## Diff. Geom, Math. Phys., PDE Seminar: Or Hershkovits

- Date: 11/08/2016
- Time: 15:30

University of British Columbia

The moduli space of 2-convex embedded spheres

The space of smoothly embedded n-spheres in Rn+1 is the quotient space Mn:=Emb(Sn,Rn+1)/Diff(Sn). In 1959 Smale proved that M1 is contractible and conjectured that M2 is contractible as well, a fact that was proved by Hatcher in 1983.While it is known that not all Mn are contractible, for n\get 3 no single homotopy group of Mn is known. Even knowing whether the Mn are path connected or not would be extremely interesting. For instance, if M3 is not path connected, the 4-d smooth Poincare conjecture can not hold true. In this talk, I will first explain how mean curvature flow can assist in studying the topology of geometric relatives of Mn.I will first illustrate how the theory of 1-d mean curvature flow (aka curve shortening flow) yields a very simple proof of Smale's theorem about the contractibility of M1. I will then describe a recent joint work with Reto Buzzno and Robert Haslhofer, utilizing mean curvature flow with surgery to prove that the space of 2-convex embedded spheres is path connected.

Location: ESB 2012