Workshop on Representation Theory and Noncommutative Geometry
Nov 18th- Nov 19th 2023
Washington University in St. Louis
The goals of the workshop are to highlight some of the most significant recent advances in noncommutative geometry and representation theory, identify promising new research directions, and acquaint graduate students and postdocs with the most current research in the field.
The workshop is supported by NSF and the Department of Mathematics at Washington University in St. Louis.
Invited Speakers:
Jacob Bradd (Penn State University)
Pierre Clare (College of William & Mary)
Tyrone Crisp (University of Maine)
Nigel Higson (Penn State University)
Joel Villatoro (Washington University in St. Louis)
Hao Zhuang (Washington University in St. Louis)
Program:
Location: Cupple I, Room 199
Saturday:
10:00-10:50 Nigel Higson
11:10 - 12:00 Hao Zhuang
2:00-2:50 Pierre Clare
3:10 - 4:00 Joel Villatoro
Sunday:
9:30- 10:20 Tyrone Crisp
10:40-11:30 Jacob Bradd
Title: The similar structures of the Casselman-Wallach algebra and the reduced group C*-algebra
Speaker: Jacob Bradd
Abstract: I will discuss how, for a real reductive group, the Casselman algebra and the reduced group C*-algebra are assembled from very similar elementary components in very similar ways. Additionally, these components have equal K-theory, which implies that the Casselman algebra and the C*-algebra have equal K-theory (when the K-types are restricted to a finite set). These components are inspired by the work of Delorme on the Paley-Wiener theorem (a description of the Fourier transform for compactly supported smooth functions of the group), and from the Clare-Crisp-Higson description of the reduced group C*-algebra. I will discuss the decomposition for SL(2,R) and then discuss some aspects of the general case.
Title: C*-algebraic pictures of intertwining operators
Speaker: Pierre Clare
Abstract: Intertwining operators play a fundamental role in representation theory and explicit constructions in concrete models, when available, often lead to valuable insight into delicate aspects of the theory. On the other hand, while these operators play an equally central role in the operator algebraic approach to the representation theory of Lie groups, explicit constructions remain rare, in spite of evidence for potentially useful applications. The goal of this talk will be to present known partial results and possible directions for this line of research.
Title: Trace-class operators on Hilbert modules
Speaker: Tyrone Crisp
Abstract: When H is a Hilbert space, the Haagerup tensor products H\otimes H^* and H^*\otimes H are completely isometrically isomorphic, respectively, to the space of compact operators and the space of trace-class operators on H. Blecher has shown that the result about compact operators extends to the setting of Hilbert modules over arbitrary C*-algebras. I will present recent joint work with M. Rosbotham in which we extend the result about trace-class operators to countably generated Hilbert modules over commutative C*-algebras. I will also explain what this result says about unitary representations induced from central subgroups.
Title: Vogan’s theorem on tempiric representations and C*-algebra K-theory
Speaker: Nigel Higson
Abstract: This will be a very informal presentation - on what I understand about Vogan’s theorem on the K-types of tempered irreducible representations with real infinitesimal character; on what parts of Vogan’s theorem I can prove using K-theory for C*-algebras; and on what parts I would like to prove (all of them, of course). I hope the talk will be a starting point for useful discussions over the weekend.
Title: On convolution algebras of double groupoids and 2-groups
Speaker : Joel Villatoro
Abstract: This talk is about a joint work with Angel Roman. Double groupoids and strict 2-groups are characterized by having more than one composition operation. These operations are compatible in the sense that they satisfy an interchange law. In this talk I will go over a few different observations about the convolution algebras that arise from these two operations. We will place a particular emphasis on investigating how the compatibility condition at the level of composition operations is reflected by the convolution structures. I will also briefly discuss some of the motivations for investigating this topic that comes from the study of non-integrable algebroids.
Title: Invariant Morse-Bott-Smale cohomology and the Witten deformation
Speaker : Hao Zhuang
Abstract: In this talk, we will introduce an invariant Morse-Bott-Smale chain complex for closed T-manifolds with a special type of T-invariant Morse-Bott functions. Then, we will establish a quasi-isomorphism between the invariant Morse-Bott-Smale complex and the Witten instanton complex. Finally, we will generalize Mohsen's deformation to normal cone method to prove Morse type inequalities associated with the invariant Morse-Bott-Smale complex.
Local Organizer:
Angel Roman (angelr@wustl.edu)
Yanli Song (yanlisong@wustl.edu)
Xiang Tang (xtang@wustl.edu)