Of interest to people who study both hyperplane arrangements and commutative algebra are the homological properties of the module of logarithmic derivations of a hyperplane arrangement A. I will introduce the "ideal of pairs", which is a sort of "symmetrization" of this module of logarithmic derivations with respect to matroid duality. This is an ideal which simultaneously "sees" many of the homological properties of both the arrangement and its dual.

Zeta functions of integral quiver representations enumerate subrepresentations of finite additive index. In my talk I will explain how this subject combines and generalizes beautiful, well-known objects and ideas from algebraic number theory, algebraic geometry and combinatorial algebra. Aspects of this talk are based on joint and ongoing work with Seungjai Lee and Joshua Maglione.

Generalized permutahedra appear in various areas from submodular function optimization to the study of Grassmannians and Flag varieties. I will give an introduction to some of their geometric and combinatorial properties leading to the study of their subdivisions. Those arise naturally in Tropical Geometry and Discrete Convex Analysis. I present a new characterization of the corresponding height functions and show how this relates to the Bruhat order of the symmetric group. The talk is based on joint work with Michael Joswig, Dante Luber and Jorge Alberto Olarte (https://arxiv.org/abs/2111.13676).

Multiple zeta values are the building blocks for products of values of the Riemann zeta function. In my talk I will explain briefly some of the (conjectural) algebraic structures of these objects. This will then motivate q-analogs of multiple zeta values, in particular I will compare their algebraic structure to the ones of the multiple zeta values. A key ingredient will be a particular nice spanning set for the space of multiple q-zeta values.

It is well-known that the numerator of the Ehrhart series, the so-called h*-polynomial, of a lattice polytope has nonnegative coefficients. Only in a few cases (e.g., for simplices) we are able to interpret the coefficients in the context of another counting problem. We present two further families of polytopes whose h*-polynomials can be interpreted in terms of permutation statistics. In this context we give new definitions of the major index and descent statistic on signed multiset permutations. In this talk we give a brief introduction to Ehrhart theory and permutation statistics (and their generalisations).

This talk will be devoted to rank distributions in spaces of matrices and the enumeration of orbits. In particular, I will discuss the effects of imposing linear relations on certain spaces of matrices. This turns out to be related to counting linear orbits of matrix groups. I will also discuss connections with polynomiality questions and the combinatorics of permutation statistics. This is based on joint work with T. Rossmann.

I will report on recent work which shows that for a given exponent $n$, counting linear orbits or conjugacy classes of $Z$-defined unipotent groups over a finite field of size $q$ is as difficult as counting $GF(q)$-rational points on arbitrary schemes, at least when all counts are considered modulo $q^n$.

Given a polynomial map $f$ with integer coefficients into an algebraic group $G$, we study families of random walks induced by f on the compact groups ${G(Z_p)}_p$, as $p$ varies. In order to do so, we explore a natural algebraic convolution operation between maps into $G$, as well as connections between the singularities of the map $f$, and the analytic behavior of pushforwards by $f$ of nice measures over local fields (e.g. $Q_p$). A case of articular interest is when f is a word map, such as the commutator map $w_G(x,y)=xyx^{-1}y^{-1}$, in which one may study a larger family of random walks by varying over both $p$ and the group $G$. Based on joint works with Itay Glazer (mainly arxiv:1811.09838 and arxiv:1912.12556).

Matroids are combinatorial structures that generalize the concept of linear independence. The linear representations and characteristic sets of matroids are well known. But the algebraic representations and characteristic sets received much less attention, and the possible algebraic characteristic sets are still not completely known. We look at possible pairs of linear-algebraic characteristic sets of matroids and algebraic characteristic sets, which are infinite but not co-finite. Based on joint works with Dustin Cartwright (arXiv:2212.00095).

Parabolic subgroups are essential in the study of Artin groups. Motivated by what happens in Coxeter groups and braid groups we can ask the following question: is the intersection of parabolic subgroups of an Artin group a parabolic subgroup? In this talk we will show how to study this question in the two-dimensional case. To do so, we will introduce systolic-by-function complexes, which are a generalization of systolic complexes.