MARK KAC SEMINAR

We present recent work on characterizing Ricci curvature and Ricci flow in terms of functional inequalities for heat semigroups. The talk includes extensions of these methods to geometric flows on manifolds, as well as to the path space of Riemannian manifolds evolving under a geometric flow.

The slides of the first lecture are available here.

Analyzing the stationary distribution and the convergence to stationarity of non reversible Markov chains is often a very challenging task. In these lectures we discuss these problems for a class of random walks in random directed graphs and other sparse random structures. We first consider the configuration model with prescribed in- and out-degree sequences. The mixing time is identified explicitly in terms of the degree sequences, while the stationary distribution is shown to have a nontrivial shape, characterized via recursive distributional equations. Moreover, the chain has a sharp cutoff behavior around the mixing time, with a universal Gaussian shape inside the cutoff window. We then discuss the extension of these results to a larger family of sparse Markov chains whose transition matrices have exchangeable random rows satisfying a sparsity assumption. We show that the cutoff phenomenon holds in this general setting, with a mixing time characterized in terms of the average one step entropy of the Markov chain. Examples include various models of random directed graphs and random walks among heavy-tailed directed conductances.