Unless specified, all seminars are Wednesday 4-5pm at

September 5th

Local-triviality dimension of actions of compact quantum groups

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, DÄ…browski 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

Demystifying Rokhlin dimension and related notions

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

Asymptotic stable division property and the Arveson-Douglas Conjecture

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

Local weak* convergence and the entropy of algebraic actions

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

Incompressibility of compact groups

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