace eigenfunctions on Riemannian symmetric spaces and the Borel-Weil Theorem

We identify a geometric relation between the Laplace-Beltrami spectra and eigenfunctions on compact Riemannian symmetric spaces and the Borel-Weil theory using ideas from symplectic geometry and geometric quantization. This is done by associating to each compact Riemannian symmetric space, via Marsden-Weinstein reduction, a generalized flag manifold which covers the space parametrizing all of its maximal totally geodesic tori. In the process we notice a direct relation between the Satake diagram of the symmetric space and the painted Dynkin diagram of its associated flag manifold. We consider in detail the examples of the classical simply-connected spaces of rank one and the space SU(3)/SO(3). In the second part of the talk, with the aid of harmonic polynomials, we induce Laplace-Beltrami eigenfunctions on the symmetric space from holomorphic sections of the associated line bundle on the generalized flag manifold. In the examples we consider we show that our construction provides all of the eigenfunctions.

Posted October 16, 2022

Last modified October 23, 2022

Howard Cohl, National Institute of Standards and Technology

Jacobi functions of the first and second kind. II

We continue our review of properties of the Jacobi functions of the first and second kind which are generalizations of the Jacobi polynomials and the Jacobi function of the second kind for integer degrees, where the degree is now allowed to be an arbitrary complex number. We also describe the properties of the Jacobi functions of the first and second kind on-the-cut (-1,1) which are obtained from the standard functions through a limiting process. Various properties that we include are hypergeometric representations, symmetry and anti-symmetry relations (and their relations to Gegenbauer, associated Legendre and Ferrers functions), linear and quadratic transformations, connection formulas, infinite series, and expressions for evaluating these functions when all the degree and parameters are all integers.