Abstract

In the early 90s, new ideas from string theory led to exciting developments in enumerative geometry. Kontsevich proved a recursive formula for N_d, the number of degree d genus 0 plane curves passing through 3d-1 points, using the notion of quantum cohomology, which comes from topological quantum field theory. In 2004, Mikhalkin proved a recursive formula for counting curves of arbitrary genus on toric surfaces, by showing that curves in the toric surface X can be identified with certain piecewise linear graphs in R^2, which we call tropical curves. In 2008, Gathmann and Markwig gave a purely combinatorial proof of Mikhalkin’s formula for genus 0 plane tropical curves, by showing that the space parametrising plane curves in the real plane also satisfies this nice recursive structure. In my talk I will present a sketch of Gathmann and Markwig’s proof, explaining what all the terms I used above mean.