Math 426: Enumerative Geometry Beyond Numbers

The instructor for the course is me, Jim Bryan

This course is an introduction to modern enumerative algebraic geometry. It is largely concerned with quantum invariants coming from "curve counting" in various guises. "Quantum invariants" in this context is a catch-all phrase referring to deformation invariants constructed in algebraic geometry which are mathematical analogs of quantities arising in string theory. The invariants we will consider are Gromov- Witten invariants (including quantum cohommology), Donaldson-Thomas invariants, Pandharipande-Thomas invariants, and Gopakumar-Vafa invariants. They can all be regarded as theories which provide virtual counts of curves on a Calabi-Yau threefold. We will study the structure which underlies these invariants and the various relationships (many of which are conjectural) between the invariants as well as techniques for computing these invariants.

The class meets in MATH 102 on MWF from 1:00pm to 2:00pm

I will post my lecture notes here. I will update this as the term progresses. Lectures 1-6, Lectures 7-14 , Lectures 15-19 , Lectures 20-32 , Lectures 33-34 , Lectures 35-40 ,

A good overview is the paper 13/2 ways to count curves by Pandharipande and Thomas. The references within this paper provide an excellent guide to the literature.