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Julia Plavnik, Texas A&M University
The problem of classifying modular tensor categories is motivated by applications to topological quantum computation as algebraic models for topological phases of matter. These categories have also applications in different areas of mathematics like topological quantum field theory, von Neumann algebras, representation theory, and others.
In this talk, we will start by introducing some of the basic definitions and properties of fusion, braided, and modular tensor categories, and we will also give some concrete examples to have a better understanding of their structures.
The idea of the talk is to give an overview of the current situation of the classification program for modular categories. We will explain some of the techniques that we found useful to push further the classification, with a focus on new constructions of modular tensor categories. If time allows, we will mention some results for the super-modular case.
Jing Wang, UIUC
We discuss degenerate hypoelliptic diusion processes and their limiting behaviors in both large time and small time. For a diusion process on a sub-Riemannian manifold, questions related to large time behavior such as stochastic completeness and convergence to equilibrium are closely related to global geometric bounds including Ricci curvature lower bound. Small time behavior of its transition density falls into the regime of large deviation estimate, which connects to the sub-Riemannian distance.
In particular we are interested in the study of small time behavior of a general strict-degenerate diffusion process (weak Hormander's type), which has been a longstanding open problem. In this talk we will present a recent progress in this problem by developing a graded large deviation principle for diusions on a nilpotent Lie group. Parts of this talk are based on joint work with Fabrice Baudoin and Gerard Ben Arous.
Jenya Sapir, Binghamton University
Abstract: I will talk about a recent result of Athreya, Lalley, Wroten and myself. Given a hyperbolic surface S, a typical long geodesic arc will divide the surface into many polygons. We give statistics for the geometry of this tessellation. Along the way, we look at how very long geodesic arcs behave in very small balls on the surface.
There are no conferences this week.