The first characterization of groups by an asymptotic description of random walks on their Cayley graphs dates back to Kesten’s criterion of amenability. I will first review some connections between the random walk parameters and the geometry of the underlying groups. I will then discuss a flexible construction that gives solution to the inverse problem (given a function, find a corresponding group) for large classes of speed, entropy and return probability of simple random walks on groups of exponential volume growth. Based on joint work with Jeremie Brieussel.
 

During the early development of calculus, eminent mathematicians such as Leibniz and Cauchy freely used infinitesimals in their calculations. Once the mathematical community became dubious of their status, the use of infinitesimals was replaced by the now familiar epsilon-delta rigor. In the 1960s, Abraham Robinson used techniques from model theory to rescue infinitesimals from their squalid state and instead put them on a firm foundation in what he called nonstandard analysis. Since its inception, nonstandard techniques have proven useful in many diverse areas of mathematics, from geometry to functional analysis to mathematical finance. Besides allowing one to give precise meaning to intuitive, heuristic arguments involving “ideal” elements, nonstandard analysis offers new techniques such as hyperfinite approximation and Loeb measure. In this talk, I will survey some uses of nonstandard analysis in Lie theory and additive combinatorics. Some highlights of the talk will be the nonstandard solution to Hilbert’s fifth problem (and its extension to the local case), the Breuillard-Green-Tao structure theorem for approximate groups, and some progress on a sumset conjecture of Erdos.

     We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It has also several applications in optics and medical imaging among others.
     The problem can be recast as a geometric problem: Can one determine a Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will also consider the problem of determining the metric from the scattering relation, the so-called lens rigidity problem. The linearization of these problems involve the integration of a tensor along geodesics, similar to the X-ray transform.
     We will also describe some recent results, join with Plamen Stefanov and Andras Vasy, on the partial data case, where you are making measurements on a subset of the boundary. No previous knowledge of Riemannian geometry will be assumed.