I will introduce two basic problems in random geometry. A self-avoiding walk is a sequence of steps in a d-dimensional lattice with no self-intersections. If branching is allowed, it is called a branched polymer. Using supersymmetry, one can map these problems to more tractable ones in statistical mechanics. In many cases this allows for the determination of exponents governing the relationship between the diameter and the number of steps.
We show a (3/2-\epsilon)-approximation algorithm for the graphical traveling salesman problem where the goal is to find a shortest tour in an unweighted graph G. This is a special case of the metric traveling salesman problem when the underlying metric is defined by shortest path distances in G. The result improves on the 3/2-approximation algorithm due to Christofides for the case of graphical TSP.
Artist Paul Hodgson was a Director's Visitor at the Institute in 2010. In a Friends Forum, he discussed the "difficulties in making a judgement and dubtfulness in choosing one thing over another," that underlie his current practice and emerge "both in the way that I fabricate the work, and the images that I choose to present."
Let G be a connected reductive group over Q such that G(R) has discrete series representations. I will report on some statistical results on the Satake parameters (w.r.t. Sato-Tate distributions) and low-lying zeros of L-functions for families of automorphic representations of G(A). This is a joint work with Nicolas Templier.