Title: Diffusion Geometry and Local Coordinates on Sets or Manifolds

Abstract: For a finite metric space X or a smooth manifold M, one can define, in a canonical manner, a Laplace Operator. One then obtains Laplacian eigenfunctions. The study of Laplacian eigenfunctions is quite old and plays a central role in many areas of mathematics. The point of view of Diffusion Geometry is to use these eigenfunctions to provide local (or global) coordinate systems. The point here is that the object under study (X or M) might be easier to work with if one had new coordinate systems for it. This point of view has become popular recently in areas of applied mathematics. It is unclear however, why using eigenfunctions in this fashion should come with any guarantee of providing "good" representations of the original object. In this lecture we explain a theorem (joint work with Mauro Maggioni and Raanan Schul) that provides strong guarantees for local coordinate charts. For example, on a D dimensional, smooth manifold M of finite volume, we can choose D eigenfunctions to provide "good local charts". Perhaps surprisingly, in the case of simply connected, planar domains, this proof is closely related to the Riemann Mapping Theorem. Before presenting the theorem we will show some applications of diffusion geometry to various applied problems.