Title: Diffusions, exit time moments and spectral geometry
Speaker: Professor Pat MacDonald
Speaker Info: University of South Florida
Brief Description:
Special Note:
Abstract:
This talk concerns diffusions, variational principles and boundary value problems on directed graphs with natural weightings, and corresponding graph geometries. Given an appropriate subgraph, we associate to the subgraph a pair of sequences which are invariant under the action of the automorphism group of the ambient graph. The first of these sequences is defined probabilistically and plays an important role in the mechanics of elastic bodies. The purpose of the talk is to clarify the role that the invariants play in the geometry of subgraphs and the global analysis of the ambient graph. As a first step in this direction, we clarify the relationship between our invariants and the Dirichlet spectrum associated to a subgraph. Using results from the Hausdorff moment problem, we prove that for a fixed subgraph, the associated invariants determine the part of the Dirichlet spectrum corresponding to eigenvectors of nonzero mean. As a corollary, we obtain a combinatorial proof that the heat content asymptotics of a given subgraph are completely determined by either sequence of invariants. As another corollary, we prove that for a natural class of weightings, our invariants characterize the ambient graphs which admit a strong isoperimetric inequality. Time permitting, we will discuss extension of our results to noncompact Riemannian manifolds and fractals.Date: Friday, October 20, 2000