Featured Speaker:  Oscar Randal-Williams (University of Oxford)

Title: Diffeomorphisms of discs

Abstract:

In dimensions neq4neq4 the difference between groups of diffeomorphisms and of homeomorphisms of an nn-manifold MM is governed by an hh-principle, meaning that it reduces to understanding these groups for M=ℝnM=Rn. The group of diffeomorphisms is simple, by linearising it is equivalent to O(n)O(n), but the group Top(n)Top(n) of homeomorphisms of ℝnRn has little structure and is difficult to grasp. It is profitable to instead consider the nn-disc M=DnM=Dn, because the group of homeomorphisms of a disc (fixing the boundary) is contractible by Alexander's trick: this removes homeomorphisms from the picture entirely, and makes the problem one purely within differential topology. I will explain some of the history of this problem, as well as recent work with A. Kupers in this direction.