Counting planar curves using tropical geometry

Add to your list(s) Download to your calendar using vCal

• Sae Koyama, University of Cambridge
• Friday 09 May 2025, 16:00-17:00
• MR13.

If you have a question about this talk, please contact Adrian Dawid.

Tropical curves are balanced piecewise linear functions from graphs into R^n, with d “infinite legs” going in directions e_1, e_2, and -e_1-e_2. The number d is the degree of the tropical curve, while the first Betti number of the graph is the genus. We can ask enumerative questions: how many tropical curves of genus g and degree d pass through 3d+g-1 points?

It turns out that, in the case of genus 0, this number is precisely the same as the number of algebraic curves of degree d and genus g passing through 3d-1 points. This so-called Mikhalkin’s tropical correspondence has far-reaching consequences. If we want to count algebraic curves, we “simply” have to count tropical curves. I will introduce the notion of tropicalization, sketch the correspondence theorem, and suggest how these results may be generalised.

This talk is part of the Junior Geometry Seminar series.

This talk is included in these lists:

• All CMS events
• CMS Events
• DPMMS info aggregator
• Hanchen DaDaDash
• Interested Talks
• Junior Geometry Seminar
• MR13
• bld31

Note that ex-directory lists are not shown.