The famous question of M. Kac, "Can one hear the shape of a drum?", asks to determine the shape of a drumhead by hearing its eigenfrequencies. Similar questions that involve eigenfrequencies of more general objects (for instance Riemannian manifolds) appear in various places in mathematics and applications. The study of these questions involves an interesting mix of analysis, geometry, dynamical systems, algebra and number theory.
In this talk I will give an overview of the topic and discuss recent joint results with G. Paternain (Cambridge) and G. Uhlmann (Washington/Helsinki).