Title: Diffusion Geometry and Local Uniformization by eigenfunctions.

Abstract:
We will first describe recent developments in diffusion  
Geometry. The idea of this area is to study large data sets  
(physical, biological, financial, etc.) by building nonlinear  
coordinate systems that reflect the internal structures/geometry of  
the data set. This is done by building LaPlace-like operators on the  
data set and using a carefully chosen collection of the corresponding  
eigenfunctions. The robust behavior of this method is perhaps  
surprising, and one would like to understand exactly why it is so  
effective. We then present a theorem (joint work with Raanan Schul  
and Mauro Maggioni) that answers this question in the continuous  
limit, i.e. on a Riemannian manifold. Modulo details the statement is  
as follows. If the Manifold M has dimension d and volume = 1, then on  
any embedded ball in the manifold one can find exactly d (global  
LaPlace) eigenfunctions that blow up the central part of the ball to  
at least unit size. Here the eigenfunctions are normalized to have  
global L^2 norm = 1. The mapping comes with universal BiLipschitz  
constants that only display a dimensional dependence, with no  
dependence on the particular ball chosen or even the manifold. (This  
last statement needs a careful explanation.) This turns out to be an  
analogue of the so called distortion theorems from the classical  
theory of conformal mappings. Indeed one can look at the original  
physical proofs of the Riemann mapping theorem and detect reasons why  
this manifold version should hold. (The result is new even for simply  
connected planar domains and Dirichlet eigenfunctions.) There are  
several problems left open on which we are presently working and some  
of these will be discussed.