Noncommutative Geometry Seminar

Date: March 7, 2017

Time: 12:30PM - 1:30PM

Location: BLOC 506A

Speaker: Jianchao Wu, Penn State

Title: Noncommutative dimensions and dynamics

Abstract: We showcase a number of recently emerged concepts of dimensions defined for topological dynamical systems, such as the dynamical asymptotic dimension introduced by Guentner, Willett and Yu and the amenability dimension inspired by the work of Bartels, Lück and Reich. Roughly speaking, these dimensions measure the complexity of the topological dynamical system by the extent to which we can decompose the dynamical system into disjoint "towers", which are small neighborhoods of partial orbits. Having finite such dimensions often facilitates certain "algorithms" for K-theoretic computations, which makes them powerful tools for proving the Baum-Connes conjecture and the Farrell-Jones conjecture, as well as bounding the nuclear dimensions (for crossed product C*-algebras), which is a crucial regularity property in the recent breakthrough in the classification program of simple separable amenable C*-algebras. They also turn out to have close relations with the Rokhlin dimension, which was developed independently for C*-dynamical system and draws inspiration from the classical Rokhlin lemma in ergodic theory and its subsequent application in the study of von Neumann algebras. These new developments have spurred growing interests in the noncommutative dimension theory, for which a focal challenge is to find ways to control these new dimensions. This talk includes work in collaboration with Hirshberg, Szabo, Winter and Zacharias as well as further recent developments.