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Mikhalkin’s curve-counting formula for P^2

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If you have a question about this talk, please contact Alexis Marchand.

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.

This talk is part of the Junior Geometry Seminar series.

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