Analysis   Seminar    


Unless specified, all seminars are Wednesday 4-5pm at 150 Math Building. This is different from the room 250 Math Building in other semesters.                                                                


September 5th                 Mariusz Tobolski,   Institute of Mathematics Polish Academy of Sciences
                              Local-triviality dimension of actions of compact quantum groups
            
                              Abstract: We introduce the local-triviality dimension of an action of a compact quantum group on a unital C*-algebra using completely positive contractive
                              order zero maps of Winter and Zacharias. In the case of a compact Hausdorff group acting on a compact Hausdorff space our definition recovers the usual
                              local triviality of a compact principal bundle. Actions with finite local-triviality dimension are automatically free and there exists an analog of an
                              n-universal bundle (in the sense of Steenrod) for any compact quantum group G. Our main motivating examples are the Matsumoto-Hopf fibration and the antipodal
                              action on free orthogonal quantum sphere. As the main application, we prove a Borsuk-Ulam-type conjecture of Baum, Dąbrowski and Hajac in the case where the
                              compact quantum group G admits a classical subgroup whose induced action has finite local-triviality dimension.


September 12th                Jianchao Wu,     Penn State University
                              Demystifying Rokhlin dimension and related notions

                              Abstract: The theory of Rokhlin dimension was introduced by Hirshberg, Winter and Zacharias as a tool to study the regularity properties of C*-algebras in
                              relation with group actions. It was inspired by the classical Rokhlin lemma in ergodic theory. Since then, it has been greatly developed as well as simplified,
                              and connections to other areas have been discovered. In this talk, I will present some newer perspectives to help us understand this concept. In particular,
                              I will explain its relation to the Schwarz genus for principal bundles in the context of generalized Borsuk-Ulam theorems. Time permitting, I will also
                              indicate how one can extend the theory beyond residually finite groups. This includes recent and ongoing joint projects with Gardella, Hajac, Hirshberg,
                              Hamblin, Tobolski and Zacharias.


September 19th                Yi Wang,     SUNY at Buffalo
                              Asymptotic stable division property and the Arveson-Douglas Conjecture

                              Abstract: The Arveson-Douglas Conjecture concerns essential normality of submodules of the Bergman module. We will define the asymptotic stable division
                              property and show that with additional mild conditions, the asymptotic stable division property implies essential normality. We will also apply this result
                              on certain submodules. This gives us a unified proof of most known results on the Arveson-Douglas Conjecture. The proof is based on an inequality of a new
                              type, a covering lemma and some local analysis.


September 26th                Ben Hayes,    University of Virginia
                              Local weak* convergence and the entropy of algebraic actions

                              Abstract: I will discuss the entropy of probability measure-preserving actions of sofic groups, due to Bowen and Kerr-Li. I will focus on the case when the
                              action is by automorphisms of a compact metrizable group (these are called algebraic actions). I will give an abstract criterion, in terms of measures on
                              model spaces, which guarantees that the measure-theoretic entropy and topological entropy agree. Knowledge of sofic groups and sofic entropy will not be
                              assumed.


October 31th                  Alexandru Chirvasitu,   SUNY at Buffalo
                              Incompressibility of compact groups

                              Abstract: The join X*Y of two topological spaces X and Y is the set of formal convex combinations of points from the two spaces. The join is a familiar
                              construction to topologists, and arises naturally in the construction of the universal principal bundle for a topological group.

                              I will call a group G `incompressible' if there are no G-equivariant maps from higher to lower joins of copies of G. The main result is that all compact
                              groups are incompressible, generalizing unpublished work by M. Bestvina and R. Edwards in the case of 0-dimensional groups.

                              Applications include a Borsuk-Ulam-type theorem for actions whose induced principal bundle is locally trivial.

                              (joint w/ Ludwik Dabrowski and Mariusz Tobolski)



Past Analysis Seminar