Speyer's g conjecture and Betti numbers for a pair of matroids

Abstract: In 2009, looking to bound the face vectors of matroid subdivisions and tropical linear spaces, Speyer introduced the g-invariant of a matroid. He proved its coefficients nonnegative for matroids representable in characteristic zero and conjectured this in general. Later, Shaw and Speyer and I reduced the question to positivity of the top coefficient. This talk will overview work in progress with Berget that proves the conjecture.

Geometrically, the main ingredient is a variety obtained from projection away from the base of the matroid tautological vector bundles of Berget--Eur--Spink--Tseng, and its initial degenerations. Combinatorially, it is an extension of the definition of external activity to a pair of matroids and a way to compute it using the fan displacement rule. The work of Ardila and Boocher on the closure of a linear space in (P^1)^n is a special case.

algebraic geometrycombinatorics

Audience: researchers in the topic

Combinatorics and Geometry BLT Seminar