# The moduli space of two-convex embedded spheres and tori

**Date**: Wednesday 15 March 2017, 15:00 – 16:30**Location**: Roger Stevens LT 12 (10M.12)**Type**: Geometry, Seminars, Pure Mathematics**Cost**: Free

#### Reto Buzano, Queen Mary University of London. Part of the geometry seminar series.

It is interesting to study the topology of the space of smoothly embedded n-spheres in R^{n+1}. By Smale’s theorem, this space is contractible for n=1 and by Hatcher’s proof of the Smale conjecture, it is also contractible for n=2. These results are of great importance, generalising in particular the Schoenflies theorem and Cerf’s theorem. In this talk, I will explain how mean curvature flow with surgery can be used to study a higher-dimensional variant of these results, proving in particular that the space of 2-convex embedded spheres is path-connected in every dimension n. We then also look at the space of 2-convex embedded tori where the question is more intriguing and the result in particular depends on the dimension n. This is all joint work with Robert Haslhofer and Or Hershkovits.

**Reto Buzano**, *Queen Mary University of London*