CENTER FOR COMPUTATIONAL MATHEMATICS COLLOQUIUM

                  UNIVERSITY OF COLORADO AT DENVER



TITLE:   Eigenmodes of isospectral drums
 

SPEAKER: Tobin A. Driscoll, Department of Applied Mathematics, 
         University of Colorado at Boulder
         

DATE:    Monday, February 17, 1997


PLACE:   Math Conference Room 626 
         UCD Building, 1250 14th St., Denver


TIME:    2:30 pm (Refreshments served at 2:15 pm) (PLEASE NOTE UNUSUAL TIME)



ABSTRACT

In 1966 Mark Kac asked, "Can one hear the shape of a drum?"
Or, mathematically speaking, do nonisometric planar regions
always have different Laplace spectra? Twenty-five years
later, Gordon, Webb, and Wolpert answered "no" and constructed
a counterexample. Their discovery has been documented by
articles in Science, Science News, and the Math Monthly. The
simplest form of their example is a pair of eight-sided
nonconvex polygons.

While elementary analysis can be used to prove the
isospectrality of the GWW drums, determining the spectrum
itself is analytically infeasible. A physical experiment
conducted by Sridhar and Kudrolli in 1994 found the
eigenvalues to about three digits. Standard numerical methods
fail to improve on these results. However, an algorithm
originally due to Descloux and Tolley exploits the structure
of the Laplace eigenvalue problem on polygons. With a
modification that doubles its accuracy, this method is
efficient and accurate. The first 25 eigenvalues of the GWW
drums have been found to 12 digits, each eigenvalue
calculation consuming a few minutes on a workstation. With the
eigenvalues and associated eigenfunctions, animations of the
drums' vibrations have been produced.