Alex Fink (Queen Mary University of London) - Speyer's g conjecture and Betti numbers for a pair of matroids

In 2008, looking to bound the face vectors of 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 being written up with Andy Berget (Western Washington) that proves the conjecture. The first half will be introductory, aiming to define the g-invariant, state the theorem, and compare to other recent work in matroid theory. The second half will look at the proof techniques.

Geometrically, the main ingredient in the proof is a Kempf collapsing constructed from the matroid tautological vector bundles of Berget--Eur--Spink--Tseng, and an initial degeneration thereof. 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 on tropical Chern classes. The work of Ardila and Boocher on the closure of a linear space in (P^1)^n is a special case.