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Texas A&M University
Mathematics

Noncommutative Geometry Seminar

Spring 2020

 

Date:January 22, 2020
Time:2:00pm
Location:BLOC 628
Speaker:Xin Ma, SUNY Buffalo
Title:The groupoid semigroup and its application
Abstract:In this talk, I will introduce and discuss an algebraic tool called the groupoid semigroup. I will show how this semigroup help establishing the almost unperforation form for dynamical comparison. Then, I will show how it relates to the type semigroup in the ample case. Finally, I will present some result on (strongly) pure infiniteness of reduced groupoid C*-algebras.

Date:February 26, 2020
Time:2:00pm
Location:BLOC 628
Speaker:Hao Guo, Texas A&M University
Title:Functoriality of higher invariants for elliptic operators
Abstract:I will report on joint work with Zhizhang Xie and Guoliang Yu on functoriality properties of the higher index and higher rho invariant for elliptic differential operators on manifolds with symmetry. We show that given a homomorphism between the deck transformation group of a Galois cover and a quotient by a normal subgroup, the map induced on the level of group C*-algebras naturally relates the higher indices and higher rho invariants associated to the two group actions. We work with the maximal version of the group C*-algebra. Our results can be applied to the problem of computing higher invariants on a covering space, for example by relating it to higher invariants on finite-sheeted covers.

Date:April 22, 2020
Time:2:00pm
Location:Zoom Meeting ID:
Speaker:Jonathan Rosenberg, University of Maryland
Title:Positive scalar curvature on a class of spin pseudomanifolds
Abstract:By famous theorems of Gromov-Lawson and Stolz, we know precisely which simply connected closed spin manifolds (in dimensions 5 and up) admit Riemannian metrics of positive scalar curvature. We discuss a generalization of the same problem to a class of singular manifolds with one singular stratum, where the link of the singularity is a homogeneous manifold of constant positive scalar curvature (such as a compact symmetric space), and where the metric is required to have a nice cone-like structure in a neighborhood of the singularity. The obstruction theory uses index theory on singular spaces, and the existence theory uses surgery theory. This is joint work with Boris Botvinnik (Oregon) and Paolo Piazza (Rome La Sapienza).

Date:April 29, 2020
Time:2:00pm
Location:Zoom 942810031
Speaker:Boris Tsygan, Northwestern University
Title:Rigidity, the Goodwillie regulator, and the Gauss-Manin connection over p-adic integers
Abstract:Periodic cyclic homology of associative algebras generalizes in many ways DeRham cohomology and more generally crystalline cohomology of algebraic varieties over a field of characteristic zero. Among the properties of De Rham cohomology that can be so generalized are: rigidity under infinitesimal deformations and a regulator map from relative algebraic K theory to relative cyclic homology of a nilpotent ideal (Goodwillie), and the Gauss-Manin connection (Getzler). I will explain how these results generalize to p-adic completions of cyclic complexes over p-adic integers. These generalizations develop recent results of Beilinson and Petrov-Vologodsky.

Date:May 6, 2020
Time:2:00pm
Location:Zoom 942810031
Speaker:Yanli Song, Washington University, St. Louis
Title:Higher index theorem for proper actions of Lie groups
Abstract:In this talk, I will describe a cohomological formula (including a fixed point theorem) for a higher index pairing between invariant elliptic differential operators and cyclic cohomology classes associated to orbital integrals. This is joint work with Peter Hochs and Xiang Tang.

Date:May 13, 2020
Time:2:00pm
Location:Zoom 942810031
Speaker:Robin Deeley, University of Colorado, Boulder
Title:The K-theory of the stable algebra and stable Ruelle algebra of a Wieler solenoid
Abstract:Wieler has shown that every irreducible Smale space with totally disconnected stable sets is a solenoid (i.e., obtained via a stationary inverse limit construction). Through examples I will discuss how this allows one to compute the K-theory of the stable algebra, S, and the stable Ruelle algebra, S\rtimes Z. These computations involve writing S as a stationary inductive limit and S\rtimes Z as a Cuntz-Pimsner algebra. These constructions reemphasize the view point that Smale space C*-algebras are higher dimensional generalizations of Cuntz-Krieger algebras. As I mentioned the talk will be example based and hence no knowledge of Smale spaces or Wieler solenoids is required. The main results are joint work with Magnus Goffeng and Allan Yashinski.

Date:May 20, 2020
Time:2:00pm
Location:Zoom 942810031
Speaker:Rufus Willett, University of Hawai'i
Title:Almost ideal structures and K-theory
Abstract:The most basic (and powerful) tools to compute C*-algebra K-theory are probably the long-exact sequences associated to ideals. Unfortunately, many of the most interesting C*-algebras are simple, so these tools are inapplicable. I’ll discuss techniques to compute K-theory that work when one only has ‘approximate ideals’, examples coming from dynamics including simple C*-algebras, and applications to the Baum-Connes conjecture and Künneth formula.