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 projection away from the base of the matroid tautological vector bundles of Berget--Eur--Spink--Tseng, and initial degenerations of these. 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.