Counting curves in threefolds

Abstract: A complex manifold of (complex) dimension n is a topological space which looks locally, around each point, like an open subset of \C^n, and on which it makes sense to talk about holomorphic (or analytic) functions. Setting n=1 we get complex curves (also called Riemann surfaces, as they have 2 real dimensions).

A complex curve in a complex manifold X can be described in two different ways: as a holomorphic map from the curve to X, or as the zero locus of a bunch of holomorphic functions on X. Or, in physics-speak, it can be described as the worldsheet of a string or as a D-brane.

These different points of view suggest different ways to "count" such complex curves in a fixed manifold X; the first leading to "Gromov-Witten invariants" and the second to the more recent invariants of Maulik-Nekrasov-Okounkov-Pandharipande (MNOP) when X has dimension 3 (the dimension relevant to string theory). The MNOP conjecture is an extraordinary and mysterious conjecture relating these two "counts".

I will try to explain all this, and then a third method of counting curves, also partly motivated by string theory. If time allows I will also explain how this sheds light on a fourth, conjectural count of curves, due to Gopakumar and Vafa, that would be the "best" solution to this problem, in some sense.

This is joint work with Rahul Pandharipande.

Refreshments will be served at 2:45 p.m. (Math Lounge, MATX 1115).