# Some connections between algebraic geometry and mathematical physics

Hilbert scheme of points on a surface is a classical object of interest in algebraic geometry. In the early 90s, Goettsche found a remarkable generating function computing the Poincare series of the Hilbert scheme. Motivated by considerations from string theory, Vafa and Witten pointed out that this q-series is the character of the Fock space representation of a Heisenberg vertex algebra. Subsequent developments of this story by Nakajima, Grojnowski, and others, has given rise to a vibrant area of geometric representation theory. On the other hand, in the 1990s, Beilinson and Drinfeld discovered “factorization algebras” as the algebra-geometric incarnation of vertex algebras. Factorization algebras have become instrumental in recent developments of geometry and foundations of quantum field theory. After reviewing these developments, I will report on an ongoing project with Emily Cliff (Oxford) whose goal is to relate Hilbert schemes with factorization algebras.