Departmental Colloquium

Event Information The simplicity conjecture and the algebraic structure of groups of area-preserving homeomorphisms
16:10 on Wednesday February 22, 2023
17:00 on Wednesday February 22, 2023
BA6183, Bahen Center, 40 St. George St.
Sobhan Seyfaddini

Institut Mathématique de Jussieu, Paris

In an influential article from the 1970s, Albert Fathi, having proven that the group of compactly supported volume-preserving homeomorphisms of the n-ball is simple for n≥3,asked if the same statement holds in dimension 2. In a joint work with Cristofaro-Gardiner and Humilière, we proved that the group of compactly supported area-preserving homeomorphisms of the 2-disc is not simple. This answers Fathi's question and settles what is known as "the simplicity conjecture" in the affirmative.

In fact, Fathi posed a more general question about all compact surfaces: is the group of "Hamiltonian homeomorphisms" (which I will define) simple? In my talk, I will review recent joint work with Cristofaro-Gardiner, Humilière, Mak and Smith answering this more general question of Fathi. The talk will be for the most part elementary and will only briefly touch on Floer homology which is a crucial ingredient of the solution.