|Date: Tuesday, November 07, 2017
Location: 1866 East Hall (3:00 PM to 4:00 PM)
Title: Weyl's law for singular projective algebraic varieties
Abstract: It is a classical result that the spectrum of the Laplacian on a compact Riemannian manifold forms a sequence going to positive infinity and satisfies an asymptotic growth rate known as Weyl's law determined by the volume and dimension of the manifold. The spectrum of the Laplacian is an isometry invariant, and Weyl's law motivated Kac's famous question, "Can one hear the shape of a drum?" where he asked if spaces with the same spectrum are isometric. (Milnor showed the answer is no in general.) In this talk, I will explain why Weyl's law also holds for the non-singular locus of embedded, irreducible, singular projective algebraic varieties with the metric induced from the Fubini-Study metric of complex projective space. This non-singular locus is an open manifold with finite volume that comes from a very natural class of spaces studied heavily outside of differential geometry. I will discuss the difficulties that occur when considering Weyl's law for open manifolds and how they can be overcome for the case above using a heat kernel estimate of Li and Tian.
Speaker: John Kilgore
Institution: University of Michigan