In 1966, Mark Kac famously asked: “Can one hear the shape of a drum?” — that is, can the geometry of a domain $\Omega \in \RR^d$ be determined from the spectrum of the Dirichlet Laplacian on $\Omega$?
The answer, known since 1992, is no: there exist non-congruent domains that are isospectral, sharing the same set of eigenvalues.
Nevertheless, the spectrum does encode significant geometric information.
In this talk, we will explore the deep connections between geometry and spectral theory, discussing what can (and cannot) be “heard” from the eigenvalues of a drum.