We prove that, for n ≥ 5, an interval group associated with a proper quasi-Coxeter element of the Coxeter group of type Dn admits no surjective homomorphism onto the braid group on n strands. In particular, this provides an alternative proof that such a group is not isomorphic to the Artin group of type Dn . The proof relies on techniques from the theory of mapping class groups, and a large part of the paper provides an exposition of this theory for non-specialists.

In this paper we prove a free-join decomposition theorem on Gorenstein polytopes significantly improving upon results in prior joint work of the second author with Batyrev, Borger, Haase, Kretschmer, and Payne. In particular, as a strong extension of the Batyrev-Juny lattice pyramid theorem it implies that any d-dimensional Gorenstein polytope P of index larger than (d+2)/2 is a free join of Gorenstein polytopes. Here, the index (also called codegree) is the dilation factor r such that rP is reflexive. As our main application, we prove that the stringy E-polynomial of a Gorenstein polytope P vanishes if and only if P is thin (i.e., its local h^*-polynomial vanishes), and it has the expected degree otherwise. The latter was conjectured by Batyrev and the second author. The proofs of the general results were found via ChatGPT 5.6 Sol, while a proof of the extremal case of the Batyrev-Juny conjecture is contained in the master thesis of the first author.

Lattice polytopes are called very ample if for every sufficiently large k every lattice point of height k in the cone over the lattice polytope is the sum of k lattice points of height 1. This is a weakening of the well-known integer decomposition property (also called IDP). We give an example of a very ample lattice polytope whose h^-vector is non-unimodal. Here, the h^-vector is the coefficient vector of the numerator of the Ehrhart series of the lattice polytope. This answers a question of Ferroni and Higashitani, as well as a related question by Balletti. The main question whether IDP lattice polytopes have unimodal h^*-vector is still open. The example was found using ChatGPT 5.6 Sol. It is just the Cartesian square of a lattice polytope belonging to a class of very ample examples constructed by Lasoń and Michalek.

In this note we prove that in fixed dimension the Ehrhart h^-polynomial of a lattice polytope of sufficiently large lattice width is real-rooted. In particular, this implies strict log-concavity and unimodality of the h^-vector and answers a question of Averkov, Hofscheier and the author. For a lattice simplex we prove the analogous statement for its local h^*-polynomial, also called box polynomial. The proofs were found using ChatGPT 5.6 Sol and follow essentially directly from a result by Basu and Oertel that for large enough lattice width counting lattice points approximates the volume.

We prove a sharp upper bound on the number of distinct columns of a totally unimodular matrix with column sums 1 improving upon Heller’s classical bound. The proof uses Seymour’s decomposition theorem. Such matrices are closely related to unimodular polytopes: lattice polytopes where the vertices of every full-dimensional subsimplex form an affine lattice basis. This is an interesting subclass of 0/1-polytopes and contains, for instance, edge polytopes of bipartite graphs. Our main result on totally unimodular matrices implies a sharp upper bound on the number of vertices of unimodular polytopes.

Quint and Shubik (1997) conjectured that a non-degenerate n x n game has at most 2^n-1 Nash equilibria in mixed strategies. The conjecture is true for n <= 4 but false for n >= 6. We answer it positively for the remaining case n=5, which had been open since 1999. The problem can be translated to a combinatorial question about the vertices of a pair of simple n-polytopes with 2n facets. We introduce a novel obstruction based on the index of an equilibrium, which states that equilibrium vertices belong to two equal-sized disjoint stable sets of the graph of the polytope. This bound is verified directly using the known classification of the 159,375 combinatorial types of dual neighborly polytopes in dimension 5 with 10 facets. Non-neighborly polytopes are analyzed with additional combinatorial techniques where the bound is used for their disjoint facets.

Network games provide a powerful framework for modeling agent interactions in networked systems, where players are represented by nodes in a graph and their payoffs depend on the actions taken by their neighbors. Extending the framework of network games, we introduce and study semidefinite network games. In this model, each player selects a positive semidefinite matrix with trace equal to one, known as a density matrix, to engage in a two-player game with every neighboring node. The player's payoff is the cumulative payoff acquired from these edge games. Initially, we focus on the zero-sum setting, where the sum of all players' payoffs is equal to zero. We establish that, in this class of games, Nash equilibria can be characterized as the projection of a spectrahedron. Furthermore, we show that determining whether a semidefinite network game is a zero-sum game is equivalent to deciding if the value of a semidefinite program is zero. Beyond the zero-sum case, we characterize Nash equilibria as the solutions of a semidefinite linear complementarity problem.

We show that the notion of MAT-freeness for hyperplane arrangements depends on the underlying field. In particular, MAT-freeness is not combinatorial.

Cuntz and Kühne introduced the class of connected subgraph arrangements \(A_G\), depending on a graph \(G\), and classified all graphs \(G\) such that the corresponding arrangement \(A_G\) is free. We extend their result to the multiarrangement case and classify all graphs \(G\) for which the corresponding arrangement \(A_G\) supports some multiplicity \(\mu\) such that the multiarrangement \((A_G,\mu)\) is free.

Mustafin varieties provide a framework for degenerating the projective plane together with marked points, governed by the combinatorics of Bruhat-Tits buildings. Such limit objects are parametrized by the boundary points of a projective compactification of the moduli space of n points in ℙ² constructed by Schaffler-Tevelev. In this paper, we study the locus of this compactification generically parametrizing self-dual configurations, or equivalently, six points lying on a conic. We prove that this locus is a tropical compactification of the moduli space of self-dual configurations, with associated fan a refinement of the self-dual tropical Grassmannian introduced by Geiger-Hashimoto-Sturmfels-Vlad. We show that this compactification is a blow up of \(M_{0,6}\) and we classify the objects it parametrizes: degenerations of ℙ² equipped with a stable 6-pointed rational curve that records the limiting conic.