By the famous Sylvester-Gallai theorem, it is known that every line arrangement in the real projective plane which is not a pencil has a double point. But it is also known that this does not hold true for those in the complex projective plane. However, not so many such arrangements (called the Sylvester-Gallai arrangements) are known. In this talk, we investigate this problem from the viewpoint of algebra of arrangements, mainly the freeness, minimal free distance and the cokernel of the Euler restriction maps.

When an arrangement of subvarieties locally looks like an arrangement of hyperplanes, its intersection pattern is encoded in a poset which locally looks like a geometric lattice, called a locally geometric poset. We will discuss how various notions related to geometric lattices, including matroids, supersolvability, and building sets, generalize to this setting.

The recent proof of the Dowling-Wilson top-heaviness conjecture for matroids relied on the properties of a module \(IH(M)\) associated to a matroid \(M\), particularly the fact that it satisfies the hard Lefschetz property. When \(M\) is a realizable matroid, \(IH(M)\) has a topological interpretation as the intersection cohomology of the arrangement Schubert variety associated to \(M\). For general matroids, it was defined as a direct summand of the much larger augmented Chow ring \(CH(M)\) using a complicated inductive process which is difficult to work with. I will present a simple module-theoretic characterization of \(IH(M)\), which also leads to a direct construction avoiding the augmented Chow ring. (Joint work with June Huh, Jacob Matherne, Nicholas Proudfoot and Botong Wang.)

In 2017 Moseley, Proudfoot, and Young conjectured that the reduced Orlik-Terao algebra of the braid matroid was isomorphic as a symmetric group representation to the cohomology of a certain configuration space. This was proved by Pagaria in 2022. We generalize Pagaria's result from the braid arrangement to arbitrary hyperplane arrangements and recover a new proof in the case of the braid arrangement. Along the way, we give formulas for several other invariants of a hyperplane arrangement with a group action, including the class of the matroid Schubert variety in the equivariant K-theory of a product of projective lines, and the Postnikov-Shapiro algebra. Work in progress, joint with Nick Proudfoot.

The hypersurface given by the Kirchhoff polynomial is a singular projective variety with relevance to physics, and a special case of a geometric construction from a matroid realization. I will describe a resolution of singularities for such hypersurfaces using a tropical compactification involving some conormal geometry of matroids. This is based on joint work with Dan Bath, Mathias Schulze, and Uli Walther.

This presentation aims to explore the relationship between hyperfields and matroids over them, the process of dequantization, and the theory of Lorentzian polynomials. In ongoing joint work with Matt Baker, Mario Kummer, and Oliver Lorscheid, we extend the connection between Lorentzian polynomials and valuated matroids to include matroids over triangular hyperfields, as introduced by Viro. This approach seeks to provide a deeper understanding of the space of Lorentzian polynomials.

Regular semisimple Hessenberg varieties have interesting birational morphisms among them which we can exploit to compute the Tymoczko representations on their cohomology. As a consequence, we obtain elementary proofs of the modular law and the Shareshian-Wach conjecture which relates the Frobenius characteristics with the chromatic quasi-symmetric functions of associated graphs. A similar story can be told for the twin manifolds of Hessenberg varieties where the Frobenius characteristics give us certain LLT polynomials. In this talk, I will discuss these birational geometric aspects of Hessenberg varieties and some applications. Based on a joint work with Donggun Lee.

The moduli space of pointed rational curves has a natural action of the symmetric group, which permutes the marked points, inducing a graded representation of the symmetric group on the cohomology of the moduli space.
In this talk, we provide a combinatorial formula for this representation, where we discover a recursive structure that is simple enough to be implemented in computer programming. This is achieved by employing two types of birational models of the moduli space: (1) moduli of delta-stable quasimaps, which are certain triples of nodal curves, line bundles and sections, and (2) Hassett's moduli of weighted stable curves.
Restricting to the trivial representation part, we derive a new and effective formula for the Poincaré polynomials of the quotient spaces of the moduli spaces by the action of the symmetric groups. We also show that these Poincaré polynomials satisfy asymptotic log-concavity, by establishing asymptotic formulas for the Betti numbers at given degrees. This is based on joint works with J. Choi and Y.-H. Kiem.

