A well-known problem in mathematics and physics consists in
understanding how the geometry (or shape) of a musical instrument
affects it sound. This gives rise to two related types of
mathematical problems: direct spectral problems (how the shape of a
drum affects its sound) and inverse spectral problems (how one can
recover the shape of a drum from its sound). Here, we will consider
both types of problems in the context of drums with fractal (that is,
very rough) boundary. We will show, in particular, that one can “hear”
the fractal dimension of the boundary (a certain measure of its
roughness) and in certain cases, a fractal analog of its length. In
the special case of vibrating fractal strings (the one-dimensional
situation), we will show that the corresponding inverse spectral
problem is intimately connected with the Riemann Hypothesis, which is
arguably the most famous open problem in mathematics and whose
solution will likely unlock deep secrets about the prime numbers. In
conclusion, we will briefly explain how this work eventually gave rise
to a mathematical theory of complex fractal dimensions, which captures
the vibrations (or oscillations) that are intrinsic to both fractal geometries and the prime numbers.
