In 1992 Gordon, Webb and Wolpert showed that you can't hear the shape of a drum. They constructed two different planar domains (the drums) that vibrate at exactly the same infinite list of fundamental frequencies. Interestingly although this list of frequencies doesn't determine the shape of a drum, it does determine the drum's perimeter and surface area.
An active area of research in differential geometry, called inverse spectral geometry, focuses on whether or not you can hear the shape of more general objects. I will discuss what the vibrational frequencies of a Riemannian orbifold tells us about the orbifold. A Riemannian orbifold is a mildly singular generalization of a Riemannian manifold. In the first part of this talk we'll explore what orbifolds are, and in the second we'll discuss the inverse spectral geometry of these objects.