Tropicalization in Combinatorics

Tropical Geometry Seminar
Thursday, November 11, 2021 - 9:30am for 1 hour (actually 50 minutes)
Skiles 006
Greg Blekherman – Georgia Tech
Andreas Gross and Trevor Gunn

Tropicalization is usually applied to algebraic or semi-algebraic sets, but I would like to introduce a different category of sets with well-behaved tropicalization: sets with the Hadamard property, i.e. subsets of the positive orthant closed under coordinate-wise (Hadamard) multiplication. Tropicalization (in the sense of logarithmic limit sets) of a set S with the Hadamard property is a convex cone, whose defining inequalities correspond to pure binomial inequalities valid on S.

I will do several examples of sets S with the Hadamard property coming from combinatorics, such as counts of independent sets in matroids, counts of faces in simplical complexes, and counts of graph homomorphisms. In all of our examples we observe a fascinating polyherdrality phenomenon: even though the sets S we are dealing with are not semilagebraic (they are infinite subsets of the integer lattice) the tropicalization is a rational polyhedral cone. Also, the pure binomial inequalities valid on S are often combinatorially interesting.

Joint work with Annie Raymond, Rekha Thomas and Mohit Singh.