Seminar 28 May 2008: Abstract
Peter W. Jones (Yale University): Diffusion Geometry and Local Uniformization by eigenfunctions
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.