Tropical Mathematics and its Applications

To register, please send an e-mail to a.fink@qmul.ac.uk, f.rincon@qmul.ac.uk or b.smith9@lancaster.ac.uk.

If you are a UK-based PhD student requiring funding to support your travel, please let us know by 28th October.

Stable curves and chromatic polynomials (Rob Silversmith)

For any finite simple graph G, we introduce an associated natural class of integrals on moduli spaces of curves. We give a surprising formula for the integrals in terms of the chromatic polynomial of G, establishing a new connection between algebraic graph invariants and algebraic curves. I'll discuss the background/context, the formula, and two proofs, one via classical intersection theory and the other using the theory of hyperplane arrangements and algebraic statistics. I’ll also discuss some related speculations and open problems. Joint work with Bernhard Reinke.

Tropical enumerative geometry on ruled surfaces (Victoria Schleis)

Gromov-Witten invariants, providing counts of curves on varieties that satisfy point and tangency conditions, are important invariants in algebraic geometry that are hard to compute in classical enumerative geometry. In the last two decades, various methods in tropical geometry have been developed that break this geometric problem down into a combinatorial one.

We study the problem of counting curves on ruled surfaces. To this end, we develop tropical tools that allow us to count curves on non-orientable surfaces. For two of these surfaces, we establish a tropical curve count and prove its correspondence to the algebraic-geometric Gromov-Witten invariant. Further, we prove regularity results on generating series of the Gromov-Witten invariants — we show that they are quasi-polynomial and quasi-modular. This is joint work with Thomas Blomme.

An algebraic interpretation of Eulerian polynomials, derangement polynomials, and beyond, via Gröbner methods (Basile Coron)

I will explain how to use various Gröbner bases of Chow rings of permutohedral varieties to recover the well-known connection between Eulerian polynomials/derangement polynomials on the one hand and Chow polynomials of boolean matroids/corank 1 uniform matroid on the other hand. We will then see how this generalizes in any corank.