Harmonic Analysis Seminar
Abstract and additional information

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.

Harmonic Analysis Seminar
Abstract and additional information

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.

Harmonic Analysis Seminar
Abstract and additional information

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.

Harmonic Analysis Seminar
Abstract and additional information

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.

Harmonic Analysis Seminar
Abstract and additional information

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

Harmonic Analysis Seminar
Abstract and additional information

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.

Harmonic Analysis Seminar
Abstract and additional information

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.

Harmonic Analysis Seminar
Abstract and additional information

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.

Harmonic Analysis Seminar
Abstract and additional information

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.

Harmonic Analysis Seminar
Abstract and additional information

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.

Harmonic Analysis Seminar
Abstract and additional information

Posted October 8, 2020

3:30 pm - 4:30 pm https://lsu.zoom.us/j/99528757270?pwd=SUprWlJyczd3VUhEZ3Z3MTJjdjlwdz09 Originally scheduled for 3:30 pm, Wednesday, October 14, 2020
Nicholas J Christoffersen, LSU

TBA

Harmonic Analysis Seminar
Abstract and additional information

Posted October 8, 2020

3:30 pm - 4:30 pm https://lsu.zoom.us/j/99528757270?pwd=SUprWlJyczd3VUhEZ3Z3MTJjdjlwdz09
Nicholas J Christoffersen, LSU

TBA