We will start with an overview of buildings, which are geometric objects used to study algebraic groups. We will discuss various examples of non-discrete affine buildings, all of which arise from algebraic groups over fields with non-discrete valuations, such as the hyperreals. We introduce a new notion of morphisms between buildings and say how the various examples are related to each other.

The study of the box dimension of commutator sets is motivated by fundamental questions concerning the commutator width of groups. The commutator width of a group $G$ is the minimal integer $n$ such that every element of the derived subgroup $G'$ can be written as a product of $n$ commutators. In any $d$-generated pro-$p$ group, the commutator width is bounded above by $d$. Notably, certain branch groups--including some GGS groups and the Grigorchuk group-- provide concrete examples where the commutator width exceeds 1; that is, not every element can be expressed as a single commutator. Our approach aims to measure the proportion of commutators in certain pro-$p$ groups. Let $G\leq \Gamma$ be a (weakly) regular branch group, where $\Gamma$ is the standard Sylow pro-$p$ subgroup of $\mathrm{Aut}(\mathcal{T})$. We prove that in $G$ the lower box dimension of the set of commutators with a particular fixed element--computed with respect to the filtration given by the level stabilizers-- equals 1. If we assume the subgroup congruence property, this result can be extended to all the filtrations. By contrast, we observe that the lower box dimension of the set of all commutators in a non-abelian finitely generated free pro-$p$ group, computed with respect to the lower $p$-series, is strictly less than 1.

Given a $p$-adic algebra, its zeta function is defined as the power series of which the $i$-th coefficient is the number of subalgebras with index $p^i$. The poles of the zeta function describe the growth of the number of index-$p^i$ subalgebras as $i$ increases. A lot of research has gone into understanding the largest pole, since it dominates the asymptotic growth. $p$-Adically, however, it is the smallest pole which is most important. In this talk, we will discuss recent progress on this front.

Attached to an extension K/F of algebraically closed fields there is a combinatorial geometry called the "full algebraic matroid" A(K/F). It can be thought of as a geometric space whose "points" are algebraically closed fields between K and F of transcendence degree 1 over F, "lines" are algebraically closed intermediate fields of transcendence degree 2, and so on. Given six points in A(K/F) which satisfy certain matroid-theoretic conditions, Hrushovski's Group Configuration Theorem reconstructs an entire one-dimensional algebraic group in K. This construction was used by Evans and Hrushovski (1989) to detect subgeometries of A(K/F) which are projective planes and to prove that they are all coordinatized by a small list of possible skew fields. In this talk I will introduce the main ideas of their paper which draw on connections between model theory, algebraic groups, and classical projective geometry. I will then explain how these techniques can be developed further to prove that the recognition problem for algebraic matroids is algorithmically unsolvable. This is based on ongoing joint work with Geva Yashfe.

A positive geometry is, roughly speaking, a complex projective variety with a distinguished semialgebraic set in its real points, considered as the nonnegative part. Important to a positive geometry is a differential form, called the canonical form, that is compatible with the combinatorics of the boundary of the nonnegative part. Positive geometries are first studied by physicists and continue to have close connections to physics. On the other hand, del Pezzo surfaces are classical objects in algebraic geometry that are known for their rich combinatorics. In this talk, I will discuss del Pezzo surfaces and their moduli spaces and show that in many cases, they can be equipped with the structure of a positive geometry. Special attention will be given to the moduli space of marked del Pezzo surfaces of degree 3. This is joint work with Nick Early, Alheydis Geiger, Marta Panizzut, and Bernd Sturmfels.

Around 2013, a natural conjecture by Julia Knight anticipated that given a first order sentence, its truth value over a random group in the few relators model, is equivalent to its truth value over non-abelian free groups. Our work extends the framework of this conjecture to random groups in the Gromov Density Model, at optimal density d<1/2. We prove the conjecture for sentences that belong to the Boolean algebra of universal sentences, as well as for sentences of minimal rank with arbitrary number of quantifiers, for random groups of density d<1/2. In this talk, we will present our results and, as time permits, outline the key strategies and ideas involved in the proofs.

Perfect matchings, also called 1-factors, are classical objects with abundant connections to many areas of combinatorics. It is well-known that the complete graph on an even number of vertices can always be decomposed into perfect matchings, called a 1-factorisation. Equivalently, we can view a 1-factorisation as a set of perfect matchings such that every edge is contained in exactly one perfect matching. From this point of view, one can generalise 1-factorisations to hyperfactorisations, studied by Jungnickel and Vanstone. A hyperfactorisation is a set of perfect matchings of the complete graph such that every pair of disjoint edges is contained in a constant number of perfect matchings. Boros, Jungnickel, and Vanstone showed that the complete graph on 2n vertices has a non-trivial hyperfactorisation for every n ≥ 5.
We investigate generalisations of 1-factorisations and hyperfactorisations and show that they are special subsets of an association scheme obtained from a Gelfand pair in the symmetric group. Furthermore, we construct many new examples of hyperfactorisations using group actions and the Kramer-Mesner method.

Subgroup zeta functions encode subgroup growth in finitely generated nilpotent groups, but are difficult to compute explicitly, even for one of the simplest families of nonabelian nilpotent groups: the higher Heisenberg groups. I will discuss explicit formulae for these groups, based on recent joint work with C. Voll.

In this talk, we explore several zero-sum invariants for a finite group G: the small Davenport constant d(G), the ordered Davenport constant Dₒ(G), and the Loewy length L(G). We see that Dₒ(G) = d(G) + 1 = d(A) + 2 for every non-abelian group of the form G = A ⋊₋₁ C₂, where A is any finite abelian group. In addition, Dimitrov conjectured in 2004 that Dₒ(G) = L(G) for every finite non-abelian p-group G. We provide explicit families of finite non-abelian p-groups for which Dimitrov's conjecture holds.

Grigorchuk-Gupta-Sidki groups are key examples of groups acting on rooted trees, which are significant throughout the theory of infinite groups and beyond. There are very recent links between groups acting on rooted trees and algebraic geometry. Specifically, Gul and Uria-Albizuri showed in 2018 that quotients of the periodic Grigorchuk-Gupta-Sidki groups, GGS-groups for short, admit Beauville structures. Based on recent work by Moritz Petschick, one can show that the periodic GGS-groups, that are defined by either an inverse-symmetric vector or a symmetric vector, have quotients that admit strongly real Beauville structures. Recall that a complex surface S is called real if there exists a biholomorphism between S and its complex conjugate surface such that the biholomorphism is an involution. The concept of strongly real is slightly more restrictive, and will be defined in the talk. This is joint work with Amir Dzambic.

The representation growth of a group $G$ measures the asymptotic distribution of its irreducible representations. Whenever the growth is polynomial, a suitable vehicle for studying it is a Dirichlet generating series called the representation zeta function of $G$. One of the key invariants in this context is the abscissa of convergence of the representation zeta function. The spectrum of all abscissae arising across a given class of groups is of considerable interest and has been studied in some cases.

In the realm of $p$-adic analytic groups (with perfect Lie algebra), the abscissae of convergence are explicitly known only for groups of small dimensions. But there are interesting asymptotic results for 'simple' $p$-adic analytic groups of increasing dimensions. In this talk, I will give an overview of the main tools and ingredients in this area and I will report on recent work joint with Moritz Petschick to enlarge the class of groups for which we have explicit results.