Colloquium

Heat equation, combinatorics and geometric invariants

by Iosif Polterovich, Professor of Mathematics, Universite de Montreal

Abstract

Relations between the geometric properties of manifolds and thespectrum of the Laplacian have been studied for several decades. Heat equation asymptotics is an important source of spectrally determined geometric invariants. Referring to the well-known question "Can one hear the shape of a drum?", heat invariants tell us, for instance, that we can hear the volume and the scalar curvature. However, the complexity of heat invariants grows extremely fast, and most of the information they contain remains unaccessible. In the lecture we discuss a new approach to this problem and some of its applications, in particular to combinatorics and mathematical physics.