Events for the week of September 08 2008

Abstract. A fat graph is a family of Riemannian manifolds $M_\epsilon$, $\epsilon>0$, modelled on a finite metric graph $G$, in a way similar to an $\epsilon$-neighborhood of a straight-edge embedding of $G$ in some Euclidean space. The behavior of the spectrum and of spectral invariants of various geometric differential operators on $M_\epsilon$ as $\epsilon$ tends to zero has been studied by many authors in different contexts. We focus on a question arising in mathematical physics which has attracted much attention in the quantum graphs community recently: Which operator on the singular limit $G$ describes the asymptotic behavior of the eigenvalues of the Laplacian on $M_\epsilon$ appropriately? While for Neumann boundary conditions (or closed manifolds) the answer has been known for some time and can be obtained by quite elementary methods, the case of Dirichlet conditions is harder and was solved only recently. This will be explained in the talk.