Thin tubes in mathematical physics, global analysis and spectral geometry

TL;DR

This survey explores the mathematical analysis of thin tubes, focusing on the Laplacian and related operators in various contexts, aiming to foster interdisciplinary collaboration in spectral geometry and global analysis.

Contribution

It synthesizes existing research on thin tubes in mathematical physics, highlighting methods and results to promote cross-disciplinary understanding.

Findings

Analysis of Laplacian operators on thin tubes

Connections between geometry and spectral properties

Encouragement of interdisciplinary research

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.