I will share with you a connection between multimatroids and moduli spaces of rational curves with cyclic action. Multimatroids are generalizations of matroids and delta-matroids that naturally arise from topological graph theory. The main result is a combinatorial formula for certain intersection numbers on the moduli space by relating to the volumes of independence polytopal complexes of multimatroids. Based on past and on-going joint works with Emily Clader, Chiara Damiolini, Chris Eur, Daoji Huang, and Rohini Ramadas.

We give formulas for \(L^2\) type invariants of hyperplane arrangement complements at degree one and study their connections with combinatorics. If time allows, some similar results for smooth complex quasi-projective varieties will be discussed.

In this talk, I will associate a "Chow polynomial" to any pair \((P,\kappa)\) consisting of \(P\) a finite, graded, bounded poset and \(\kappa\) a certain element of the incidence algebra \(I(P)\) called a \(P\)-kernel. For the pair where \(P\) is the lattice of flats of a matroid and \(\kappa\) is the characteristic polynomial, we recover the Poincaré polynomial of Feichtner-Yuzvinsky's Chow ring of a matroid (the so-called Chow polynomial of a matroid).
I will discuss general properties of these Chow polynomials as well as connections to other settings such as polytopes and Coxeter groups. In addition, I will explain some relations between these Chow polynomials and the Kazhdan-Lusztig-Stanley polynomials that can also be constructed from such a pair \((P,\kappa)\). This is ongoing joint work with Luis Ferroni and Lorenzo Vecchi.

Classical Ehrhart theory for a lattice polytope encodes the relation between the volume of the polytope and the number of lattice points the polytope contains. In this talk, I will discuss a geometric interpretation, via the (equivariant) Hirzebruch-Riemann-Roch formalism, of a generalized weighted Ehrhart theory depending on a homogeneous function on the polytope and with Laurent polynomial weights attached to each of its faces. In the special case when the weights correspond to Stanley's g-function of the polar polytope, we recover in geometric terms a recent combinatorial formula of Beck-Gunnells-Materov. (Based on archive preprints arXiv:2403.17747 and arXiv:2405.02900, joint work with Jörg Schürmann.)

We introduce a stratified singular variety whose geometry captures the polymatroid associated to an arrangement of linear subspaces. This generalizes the "matroid Schubert varieties" associated to hyperplane arrangements. (Joint with C. Crowley and B. Wang)

We discuss the submodularity of numerical dimensions, which induces a polymatroid structure on nef classes on a projective variety. We also explain how the submodularity of the rank function in a polymatroid can be a posteriori put in this geometric framework. The key ingredient is a kind of intersection-theoretic inequalities via Lorentzian polynomials. (Based on joint work with J. Hu)

Elias, Proudfoot and Wakefield conjectured that the Kazhdan-Lusztig polynomial of every matriod is log-concave. This interesting conjecture remains widely open. In this talk I will show how to prove this conjecture for uniform matroids and q-niform matroids. This is based on my joint works with Alice Gao, Ethan Li, Matthew Xie, Philip Zhang, and Zhong-Xue Zhang.

This talk is based on the preprint: arXiv:2405.20010. Since the discovery of the \(K(\pi, 1)\) property for braid arrangements, the homotopy group has called attention of lots of research in the topology of hyperplane arrangements. One of the reasons for the difficulty of studying homotopy groups is that the Hurewicz map always vanishes no matter how the homotopy group is non-vanishing (Randell). In this talk, we will discuss how to detect non-trivial elements of the homotopy group using the twisted intersection numbers.

A transversal of a uniform hypergraph \(H\) is a subset of the vertex set that intersects all edges of \(H\). The transversal number \(T(H)\) of \(H\) is the minimum cardinality of a transversal of \(H\). Transversal numbers of interesting families of hypergraphs have been studied extensively. For example, a classical result of Alon shows that if \(H\) ranges over all k-uniform hypergraphs, then \(\sup T(H)/(v(H)+e(H)) = (1+o(1))\frac{\ln k}{k}\), where \(v(H)\) denotes the number of vertices and \(e(H)\) the number of edges.
In this talk, I will discuss the transversal numbers of hypergraphs that arise from pure simplicial complexes, simplicial polytopes, and simplicial spheres. In particular, I will discuss some new upper and lower bounds, including a few constructions of complexes with relatively large transversal numbers. Many of these constructions are closely related to cyclic polytopes. Joint work with Isabella Novik.