Posters

Keita Aso     (Kyushu University)

SPOG Graphs and Strictly Plus-One Generated Arrangements

Some properties of graphs correspond to those of hyperplane arrangements. For example, chordal graphs correspond to free arrangements. However, it is still unknown which graphs correspond to strictly plus-one generated arrangements, a notion introduced by Abe (2021). In this poster, we introduce SPOG graphs and show one implication toward a characterization of strictly plus-one generated arrangements. The converse implication is currently under revision.

Kashu Ito     (Ritsumeikan University)

Subgroups of the Projective Linear Group Realized by wild Galois Points

We work over fields of positive characteristic and focus on the study of Galois points on hypersurfaces with wild action. In this poster, we examine how the associated Galois groups admit linear representations and show that they can be lifted to the general linear group. We further analyze their actions on vector spaces and give explicit necessary and sufficient conditions for subgroups of the projective linear group to be realized as Galois groups of wild Galois points. Our results provide new insights into the structure of Galois groups in positive characteristic and clarify the role of wild action in algebraic geometry.

Leo Jiang     (University of Toronto)

Topology of real matroid Schubert varieties

Matroid Schubert varieties are closures of linear spaces in products of projective lines. When the linear space is over the real numbers, we show that the topology of the variety is controlled by the combinatorics of real hyperplane arrangements. More precisely, we exhibit homeomorphisms from real matroid Schubert varieties to quotients of zonotopes. Further, this combinatorial model can be generalised to define a topological space for any oriented matroid. As a consequence, we are able to compute the fundamental group and integral cohomology of these spaces, obtaining virtual Coxeter groups and signed analogues of the graded Möbius algebra respectively. This is joint work with Yu Li.

Yoh Kitajima     (Kyoto University)

Filtered order complexes and magnitude homology of pure shellable posets

Magnitude homology was first introduced for undirected graphs by Hepworth and Willerton, later Leinster and Shulman extended to generalized metric spaces. Asao then treated the magnitude homology of ranked posets as a special case within this framework. In this poster, we introduce a new family of subcomplexes of the order complex of a ranked poset, defined via its rank function, in connection with the magnitude homology of ranked posets. This construction may be viewed as a ranked-poset analogue of the Asao–Izumihara complex developed for the magnitude homology of undirected graphs. We show that when the underlying finite poset is pure and shellable, each of these subcomplexes are shellable as well. Moreover, for a geometric semilattice, each subcomplex of the order complex of its proper part is homotopy equivalent to a nontrivial wedge sum of spheres.  

Masato Konoike     (The University of Osaka)

Polytopes Associated with Hyperplane Arrangements via Chambers

In this poster, we introduce a new class of polytopes arising from the chamber structure of hyperplane arrangements. We first present the basic properties of these polytopes and then discuss their reflexivity and the freeness of the corresponding hyperplane arrangements.

Wangyang Lin     (Fudan University)

A weighted version of Miyaoka-Yau inequality for matroids

We proved the Bogomolov--Gieseker inequality for Calabi--Yau-weighted matroids whose 3-element sets are all of rank 3. With the same method, we proved the weighted Bogomolov--Miyaoka--Yau inequality for the same range of matroids with any weight. A key step to these inequalities is a miracle result: the quadratic form involved in these inequalities either has Lorentzian signature (i.e. is $\mathcal{K}$-Lorentzian)  or is negative semi-definite. With this surprising result, we offer a sufficient and necessary condition for this inequality.

Koki Maeda     (Kyushu University)

MAT-freeness of ψ-graphical arrangements in complete graphs

In the mathematics of hyperplane arrangements, there is a class of arrangements, ψ-graphical arrangements, which are close to graphical arrangements. As a classical result, there exists a characterization in graph terms of the freeness of graphical arrangements. For MAT-freeness, it was characterized in graph terms in 2023. On the other hand, for ψ-graphical arrangements, the freeness was characterized in graph terms in 2019. But, let alone the MAT-freeness characterized in graph terms, the example of ψ-graphical arrangements that is free but not MAT-free was undiscovered. This poster presentation shows the equivalence of ψ-graphical arrangements and MAT-freeness in complete graphs, and gives the example of ψ-graphical arrangements that is free but not MAT-free.

