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.