The mathematics of hearing the shape of a drum and filling Kac's holes

Abstract

In 1966 Mark Kac asked whether it is possible to hear the shape of a drum, that is, if the set of eigenvalues of the Laplace operator on a smoothly bounded domain in the Euclidean plane determine the domain up to isometries. In fact, Kac credited his colleague, Professor Bers, for this picturesque language. The article not only explored this question, but also impressed upon readers the multitude of connections between this problem and various parts of mathematics and physics. Kac was forthcoming about his arguments being mostly heuristics justified by physics and as such not fully rigorous. However, it turns out that all of his arguments can be made rigorous using more recent techniques like microlocal analysis. Here we show how to make all of Kac’s arguments rigorous. This includes demonstrating that if a sequence of convex polygons in the plane, with or without convex polygonal holes, converges in the sense of Hausdorff convergence to a convex smoothly bounded domain, then the first three coefficients in the asymptotic expansion of the heat trace of the polygons converge to those of the smoothly bounded domain. This appears to be the first time this has ever been done, showing that all of Kac’s heuristics were in fact correct. Finally, we give a survey of Kac’s legacy, the wealth of mathematics that Kac helped to inspire. Keywords: