Schedule

Questions about when group rings embed into division rings and how well-behaved these embeddings can be play a role in problems which at first sight seem unrelated. Most prominent among these is the Atiyah Conjecture. In this talk I will give a survey of the progress that has been made so far on these problems, and explain how groups acting on rooted trees provide new examples of interest.

For a compact Lie group $G$, its representation zeta function (also known as Witten zeta function) is the Dirichlet series formed from the dimensions of its irreducible representations. The simplest example is when $G = SU(2)$, giving the Riemann zeta function. In this talk, I will give a gentle overview of several intriguing arithmetic properties that have been only discovered recently. These include special values at negative integers, special values of residues, $p$-adic congruences, and connections with Eisenstein series.

Let $R$ be a commutative ring and $\mathcal{A} \subseteq \mathrm{End}(R^n)$ a subalgebra of the endomorphisms of a free module over $R$. If $\mathfrak{U}$ is the set of finite index $\mathcal{A}$-invariant submodules of $R^n$, the submodule zeta function for $\mathcal{A}$ is formally defined as: $$\zeta_{\mathcal{A}, R} (s) = \sum_{M \in \mathfrak{U}} [R^n:M]^{-s}$$ Earlier work has focused on the case where $R = \mathfrak{o}$ is compact discrete valuation ring and $\mathcal{A}$ is generated by a single element of $\mathrm{End}(R^n)$ and this situation is relatively well understood. This talk discusses recent work in which we computed explicit formulae for submodule zeta functions after extending the base ring to $\mathfrak{o}[X]/(X^m)$ for positive integer $m$ and efforts towards developing a general theory for the effect of such base extensions on zeta function.

Given a group $G$, any word in $k$ variables gives rise to a word map from $G^k$ to $G$. The study of word maps is an active field of research in group theory, and a natural question to consider is how many elements (in a suitable sense) of a given group can be written as images of a given word map. In this talk I will report on joint work with Matteo Vannacci in which we investigate the (Haar) measure of the images of word maps in profinite groups, especially in free pro-$p$ and free profinite groups. More generally, we also consider the measure of certain existentially definable sets in the same classes of groups.