Probability Seminar: Juan Souto

Abstract

Studying Gromov-Hausdorff limits of sequences of Riemannian manifolds $(M_i)$ satisfying suitable conditions on their local geometry is an extremely fruitful idea. However, in the most interesting case that the diameter of $M_i$ grows without bounds, one is forced to choose base points $p_i\in M_i$ and consider limits of the pointed spaces $(M_i,p_i)$ in the pointed Gromov-Hausdorff topology. The choice of the base points $p_i$ influences enormously the obtained limits. Benjamini and Schramm introduced the notion of distributional limit of a sequence of graphs; this basically amounts to "choosing the base point by random". In this talk I will describe the distributional limits of sequences $(M_i)$ of manifolds with uniformly pinched curvature and satisfying a certain condition of quasi-conformal nature. I will also explain how these results yield a modest extension of Benjamini's and Schramm's original result. This is joint work with Hossein Namazi and Pekka Pankka.