Speaker
Matthieu Piquerez
Abstract
On the introduction to the present symposium, one can read
“But the [Heron-Rota-Welsh] conjecture was for an arbitrary matroid, which might not be associated to any type of geometry at all! The proof by Adiprasito–Huh–Katz builds an object from combinatorics, which ought
to play the role of the cohomology ring, and proves Poincaré duality, Hard Lefschetz and the Hodge–Riemann bilinear relations for this object directly.”
In this talk, I will show that, on the contrary, these results has a geometric interpretation for any matroid… in the tropical world. Indeed, one can show that the *tropical* cohomology of the canonical compactification of so-called tropically shellable quasi-projective fans verifies the three above properties. In particular, Bergman fans of
matroids belong to those fans, hence we get a generalization of the result of Adiprasito-Huh-Katz. This is a joint work with Omid Amini.