There are certain natural cohomology classes on the moduli space M_{g,n}-bar of n-pointed, genus g stable curves known as "tautological classes." I will define these classes and explain the significance of the special situation when all classes are tautological. In joint work with Samir Canning, we find many new small values (g, n) for which all classes on M_{g,n}-bar are tautological. Via an inductive argument, these results lead to applications for arbitrary (g, n). For example, in joint work with Samir Canning and Samuel Payne, we show that the point counts of M_{g,n}-bar over finite fields are surprisingly well approximated by polynomials.
Moduli spaces of stable curves
Nov. 10, 2022 2:30pm (MATH 3…
Functional Analysis
Alex Kumjian (University of Nevada)
X
When is the canonical map from a k-graph into its fundamental groupoid an injection? We also discuss a class of examples, associated to actions of triangle groups on buildings, that led Robertson and Steger to introduce higher rank Cuntz-Pimsner algebras and thereby inspired the notion of higher rank graph (or k-graph) C*-algebras. The fundamental groupoid of a k-graph was introduced by Pask, Raeburn and Quigg in '04 and an example was given that shows that a k-graph need not embed. This is an interim report on continuing work with my colleagues Nathan Brownlowe, David Pask and Aidan Sims.
Embeddability of k-graphs and a class of examples associated to triangle group actions on buildings. Sponsored by the Meyer Fund
Nov. 10, 2022 3:35pm (MATH 3…
Probability
Ping Zhong (University of Wyoming)
X
Random variables in Voiculescu’s free probability theory can model the limits of suitable random matrix models. The Brown measure of a free random variable is a replacement for the eigenvalue counting measure of square matrices. The R-diagonal operators are a large family of free random variables that are natural limit operators of various well-known non-normal random matrix models. I will report some recent progresses on the Brown measure of the sum of two free random variables, one of which is R-diagonal. We show that subordination functions in free additive convolution can detect information about the Brown measure. These Brown measures are related to the limit eigenvalue distributions of deformed i.i.d. and deformed single ring random matrix models. Part of this talk is based on joint work with Hari Bercovici.
Brown measure and eigenvalue distributions of deformed random matrices
Moduli spaces of stable curves