Products of Chern Classes of Matroid Tautological Bundles - Alex Fink

Special Year Seminar I
2:00pm|Simonyi 101
Topic: Products of Chern Classes of Matroid Tautological Bundles
Speaker: Alex Fink
Affiliation: Institute for Advanced Study
Date: January 29, 2025
In 2008, looking to bound the face vectors of tropical linear spaces, Speyer introduced the g-invariant of a matroid, defined in terms of exterior powers of tautological bundles on Grassmannians. He proved its coefficients nonnegative for matroids representable in characteristic zero and conjectured this in general. In a pair of recent papers, with Kris Shaw and David Speyer we reduce the question to positivity of the top coefficient, and with Andy Berget we then prove the conjecture for this coefficient.
This pair of talks will aim at presenting, with context, the central combinatorial computation of my proof with Berget: a computation by the fan displacement rule of the product of the total Chern classes of two of the tautological quotient bundles of Berget--Eur--Spink--Tseng. Strata in these intersections are governed by the external activity of a pair of matroids. This provides the relationship between the definition of the g-invariant and the external activity complex that we rely on for handling the nonrepresentable case.