Thin tubes in mathematical physics, global analysis and spectral geometry

Abstract

A thin tube is an $n$-dimensional space which is very thin in $n-1$ directions, compared to the remaining direction, for example the $\epsilon$-neighborhood of a curve or an embedded graph in $\R^n$ for small $\epsilon$. The Laplacian on thin tubes and related operators have been studied in various contexts, with different goals but overlapping techniques. In this survey we explain some of these contexts, methods and results, hoping to encourage more interaction between the disciplines mentioned in the title.