Hearing the Shape of a Drum: Weyl's Law and Spectral Geometry -
In 1966, Kac popularized the problem of hearing the shape of a drum: given the natural frequencies at which a drumhead vibrates, can one determine the geometry of the drumhead? A partial answer is provided by Weyl's law (1911-1915), which relates the asymptotic growth of eigenvalues of the Laplacian (i.e., `natural frequencies') on a bounded domain to the area of the domain. We will sketch an elementary proof of Weyl's result in a simple setting, and explore generalizations, advanced techniques, and areas of modern research in spectral geometry.