Abstract of Talk by Dr. Kevin D. Oden

Partial Differential Equations and Spectral Geometry

Dr. Kevin D. Oden 
Department of Mathematics
Harvard University  
One Oxford Street
Cambridge, MA 02138

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Spectral Geometry is an enormous field which one might say began in earnest with the, still fascinating, two volume Theory of Sound by Lord Rayleigh. The basic problems deal with determining the relationship between the geometry of the vibrational medium and its vibrating frequencies ( eigenvalues) which is the, now famous, conjecture of M. Kac: "Can you hear the shape of a drum. In recent years this question has been answered in the negative. That is there exist non-isometric domains which are isospectral. However, there are still many important questions to ask. For instance, to what degree can one understand gaps between eigenvalues? What are the topological and/or geometric constraints on eigenvalue gaps?

There is also a notion of "frequency" associated with a graph. One can define a discrete Laplace operator, which in the limit (when also suitably) defined is the continuous analogue. Then, we may ask the same questions in the discrete setting.

In this talk we will give some physical background for eigenvalue problems and then sketch some joint research with S.Y. Cheng on eigenvalue gaps on domains in Euclidean space. We will then discuss the analogous problems in the discrete setting (joint research with F.R.K. Chung).

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