Talks

I will present another Finsler metric on Teichmüller space, which is called the earthquake metric. After giving the definition of this metric, I will show that there is a kind of ‘duality’ between this metric and Thurston’s metric, and that the metric is incomplete, just like the Weil-Petersson metric.

A hyperbolic quasifuchsian (or more generally convex co-compact) manifold $M$ contains a smallest non-empty geodesically convex subset, its convex core. The boundary of this convex core has a hyperbolic induced metric, and is pleated along a measured geodesic lamination. Thurston asked whether the induced metric, or the the measured pleating lamination, uniquely determine $M$. In the first part, we will explain why the answer is positive for the measured pleating lamination (joint w/ Bruno Dular). In the second part, we will put this problem in a more general frramework concerning the boundary data of convex subsets in hyperbolic manifolds or in hyperbolic space.
 

I will describe some joint work with Karen Uhlenbeck on best Lipschitz maps between surfaces. While our original motivation was to understand Thurston’s theory in Teichmueller space, it has connections with older ideas. I will remind the listeners about infinity harmonic functions, and describe our theory of infinity harmonic mappings and their dual laminations. The goal is to motivate several interesting, new and I believe hard questions in analysis and their relation to topology.

It is usually assumed that 4D instantons can only arise in non-Abelian theories. One can, however, explicitly construct instantons for QED in the background of a Dirac monopole. This is the low-energy effective field theory for fermions interacting with a 't Hooft-Polyakov monopole. This theory posesses both a topological instanton number and 't Hooft zero modes. I will show how such instantons provide the underlying mechanism for the Callan-Rubakov process: monopole-catalyzed baryon decay with a cross section that saturates the unitarity bound. 

In the first part, I will give an introduction to Thurston's metric on Teichmuller space. In the second part, I will talk about convex structures on tangent spaces of Teichmuller space with respect to the norm induced by Thurston's metric. The latter part includes my joint work with Assaf Bar-Natan and Athanase Papadopoulos.