In 1962 Ehrhart proved that the number of lattice points in integer dilates of a lattice polytope is given by a polynomial — the Ehrhart polynomial of the polytope. Since then Ehrhart theory has developed into a very active area of research at the intersection of combinatorics, geometry and algebra. The Ehrhart polynomial encodes important information about the polytope such as its volume and the dimension. An important tool to study Ehrhart polynomials is the h*-polynomial, a linear transform of the Ehrhart polynomial which is given by the numerator of the generating series. By a famous theorem of Stanley the coefficients of the h*-polynomial are always nonnegative integers. In this talk, we discuss generalizations of this result to weighted lattice point enumeration in rational polytopes where the weight function is given by a polynomial. In particular, we show that Stanley’s Nonnegativity Theorem continues to hold if the weight is a sum of products of linear forms that a nonnegativ

Vortrag im Stochastic Afternoon: https://www.sfb1283.uni-bielefeld.de/talks/view/17586

q-Matroids, the q-analogue of matroids, have been intensively investigated in recent years in coding theory due to their close connection with rank metric codes. Indeed, in 2018 it was shown by Jurrius and Pellikaan that a rank metric code induces a q-matroid that captures many of the code's invariants. Many results from classical matroid theory were found to have natural q-analogues. In this talk we will consider the notions of direct sum and introduce the free product of two q-matroids and beside the interesting properties that we can derive, we will focus on the representability issue of these products. We will show that representing the direct sum of uniform q-matroids is equivalent to constructing special linear sets which are almost scattered with respect to the hyperplanes. Moreover, we will provide an interesting link between the free product of uniform q-matroids and the problem of finding i-clubs on the projective line. This talk is based on a joint work with Relinde Jurriu

Interesting combinatorial structures can often be characterised as special subsets of association schemes. In his PhD thesis, Philippe Delsarte developed powerful linear programming techniques to prove non-existence and uniqueness results for such structures. Ideas of this type were fundamental in the work for which Marina Viazovska was awarded the Fields medal in 2022. This talk will begin with an overview of Delsarte theory. We will then study generalised permutations. They act on the set {1,2,...,n}, whose elements are coloured with one of r possible colours. We consider different notions of transitivity and interpret these in the appropriate association scheme using Delsarte Theory. No particular knowledge of association schemes will be required to appreciate this talk

Matroids are generalization of both graphs and hyperplane arrangements. Many interesting invariants of these combinatorial objects are valuative. Two prominent examples of valuative matroid invariants are the Tutte polynomial and the \mathcal{G}-invariant. The relevance of the \mathcal{G}-invariant steams from its universal property that any other valuative invariant can be obtained as a specialization. Nevertheless, the most intense studied invariant of matroids is clearly the Tutte polynomial as it respects deletion and contraction. An interesting question therefore is on which minor and duality closed classes of matroids is the Tutte polynomial universal. In my talk I will give a complete answer to this question. This talk is based on joint work with Luis Ferroni.

We are interested in polytopes with a prescribed normal fan. More precisely, for a fixed matrix $A$ with rows $a_1, \dots, a_n \in \mathbb{R}^d$ we consider the space of all right-hand sides $b \in \mathbb{R}^n$, such that $P_A(b) = \{x \in \R^d: Ax \le b\}$ is a non-empty polytope and all hyperplanes spanned $a_i \cdot x = b_i$ are supporting. This space called support cone, also known as closed inner region or closed irredundancy domain, carries a natural polyhedral structure. It decomposes into type cones which describe the various deformations of the polyhedra. In my presentation we take a closer look at these structures, and focus on the case that the polytopes $P_A(b)$ are alcoved, i.e., the rows of $A$ are differences of standard unit vectors. Furthermore, for a certain class of alcoved polytopes, we draw a connection to simple games. This is based on joint work with Raman Sanyal and Benjamin Schröter.

The Baer-Suzuki theorem, a classical result in finite group theory, states that if $C$ is a conjugacy class of a finite group $G$ and if every two elements in $C$ generate a nilpotent subgroup, then $C$ generates a nilpotent normal subgroup of $G$. This gives a nice characterization of the Fitting subgroup of $G$, that is, the largest nilpotent normal subgroup of $G$. This theorem was originally proved by Baer in 1957 and later by M. Suzuki in 1965. A more direct and elementary proof was obtained by Alperin and Lyons in 1971. This theorem was used by M. Suzuki in the classification of finite simple groups. Especially, it was used to show that every finite nonabelian simple group possesses a nontrivial real element of odd order. Many generalizations of this theorem have been proposed and studied over the years. In this talk, I will discuss some new variants of this theorem, especially, the so-called Baer-Suzuki type problems, and give an application to the character theory of fi

The present talk discusses a connection between two concepts arising from different fields of mathematics. The first concept, called vine, is a graphical model for dependent random variables. This first appeared in a work of Joe (1994), and the formal definition was given later by Cooke (1997). Vines have nowadays become an active research area whose applications can be found in probability theory and uncertainty analysis. The second concept, called MAT-freeness, is a combinatorial property in the theory of freeness of the logarithmic derivation module of hyperplane arrangements. This was first studied by Abe-Barakat-Cuntz-Hoge-Terao (2016), and soon afterwards developed systematically by Cuntz-Muecksch (2020). In the particular case of graphic arrangements, Tsujie and the speaker recently proved that the MAT-freeness is completely characterized by the existence of certain edge-labeled graphs, called MAT-labeled graphs. In this talk we show that, interestingly, there exists an expl

The Euler characteristic, initially defined for polyhedra and simplicial complexes, has become a prominent invariant for groups, especially when they act "nicely" on CW-complexes. In this talk, we explore the connections between the Euler characteristic of groups acting on trees or buildings and some generating functions emerging from the geometric structure on which the groups act.

: Let G be an affine algebraic group over a field k. We show there exists a unipotent normal subgroup of G which contains all other such subgroups; we call it the restricted unipotent radical Rad_u(G) of G. We investigate some properties of Rad_u(G), and study those G for which Rad_u(G) is trivial. In particular, we relate these notions to their well-known analogues for smooth connected affine k-groups.