Shota Maehara     (Kyushu University)

Free multiarrangements without any free extensions

The term extendability was introduced by Masahiko Yoshinaga in his paper, which reflects the idea whether a free multiarrangement can be obtained as a natural restriction of some free simple arrangements of one more higher dimension. The concept of extendability should be important for a very famous conjecture by Hiroaki Terao, based on the theory of multiarrangements started by Günter M. Ziegler, but it seems that there are still few results known about extendability except for the first work by Yoshinaga. In this poster, we present an infinite family of non-extendable multiplicities on the Coxeter arrangement of type B_2, which provides the first example of free multiplicities on a Coxeter arrangement that do not admit any free extensions. In addition, we can also obtain some free Coxeter multiarrangements of type B_3 that do not have any free extensions as an easy corollary. 

This is a joint work with Torsten Hoge and Sven Wiesner. 

Hung Manh Tran     (Phenikaa University)

Signatures of Type A Root Systems and Eulerian numbers

Given a type $A$ root system $\Phi$ of rank $n$, we introduce the notion of a signature for each subset $S \subseteq \Phi$ consisting of $n+1$ positive roots. If $S$ is represented by a tuple $(\beta_1, \ldots, \beta_{n+1})$, its signature is defined as the unordered pair ${a, b}$, where $a$ and $b$ are the numbers of $1$'s and $-1$'s, respectively, among the cofactors $(-1)^k \det(S \setminus {\beta_k})$ for $1 \le k \le n+1$. We show that the number of such tuples with a fixed signature can be expressed in terms of Eulerian numbers. The motivation for studying signatures arises from their connections with the arithmetic and combinatorial structure of cones over deformations of $\Phi$, such as the Shi, Catalan, Linial, and Ish arrangements. As an application, we use our main result to compute two invariants of these arrangements: the minimum period of the characteristic quasi-polynomial and the evaluation of both the classical and arithmetic Tutte polynomials at $(1,1)$.

Ryo Uchiumi     (The University of Osaka)

Equivariant version of characteristic quasi-polynomials

In this poster, we introduce an equivariant version of the characteristic quasi-polynomials as the permutation characters on the complement of mod q hyperplane arrangements.  We show that its character is a quasi-polynomial in q and can be expressed as a sum of the induced characters of an equivariant version of the Ehrhart quasi-polynomials.  In addition, we present the results for the Coxeter arranegements with Weyl group actions.

Prajwal Dhondiram Udanshive     (University of Western Ontario)

Resonance Varieties of Schubert Matroids

Resonance varieties of Orlik--Solomon algebras are cohomologically-defined matroid invariants motivated by geometry. While the structure of degree-1 resonance has been extensively studied, understanding the resonance in higher degrees is elusive. In this poster, we observe that for Schubert matroids, the cohomology of its Aomoto complex admits a decomposition indexed by cyclic flats. This decomposition results in a complete characterization of resonance jump loci in all degrees using lattice path combinatorics. This presentation is based on an ongoing joint work with Graham Denham.

Yukino Yagi     (The University of Osaka)

Reconstruction of oriented matroids from Varchenko--Gelfand algebras

The algebra of R-valued functions on the set of chambers of a real hyperplane arrangement is called the Varchenko--Gelfand (VG) algebra. This algebra carries a natural filtration by the degree with respect to Heaviside functions, giving rise to the associated graded VG algebra.  When the coefficient ring R is an integral domain of characteristic 2, the graded VG algebra is known to be isomorphic to the Orlik--Solomon algebra. We have been studying  VG algebras over coefficient rings of characteristic different from 2, and investigating  to what extent VG algebras determine the underlying oriented matroid structures. We prove that the oriented matroid can be recovered from the filtered and graded VG algebras when the arrangement is generic in codimension 2. This result is presented in this poster. This is joint work with Masahiko Yoshinaga.