Imagine a bounded region in the Euclidean plane with fixed boundary, that is a drum. If we knew all the frequencies at which this drum vibrates, would we be able to tell its shape? Or are there two drums with different shapes that vibrate at the same frequencies?

The celebrated paper of M. Kac “Can One Hear the Shape of a Drum?” posed this question in 1966. In fact, problems related to this question had captured mathematicians’ attention since at least the work of Rayleigh in the nineteenth century and they continue to inspire the field of Spectral Geometry today.

In this talk we will explore some of the fascinating results about hearing - or not hearing? - the shape of a drum.