Can you hear the shape of a drum? This question was asked some 60 years ago by mathematician Mark Kac, trying to reconstruct the shape of a vibrating membrane (the drum) from its (audible) vibrational spectrum. In the first part of the lecture we will study this spectral approach to geometry, while illustrating it using drums of different shapes.

We will then turn to the applications of spectral geometry in physics and astrophysics. We come to the conclusion that actually all our information about, say, the universe comes to us through spectra: instead of sound waves, the observed spectrum now ranges from radio waves, to electromagnetic waves, to gravitational waves. This means that the mathematical question of whether you can reconstruct the shape (of, say, the universe) from a spectrum is directly applicable in physics and astronomy.

We end with a brief impression of current research, which deals with the question of how geometry is an emergent phenomenon, when an increasing part of the vibrational spectrum becomes available.