Talks

Let O be a compact reflection n-orbifold whose underlying space is homeomorphic to a truncation n-polytope, i.e. a polytope obtained from an n-simplex by successively truncating vertices. In this talk, I will give a complete description of the deformation space of convex real projective structures on the orbifold O of dimension at least 4. Joint work with Suhyoung Choi and Ludovic Marquis.

Local subsystems play a fundamental role in theoretical physics, but they are usually defined in terms of background spacetime structures, in particular a Lorentzian metric. In gravity there are no such structures and the metric becomes dynamical. I will argue that nevertheless subsystems can be constructed in relation to other degrees of freedom, subject to the requirement that those variables reside within the subsystem itself. In operational terms an observer located within the system should be able to determine its edge by taking measurements within that region only. Within the context of classical field theory, I will demonstrate that this guarantees that the observables describing the system constitute a closed Poisson algebra. I'll discuss some examples, explain how surface charges generating subsystem symmetries, including the Bekenstein-Hawking area entropy, are incorporated, and make some remarks about how this can be extended to a fully quantum treatment within the framework of deformation quantization.

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.