Combinatorics & Geometry BLT seminar

Chow rings of toric varieties, which originate in intersection theory, feature a rich combinatorial structure of independent interest. We survey four different ways of computing in these rings, due to Billera, Brion, Fulton–Sturmfels, and Allermann–Rau. We illustrate the beauty and power of these methods by sketching four proofs of Huh and Huh–Katz’s formula µ^k (M) = deg(α^{r−k}β^k) for the coefficients of the reduced characteristic polynomial of a matroid M as the mixed intersection numbers of the hyperplane and reciprocal hyperplane classes α and β in the Chow ring of M. Each of these proofs sheds light on a different aspect of matroid combinatorics, and provides a framework for further developments in the intersection theory of matroids. 

Our presentation is combinatorial, and does not assume previous knowledge of toric varieties, Chow rings, or intersection theory.

The tropicalization of a subscheme of P^n is given by a homogeneous ideal in the semiring of tropical polynomials that satisfies some matroidal conditions.  This can be thought of as a "tower of valuated matroids".  In this talk I will highlight what we currently know about the connection between these matroids and the geometry of the subscheme, including recent progress on the Nullstellensatz with Felipe Rincon, and some connections still to be understood.

Valuations of matroids are very useful and very mysterious.  After taking some time to explain this concept, I will categorify it, with the aim of making it both more useful and less mysterious.

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

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Tropicalization is a way to understand the asymptotic behavior of algebraic (or semi-algebraic) sets through polyhedral geometry. In this talk, I will describe the tropicalization of the principal minors of real symmetric and Hermitian matrices. This gives a combinatorial way of understanding their asymptotic behavior and discovering new inequalities on these minors. For positive semidefinite matrices, the resulting tropicalization will have nice combinatorial structure called M-concavity and be closely related to the tropical Grassmannian and tropical flag variety. For general Hermitian matrices, this story extends to valuated delta matroids. 

This is based on joint works with Abeer Al Ahmadieh, Nathan Cheung, Tracy Chin, Gaku Liu, Felipe Rincón, and Josephine Yu.