Abstracts for Carolyn Gordon, Clanton Visiting
Mathematician, 1998-1999
WHEN YOU CAN'T HEAR THE SHAPE OF A MANIFOLD
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ABSTRACT:
Associated to every compact Riemannian manifold (e.g., spheres, tori) is a
Laplace operator, analogous to the Euclidean Laplacian. The analog of the
problem "Can one hear the shape of a drum?" asks how much geometric
information about the manifold can be determined from the eigenvalue
spectrum of this operator. We will discuss various methods for constructing
Riemannian manifolds with the same Laplace spectrum, and we will compare
the geometry of these isospectral manifolds.
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YOU CAN'T HEAR THE SHAPE OF A DRUM
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ABSTRACT:
In spectroscopy one attempts to recover the chemical composition or shape
of an object from the frequency spectrum of light or sound emitted by the
object. In the case of a vibrating membrane, viewed as a drumhead, Mark Kac
appealingly phrased this problem as "Can one hear the shape of a drum?" We
will answer Kac's question in the negative by constructing a pair of
exotically shaped sound-alike drums. We will also listen to a simulation
of the sounds of these drums produced by Dennis DeTurck.
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