Calendar

Posted September 6, 2013

Student Harmonic Analysis/Representation Theory Seminar

Matthew Dawson, Centro de Investigacion en Matematicas
Introduction to invariant means and applications to harmonic analysis, Part I

The study of harmonic analysis and representation theory on locally compact groups depends heavily on the existence of translation-invariant measures, called Haar measures, for such groups. Unfortunately, many interesting topological groups (e.g., infinite-dimensional Lie groups) are not locally compact and do not possess Haar measures. In this talk we discuss an alternate structure, called an invariant mean, which in some ways forms as a replacement for Haar measure. Groups which possess invariant means are called ``amenable\'\' and have many interesting properties related to such varied topics as the Banach-Tarski paradox, fixed point theorems, and the existence of a sort of ``Plancherel Theorem\'\' for some infinite-dimensional Lie groups.

Posted September 6, 2013
Last modified September 30, 2013

Student Harmonic Analysis/Representation Theory Seminar

Benjamin Harris, LSU
What is microlocal analysis? Part I

It had long been realized that continuous or measurable functions were not large enough collections of functions to solve many classical analysis questions so in the mid 20th century analysts sought to formalize definitions of generalized functions. Of course, analysts also needed new ways of studying these new generalized functions. Perhaps the most important of these theories is called microlocal analysis. Roughly speaking, there are two schools of microlocal analysis, sometimes called ``algebraic microlocal analysis'' and ``analytic microlocal analysis''. The so-called algebraic school began with two papers by Sato in 1958 and 1959. Sato quickly gained many collaborators in Japan, and his theory was already quite well developed when the so-called analytic school was begun by a paper of Hormander in 1971 (Hormander was aware of Sato's earlier work). Sato's ideas also inspired Kashiwara, who was Sato's student in the early years of the theory, to work on the theory of D-modules. In this series of two talks, we will give a brief and rough overview of Sato's theory of microlocal analysis. This series is really a tale of three sheaves on an analytic manifold: the sheaf of analytic functions, the sheaf of hyperfunctions, and the sheaf of microfunctions. Roughly speaking, this is a sheaf of classical functions, a sheaf of generalized functions, and a sheaf to help us study the differences between the two (that is where the microlocal analysis comes in). If time permits, we will show how to use this theory to say (more or less) concrete things about linear partial differential equations on analytic manifolds.

Posted September 6, 2013
Last modified September 30, 2013

Student Harmonic Analysis/Representation Theory Seminar

Benjamin Harris, LSU
What is microlocal analysis? Part II

It had long been realized that continuous or measurable functions were not large enough collections of functions to solve many classical analysis questions so in the mid 20th century analysts sought to formalize definitions of generalized functions. Of course, analysts also needed new ways of studying these new generalized functions. Perhaps the most important of these theories is called microlocal analysis. Roughly speaking, there are two schools of microlocal analysis, sometimes called ``algebraic microlocal analysis'' and ``analytic microlocal analysis''. The so-called algebraic school began with two papers by Sato in 1958 and 1959. Sato quickly gained many collaborators in Japan, and his theory was already quite well developed when the so-called analytic school was begun by a paper of Hormander in 1971 (Hormander was aware of Sato's earlier work). Sato's ideas also inspired Kashiwara, who was Sato's student in the early years of the theory, to work on the theory of D-modules. In this series of two talks, we will give a brief and rough overview of Sato's theory of microlocal analysis. This series is really a tale of three sheaves on an analytic manifold: the sheaf of analytic functions, the sheaf of hyperfunctions, and the sheaf of microfunctions. Roughly speaking, this is a sheaf of classical functions, a sheaf of generalized functions, and a sheaf to help us study the differences between the two (that is where the microlocal analysis comes in). If time permits, we will show how to use this theory to say (more or less) concrete things about linear partial differential equations on analytic manifolds.

Posted February 14, 2014

Student Harmonic Analysis/Representation Theory Seminar

Ambar Sengupta, Mathematics Department, LSU
Representing the Heisenberg Group

In his foundational 1925 paper on quantum mechanics Heisenberg introduced a relation that was later formulated in the form PQ-QP=icI, where P and Q are infinite matrices corresponding to momentum and position, c is a positive physical constant, and I is the infinite identity matrix. This canonical commutation relation (CCR) led to a great journey, continuing today, involving the study of unbounded operators, the spectral theorem, and much else. In this talk we will explore some of these ideas, focusing on Weyl\'s reformulation of the CCR in terms of unitary operators and how this can be viewed as a representation of a group, called the Heisenberg group, that encodes the essential structure of the CCR. We will also look at von Neumann\'s formulation and proof of the equivalence of the CCR method with the wave mechanics introduced by Schroedinger in 1926.

Posted March 7, 2014

Student Harmonic Analysis/Representation Theory Seminar

Ambar Sengupta, Mathematics Department, LSU
Representing the Heisenberg Group (Part II)

Posted March 16, 2014
Last modified March 31, 2014

Student Harmonic Analysis/Representation Theory Seminar

Doug Pickrell, University of Arizona
Factorization in Lie groups

A periodic function can be decomposed into a Fourier series (which has many applications). In this talk I will present a somewhat analogous product factorization for a periodic function with values in a compact Lie group, such as SU(2) (and some of its applications, which are more specialized). At the end I will also briefly describe an analogous factorization for homeomorphisms of a circle.