University of Rochester

Title:  Limit laws  on metric measure spaces

We will survey recent results on limit laws for stochastic processes on metric measure spaces. The main object is a Hunt process corresponding to a Dirichlet form on such a space.  Limit laws include small deviations,  large deviations principle, heat content asymptotics, Chung’s law, as well as finding an Onsager-Machlup functional. Many of these results are closely related to the boundary problems for the corresponding infinitesimal generator in a metric ball.  This setting includes a number of examples: Riemannian manifolds, sub-Riemannian manifolds including Carnot groups, singular spaces such as fractals, diffusions and fractional sub-Laplacians.

Uppsala University and University of Geneve

Title: Metric functionals and nonexpansive maps

In remarkably many contexts — such as linear transformations, holomorphic maps, group theory, surface homeomorphisms, optimal transport, and machine learning — the transformations involved act by nonexpansive maps on an associated metric space. An important notion, inspired by functional analysis, is that of metric functionals, which extend Busemann functions. Using metric functionals one can define weak topologies with compactness properties on any metric space. I will also explain the corresponding metric analogue of spectral theory with extension to random products.  One consequence is a general fixed-point theorem: every isometry of a metric space has a fixed point in a natural and nontrivial compactification of that space.

Karlsruhe Institute of Technology

Title: Submetries

Submetries are maps between metric spaces which send balls onto balls of the same radii. They generalize Riemannian submersions and quotient maps for isometric group actions, appear sometimes  in rigidity considerations and have a variety of different facets ranging from metric geometry to commutative algebra.   In the course  we will discuss basics and structural results for submetries and review the theory of  metric spaces arising in the course of the study,  such as Alexandrov spaces and sets of positive reach.  The participants are expected to know basics of Riemannian geometry such as second fundamental form, injectivity radius, O'Neills formula.

Princeton University

Title: Distortion growth

Infinite metric spaces of interest typically do not admit any bi-Lipschitz embedding into a Hilbert space. The pertinent question thus becomes understanding the "Euclidean distortion growth" of a given infinite metric space X, which is defined to be the rate at which the most non-Euclidean n-point subset of X differs from a subset of R^n. Even though this is natural and useful, determining the Euclidean distortion growth rate of classically studied spaces (e.g. Banach spaces, groups) has proved to be very difficult. One inspiration here is the search for nonlinear versions of a classical theorem of John, as initiated by visionary work of Johnson and Lindenstrauss in the early 1980s, which led to a lot of subsequent work. These lectures will describe the large body of knowledge that accumulated on this topic over the past four decades, including some very recent progress and full proofs of key results. It will also describe a range of important longstanding open questions, and it will end with a tantalizing new conjecture that I call the "gap possibility." No background beyond undergraduate mathematics will be assumed.