Title: Mikhalkin’s curve-counting formula for P^2
Speaker: Hannah Tillmann-Morris
Abstract: In the early 90s, new ideas from string theory led to exciting developments in enumerative geometry. In particular, Kontsevich proved a recursive formula for the number N_d of degree d genus 0 curves in P^2 passing through 3d-1 points in general position, using the notion of quantum cohomology from topological quantum field theory.
The formula is a corollary of the associativity of the quantum product, which was proved by showing that a generating function for the Gromov-Witten invariants of P^2 satisfies a set of PDEs called the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations.
In 2004, Mikhalkin proved a recursive formula for counting curves of arbitrary genus on toric surfaces by showing that, given an SYZ fibration X->B, curves in the toric surface X can be identified with certain piecewise linear graphs in the base B, which we call tropical curves. In 2008, Gathmann and Markwig gave a purely combinatorial proof of Mikhalkin’s formula for genus 0 curves in P^2, by showing that the counts of tropical curves satisfy the WDVV equations.
In my talk I will explain what all the terms used above mean, and present a sketch of Gathmann and Markwig’s proof. If I have time, I will discuss how log geometry gives us a way of interpreting these tropical curves as curves in P^2.
Some snacks will be provided before and after the talk.