Salomon Bochner Lectures in Mathematics, 2007-2008

In this series of lectures, we discuss two theories concerning families of K3 surfaces (a special class of algebraic surface with trivial canonical bundle). Gromov-Witten theory involves counting pseudoholomorphic curves on a symplectic manifold and is closely related to ideas from mirror symmetry and hypergeometric series. Noether-Lefschetz theory, on the other hand, arises from classical geometric questions of Hodge theory but relates to modern work of Borcherds on automorphic products. In these talks, I will introduce these circles of ideas and explain the precise quantitative relationship between them along with applications to each side.