# P. N. Shivakumar

## 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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