# 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

## Office Telephone Number:

(617)-495-5377
## Fax Number:

## Electronic Mail Address:

oden@math.harvard.edu

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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