To hear the shape of a drum is to infer information about the shape of the drumhead from the sound it makes, i.e., from the list of basic harmonics, via the use of mathematical theory. "Can One Hear the Shape of a Drum?" was the witty title of an article by Mark Kac in the American Mathematical Monthly 1966 (see References below), but these questions can be traced back all the way to Hermann Weyl.
The frequencies at which a drumhead can vibrate depend on its shape. Known mathematical formulas tell us the frequencies if we know the shape. A central question is: can they tell us the shape if we know the frequencies? No other shape than a square vibrates at the same frequencies as a square. Is it possible for two different shapes to yield the same set of frequencies? Kac did not know the answer to that question.
Somewhat more formally, we are given a domain D, typically in the plane but sometimes in higher dimension, and the eigenvalues of a Dirichlet problem for the Laplacian, which we will denote by λn. The question is: what can be inferred on D if one knows only the values of λn? Two domains are said to be isospectral (or homophonic) if they have the same eigenvalues. Another way to pose the question is: are there two distinct domains that are isospectral?
On the other hand, Zelditch proved that the answer to Kac' question is positive if one imposes restrictions to certain convex planar regions with analytic boundary. It is not known whether two non-convex analytic domains can have the same eigenvalues.
where d is the dimension. Weyl also conjectured that the next term in the approximation below would give the perimeter of D. In other words, if A denotes the length of the perimeter (or the surface area in higher dimension), then one should have
where c is some constant that depends only on the dimension. For smooth boundary, this was proved by V. Ja. Ivrii in 1980.
where D is the Hausdorff dimension of the boundary. This was disproved by J. Brossard and R. A. Carmona, who then suggested one should replace the Hausdorff dimension with the upper box dimension. In the plane, this was proved if the boundary has dimension 1 (1993), but mostly disproved for higher dimensions (1996). Both results are by Lapidus and Pomerance.