THE SECOND LECTURE: Can You Hear the Shape of a Lattice?

Summary

Introduction

Our title is intended to recall Mark Kac's famous question “Can one hear the shape of a drum?” Kac's article [Kac] drew wide attention to an important old problem which was first raised about 100 years ago. In physical language, we may state this as “do the frequencies of the normal modes of vibration determine the shape of the drum?” Of course this is a purely mathematical problem:—do the eigenvalues of the Laplacian for the Dirichlet problem determined by a planar domain determine the shape of that domain?

When the titles for the lectures on which this book is based were chosen, this problem was still unsolved. By the time they were given, it had been solved by Gordon, Webb, and Wolpert, who made use of some previous work by Sunada and Buser.

It is always exciting to see a classical problem solved, and in this case the solution can be made particularly easy, so although it has little to do with the main topic of these lectures, we give a simple solution to the Kac problem in this lecture.

It is perhaps fortunate that the solution took so long to find, because the consideration of the problem has led to a lot of interesting mathematics. In particular, one can consider the problem for arbitrary Riemannian manifolds; that is, surfaces of possibly arbitrary dimension with an appropriate metric.