Event

Talk
Whitney and Thom-Mather spaces with applications to spectral theory
9 Feb 2015, 16:00 – 17:00

The systematic study of stratified spaces was initiated by Hassler Whitney in the 1940's. He defined them as special subsets of smooth manifolds, with a deceptively simple description, and showed their applicability to analytic sets (to be referred to as W-spaces). In the 1960's, René Thom generalized Whitney's approach to abstract spaces, with a view of applications to structurally stable smooth maps. For these spaces he conjectured a lot more structure, decoding Whitney's famous "condition (b)"; Thom sketched proofs of all his assertions which were impossible to understand, though.

In solving the problem of structural stability, John Mather gave eventually a careful and complete proof of all important results conjectured by Thom such that I will refer to them as
TM-spaces. In the talk, I will briefly introduce the main notions with some illustrations, and then indicate a proof of the following generalization of Whitney's embedding result for manifolds.

Theorem: Every compact TM-space can be embedded into some Euclidean space in such a way that the image
(which inherits the structure of a TM space) is actually a W-space.

Applications to spectral theory arise from considering the top-dimensional stratum in W, which is an open and dense manifold, as a Riemannian manifold with the metric induced from the above embedding. For certain such spaces, we can establish the analogue of Weyl's law for the Laplacian on differential forms.