It is a classical result that the spectrum of the Laplacian with Dirichlet boundary condition on a compact Riemannian manifold forms a sequence going to positive infinity, and the asymptotic growth rate is determined by the volume and dimension of the manifold. This asyptotic growth rate is known as Weyl's Law. Historically, this motivated Kac to ask the question "Can one hear the shape of a drum?" In general, we can ask what geometric properties of a manifold are determined by the spectrum of its Laplacian.

I plan to give an introduction to Weyl's law for compact manifolds and then show how it also holds for singular, projective, complex algebraic varieties. Speaker(s): John Kilgore (UM)