Lorenzo Vecchi: Categorical valuative invariants for polyhedra

Abstract

In this talk we study polyhedra and their valuative invariants, i.e. functions that are well behaved under inclusion-exclusion relations on polyhedral decompositions. By replacing the alternating sum in the definition with complexes, we are able to define a new category of polyhedra and a notion of valuativity for functors, where now we work with (graded) vector space invariants that form split exact sequences.

As an application, we restrict to the subclass of matroid polytopes and prove that several cohomology rings of matroids are valuative. This can be seen as a process of categorification: we knew from literature that some polynomial invariants with non-negative integer coefficients were a valuative function, and we interpret them as the Hilbert series of some graded vector spaces, which turn out to be valuative in the new sense.

This not only produces a new general theoretical setting, but it also can be employed as a very concrete hands-on tool to perform computations.

This is based on a joint project with Ben Elias, Dane Miyata and Nicholas Proudfoot.