The 1-skeleton of an $A_2$ Bruhat-Tits building is isomorphic to the Cayley graph of an abstract group with relations coming from ”triangle presentations.” This abstract group either embeds into $PGL(3, F_q((x)))$ or $PGL(3, Q_q)$, or else is exotic. Currently, the complete list of triangle presentations is only known for projective planes of orders $q=2$ or $3$. However, one abstract group that embeds into $PGL(3,F_q((x)))$ for any prime power $q$ is known via the trace function corresponding to the finite field of order $q^3$. I have found a different method to derive this group as well as others via perfect difference sets. This new method demonstrates a previously unknown connection between difference sets and $A_2$ buildings. Moreover, this new method makes the final computation of triangle presentations easier, which is computationally valuable for large $q$.

The Residual finiteness growth function and conjugacy separability function of a group express how well one can separate two distinct elements/conjugacy classes from one another in finite quotients. As of 2016, it is known that the residual finiteness growth function can become arbitrarily large. We explore further precisely which functions can appear as residual finiteness growth/conjugacy separability growth and how they relate to one another.

In 2004, Gunter Malle published his famous conjecture on the asymptotics of extensions of number fields with a fixed Galois group, counted by discriminant. His conjecture can be generalised directly to global function fields if the order of the Galois group is coprime to the characteristic of the field (the so-called tamely ramified case). For the wildly ramified case, in which the characteristic of the global field divides the order of the Galois group, Thorsten Lagemann showed that the naive generalisation of Malle's conjecture (as in the tamely ramified case) does not apply. In this talk, we consider local and global function fields and determine the precise asymptotics of wildly ramified elementary-abelian extensions of these fields, counted by discriminant. We mainly focus on local and rational function fields for which a local-global principle in counting holds. To determine the asymptotics, we follow the classical approach of analysing the corresponding Dirichlet series of the counting function.

Infinite non-affine Coxeter groups are Coxeter groups that do not split into direct products of spherical and Euclidean reflection groups. For a fixed element in a infinite non-affine Coxeter group, there just exists sparse knowledge how to determine the minimal number of conjugates of the standard generators needed to factor this element (reflection length). In this talk, we discuss new combinatorial and geometric results regarding reflection length in infinite non-affine Coxeter groups. For Coxeter groups that are hyperbolic reflection groups, we describe certain points in the visual boundary of the n-dimensional hyperbolic space for which every neighbourhood contains large reflection length in some sense. On the combinatorial side, we establish a connection between the reflection length functions on arbitrary Coxeter groups and the reflection length functions on universal Coxeter groups of the same rank through the solution to the word problem for Coxeter groups.

Guba and Sapir introduced a family of groups associated with semigroup presentations which are called symmetric and planar diagram groups. In this talk, we discuss the idea of how right-angled Artin groups can be seen as symmetric diagram groups in contrast to the fact that not all right-angled Artin groups are planar diagram groups. This was motivated by the example of pure virtual planar braid groups being symmetric diagram groups

The Chow polynomial of a matroid is the Hilbert series of its Chow ring, defined using the maximal building set. This polynomial played a central role in the proof of the log-concavity of the Whitney numbers of the first kind by Adiprasito, Huh, and Katz. The Chow polynomial is known to be palindromic, unimodal, log-concave, and gamma-positive. One proof of its gamma-positivity arises from its connection to the Poincaré-extended ab-index, a polynomial introduced by Dorpalen-Barry, Maglione, and Stump. With the resulting gamma-expansion, we derive an explicit combinatorial formula for the Chow polynomial of uniform matroids. Finally, we discuss conjectures related to these structures, including the question of their real-rootedness.

Saito introduced the notion of freeness for a hyperplane arrangement A, and Ziegler generalized this definition to multiarrangements (A,m). Yoshinaga then showed that there exists a connection between the freeness of a multiarrangement (A,m) and the freeness of a so called extension of (A,m). It is known that if A is the Coxeter arrangement of type A_2, then for every multiplicity m there exists a free extension of (A,m). In this talk, I will address the case where A is the Coxeter arrangement of type B_2. This is joint work with Torsten Hoge and Shota Maehara.

In this talk, I would like you to meet Hall-Littlewood-Schubert series, a new class of multivariate generating functions. Their definition features semistandard Young tableaux and polynomials resembling the classical Hall-Littlewood polynomials. Their intrinsic beauty notwithstanding, Hall-Littlewood-Schubert series have many applications to counting problems in algebra, geometry, and number theory. In my talk the spotlight will be on applications to affine Schubert series. These may be seen as an integral analogue of the Poincare polynomials enumerating the rational points over finite fields of classical Schubert varieties. The latter parametrize subspaces of a given vector space by the intersection dimensions with a fixed flag of reference. This work is joint with Joshua Maglione. It pertains to TRR-projects A1, A4, and A6. I will explain things from scratch, assuming no familiarity with the advanced technical vocabulary used in this abstract.

Given an arbitrary Coxeter system $(W,S)$ and a nonnegative integer $m$, the $m$-Shi arrangement of $(W, S)$ is a subarrangement of the Coxeter hyperplane arrangement of $(W,S)$. The classical Shi arrangement $(m = 0)$ was introduced in the case of affine Weyl groups by Shi to study Kazhdan-Lusztig cells for W. As two key results, Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in $W$ and that the union of their inverses form a convex subset of the Coxeter complex. The set of $m$-low elements in W were introduced to study the word problem of the corresponding Artin-Tits (braid) group and they turn out to produce automata to study the combinatorics of reduced words in $W$. In this talk, I will discuss how to Shi’s results extend to any Coxeter system and show that the minimal elements in each Shi region are in fact the m-low elements. This talk is based on joint work with Matthew Dyer, Susanna Fishel and Alice Mark.