UTA Department of Mathematics

Special Mathematics Colloquium

Date/Time/Room: Thursday (2/28/2002) at 2:30pm in 308 Pickard Hall

Speaker: Ruth Gornet, Department of Mathematics and Statistics, Texas Tech University

``Billions of Isospectral Riemann Surfaces''

Abstract: In 1966, M. Kac popularized the question, "Can one hear the shape of a drum?" Viewing a drumhead as a plane domain D, the frequencies produced when the drum vibrates correspond to the (Dirichlet) eigenvalues of the Laplace operator L. Thus the mathematical formulation of the question posed by Kac is, "Do the eigenvalues of a plane domain determine its shape, up to congruence?" Only recently has the question been answered in the negative.

We consider this question generalized to Riemann surfaces and the Laplace-Beltrami operator. We describe Sunada's beautiful method for using finite group representations to construct isospectral Riemann surfaces. This leads to the question, "How many Riemann surfaces can be isospectral to a given Riemann surfaces?"

An upper bound was provided by P. Buser in terms of the genus g of S; in parituclar, Buser showed that the number of such surfaces has an upper bound of exp(720 g^2). In joint work with R. Brooks and W. Gustafson, we exhibit families of isospectral Riemann surfaces showing that any lower bound must be greater than g^{c log g}, ie, better than polynomial in g. We also discuss applications in spectral graph theory and number theory, and directions for future research.