I’ll provide an overview of (1) hyperplane arrangements (2) counting points on hyperplane arrangements (3) Rota’s conjecture and the idea of Huh’s proof of Rota’s conjecture (4) the connections to tropical geometry and (5) the recent idea of the “Chow ring of a matroid”. The point is to motivate you to be interested in these concepts, which we’ll be exploring throughout the term; many ideas will just be sketched.

As David Speyer discussed last week, hyperplane arrangements were a major motivation for the study of matroids. In this talk, we’ll take a more in-depth look at hyperplane arrangements with a focus on their characteristic polynomials. While initially defined in terms the point count over F_q, the characteristic polynomial can be computed entirely combinatorially from the intersection poset. Moreover, in certain situations, the characteristic polynomial has a physical interpretation; over R, it gives the number of connected components and, over C, it gives the Betti numbers. This talk will highlight these results and some of the techniques used to prove them.

In this talk, we will discuss the definition of matroids and their properties. As alluded to in previous talks, matroids generalize the notion of linear independence of subsets of some ambient set. We will discuss some examples of matroids and recipes to get them from more concrete objects. Furthermore, we will define terms such as basis, spanning sets, rank, and flats, as well as the associated lattice of flats of a matroid, generalizing ideas like the intersection poset from the theory of hyperplane arrangements.

In this talk, we will continue our introduction to matroids. We will discuss dual matroids, restriction and contraction, and realizability of matroids.

I will talk about the matroid polytope, their definition, characterization of matroids in terms of matroid polytopes, and more.

In this talk we first introduce another way of constructing new matroids from a given one M, by minimizing the linear functional \(w \cdot x\) over \(x\) in the matroid polytope of \(M\), for a fixed \(w \in \mathbb{R}^n\). We then use this to define the Bergman fan of a given matroid, which are spaces of weights \(w\) whose corresponding matroids do not contain loops. To understand the topology of the Bergman fan, we introduce a combinatorial gadget called phylogenetic trees which can be used to parametrize the Bergman fan, and give a nice subdivision of it.

I’ll start with Bergman’s 1971 definition of the logarithmic limit set of an ideal, and explain why we now call it the “tropicalization” of the ideal. I’ll then explain how, when we study tropicalizations of linear spaces, we get the Bergmann complex of a matroid, which Yucong spoke about last week. Finally, I’ll try to motivate the wonderful compactification of a hyperplane arrangement complement. Expect a fairly informal, loosely prepared, talk.

To prove log convexity of certain sequences of numbers, one wants to realize those numbers as intersection numbers of certain cohomology classes on some varieties. We will introduce one kind of such varieties called wonderful compactifications.

This construction compactifies the complement of hyperplane arrangements with nice boundary divisors. We will discuss two ways of constructing them, via ‘manually adding normal directions’ and via blow-up.

We construct divisors of the wonderful compactification of the complement of hyperplane arrangements and sketch the computation of the cohomology ring. This motivates the Chow ring of matroids, in which we introduce special degree one elements and study the multiplication behavior of flags of flats. We will also work out examples of Chow rings of matroids.

The main goal of this talk is to find the coefficients of the reduced characteristic polynomial of a matroid as the mixed intersection numbers of hyperplanes (\(\alpha\) classes) with reciprocal hyperplanes (\(\beta\) classes). I will begin by reviewing the Chow ring of wonderful compactifications and studying honest intersections. Then we will survey a few combinatorially arguments which show a similar formula holds for general matroids.

By what has been shown in previous talks, we have seen that we can show coefficients of the characteristic polynomial of a realizable matroid can be realized via specific computations in the Chow ring of its wonderful compactification. In this talk, we will introduce the notion of Poincare duality algebras, which are graded algebras with a degree function giving an isomorphism from the top degree to the base field that induces a non-degenerate pairing between complementary degrees of the algebra. Furthermore, we will introduce a notion of hard Lefschetz and Hodge-Riemann relations for such algebras. When a Poincare duality algebra satisfies a certain version of these properties, we can show that the log-concavity of its “volume polynomial” is equivalent to the eigenvalues of a symmetric form on the algebra arising from the Hodge-Riemann relations. Because the Hodge-Riemann relations in appropriate degree imply the log-concavity of the coefficients of the characteristic polynomial of the matroid, this framework gives us a program to establish the log-concavity result. Throughout this talk, I will attempt to provide intuition from the case of the Chow rings of smooth projective varieties.

We’ll work through examples of computing the lattice of flats, Mobius invariant, Poincare polynomial and Chow ring of a hyperplane arrangement. This will be an active learning class, so please come prepared to get up and work at a chalkboard!

In this talk we introduce the volume polynomial of a Poincare duality algebra and discuss what it has to do with volume. We will introduce mixed volumes of polytopes and the Alexandrov–Fenchel inequality and discuss some classical combinatorial applications of this inequality. These applications directly motivated much of the material that we’ve covered this semester.