In this introductory talk, I will present some concepts surrounding singularities and why and how we look for their resolutions.
I will talk about joint work with Shing-Tung Yau, Sergei Gukov, and earlier joint work with Gholampour and Yau on Mathematical definition of Vafa-Witten invariants on 4 and 3 dimensional manifolds.
This is the first part, consisting of four lectures, of a mini-course in which the combinatorial geometry related to toric varieties will be introduced. It will be developed to define and express properties of toric varieties and toric morphisms, and to investigate the geometry of the orbits by the torus action, in particular the orbit decomposition. Next, toric divisors, invertible and reflexive sheaves on toric variety, and their groups will be introduced and studied.
In a joint work with Frank Calegari and Yunqing Tang, we use methods from transcendental number theory to prove a conjecture that goes back to Atkin and Swinnerton-Dyer, in a special case, and generalized by Mason to the following form: A vector-valued modular form on SL(2,Z) whose components have q-expansions with bounded denominators are exactly the ones for which the underlying representation of SL(2,Z) has a finite image with kernel containing the congruence subgroup of matrices reducing to the identity modulo some positive integer N. In this talk, I will outline the basic ideas of the proof of the conjecture, describe the relation to mathematical physics and the representation theory of vertex algebras, and explain how our result in particular recovers a completely new proof of the so-called "congruence property" in rational conformal field theory.