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. 

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In 1999, Pitman and Stanley introduced the polytope bearing their name along with a study of its faces, lattice points, and volume. This polytope is well-studied due to its connections to parking functions, lattice path matroids, generalized permutahedra/polymatroids, and flow polytopes. Its lattice points correspond to plane partitions of skew shape with entries 0 and 1. Pitman and Stanley remarked that their polytope can be generalized so that lattice points correspond to plane partitions of skew shape with entries 0,1,...,m. Since then, this generalization has been untouched. We study this generalization and show that it can also be realized as a flow polytope of a grid graph. In this talk I will discuss characterizations of its vertices and give formulas for the number of vertices and faces as well as new and old formulas for the number of lattice points and volume.

This is joint work with Maura Hegarty, William Dugan, and Annie Raymond.

In a landmark paper in 1992, Stanley developed the foundations of what is now known as the Kazhdan--Lusztig--Stanley (KLS) theory. To each kernel in a graded poset, he associates special functions called KLS polynomials. This unifies and puts a common ground for i) the Kazhdan--Lusztig polynomial of a Bruhat interval in a Coxeter group, ii) the toric g-polynomial of a polytope, iii) the Kazhdan-Lusztig polynomial of a matroid. In this talk I will introduce a new family of functions, called Chow functions, that encode various deep cohomological aspects of the combinatorial objects named before. In the three settings mentioned before, the Chow function describes i) a descent-like statistic enumerator for paths in the Bruhat graph, ii) the enumeration of chains of faces of the polytope, iii) the Hilbert series of the matroid Chow ring. This is joint work with Jacob P. Matherne and Lorenzo Vecchi.

From a triangulation of a sphere, one can construct a graded ring, the generic artinian reduction of the Stanley-Reisner ring, which satisfies an analogue of Poincare duality. It is equipped with several nondegenerate forms, called Hodge-Riemann forms. This ring is defined over a field of Laurent polynomials, and these bilinear forms have remarkable properties: for example, Papadakis and Petrotou showed that they are anisotropic, i.e., the associated quadratic form has no nontrivial zeros. We compute the determinants of these bilinear forms and show that they contain enough information to recover the triangulation of the sphere. Joint with Isabella Novik and Alan Stapledon.

Much of the important combinatorial information about an arrangement of hyperplanes can be understood by studying the cohomology ring of its complement. Given an arrangement with a group action G, a fundamental but difficult question is: how does one describe the G-representations on the (graded pieces of the) cohomology ring of its complement? In this talk, I will discuss some methods of answering this question for some of my favorite hyperplane arrangements, as well as challenges that arise. This is partially based on joint work with Megan Chang-Lee and Trevor Karn.

When does a system of coupled oscillators synchronize? This central question in dynamical systems arises in applications ranging from power grids to neuroscience to biology: why do fireflies sometimes begin flashing in harmony? Perhaps the most studied model is due to Kuramoto (1975); we  analyze the Kuramoto model from the perspectives of algebra and topology. Translating dynamics into a system of algebraic equations enables us to identify classes of network topologies that exhibit unexpected behaviors. Many previous studies focus on synchronization of networks having high connectivity, or of a specific type (e.g. circulant networks); our work also tackles more general situations.

We introduce the Kuramoto ideal; an algebraic analysis of this ideal allows us to identify features beyond synchronization, such as positive dimensional components in the set of potential solutions (e.g. curves instead of points). We prove sufficient conditions on the network structure for such solutions to exist. The points lying on a positive dimensional component of the solution set can never correspond to a linearly stable state. We apply this framework to give a complete analysis of linear stability for all networks on at most eight vertices. The talk will include a surprising (at least to us!) connection to Segre varieties, and close with examples of computations using the Macaulay2 software package "Oscillator"

Joint work with Heather Harrington (Oxford/Dresden) and Mike Stillman (Cornell).

For a hyperplane arrangement in a real vector space, the coefficients of its Poincaré polynomial have many interpretations. An interesting one is provided by the Varchenko-Gel'fand ring, which is the ring of functions from the chambers of the arrangement to the integers with pointwise addition and multiplication. We will introduce the Varchenko-Gel’fand ring and use it to study hyperplane arrangements and cones defined by intersections of halfspaces defined by some of the hyperplanes. Then, we will highlight a recent application Coxeter groups.