P. N. Shivakumar

(Mathematics, University of Manitoba)

The Shape of a Drum, a Constructive Approach

Date Tuesday, March 17, 2009

In 1966, Mark Kac authored a paper with the catchy title, ``Can one hear the shape of a drum ?'' The problem is whether we can determine the region $Omega$, if we know all the eigenvalues of the eigenvalue problem $u_{xx} + u_{yy} = 0$ in $Omega$ and u = 0 on the boundary $Gamma$ of $Omega$. A large number of mathematicians and physicists have contributed to this topic. The answer is known to be ``yes'' for certain convex planar regions with analytical boundaries. The answer is also known to be ``no'' for some polygons with reentrant corners. In this talk, a brief history and a new constructive approach is given when $Gamma$ is an analytical curve, a circle, an ellipse or a square. In the case of a square, we obtain an insight into why an analytical procedure does not, as expected, yield an answer. For the Matheiu equation with an unknown parameter q and known eigenvalues, we demonstrate the theory to determine q.

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