I’ll discuss joint work of mine with with Ascencio-Martin, Britnell, Duncan, Francoeurs and Koutny to set up and study algebraic models of concurrent computation. 

Trace monoids were introduced by Mazurkiewicz as algebraic models of Petri nets, where some pairs of actions can be applied in either of two orders and have the same effect. Abstractly, a trace monoid is simply a right-angled Artin monoid. More recently Koutny et al. introduced the concept of a step trace monoid, which allows the additional possibility that a pair of actions may have the same effect performed simultaneously as sequentially. 

I shall introduce these monoids, discuss some problems we’d like to be able to solve, and the methods with which we are trying to solve them. In particular I’ll discuss normal forms for traces, comtraces and step traces, and generalisations of Stallings folding techniques for finitely presented groups and monoids.

The Aldous-Lyons conjecture from probability theory states that every (unimodular) infinite graph can be (Benjamini-Schramm) approximated by finite graphs. This conjecture is an analogue of other influential conjectures in mathematics concerning how well certain infinite objects can be approximated by finite ones; examples include Connes' embedding problem (CEP) in functional analysis and the soficity problem of Gromov-Weiss in group theory. These became major open problems in their respective fields, as many other long-standing open problems, that seem unrelated to any approximation property, were shown to be true for the class of finitely-approximated objects. For example, Gottschalk's conjecture and Kaplansky's direct finiteness conjecture are known to be true for sofic groups, but are still wide open for general groups.

In 2019, Ji, Natarajan, Vidick, Wright and Yuen resolved CEP in the negative. Quite remarkably, their result is deduced from complexity theory, and specifically from undecidability in certain quantum interactive proof systems. Inspired by their work, we suggest a novel interactive proof system which is related to the Aldous-Lyons conjecture in the following way: If the Aldous-Lyons conjecture was true, then every language in this interactive proof system is decidable. A key concept we introduce for this purpose is that of a Subgroup Test, which is our analogue of a Non-local Game. By providing a reduction from the Halting Problem to this new proof system, we refute the Aldous-Lyons conjecture.

This talk is based on joint work with Lewis Bowen, Alex Lubotzky, and Thomas Vidick.

No special background in probability theory or complexity theory will be assumed.

Simplicial volume is a homotopy invariant of manifolds introduced by Gromov to study their metric and rigidity properties. One of the strongest vanishing results for simplicial volume of closed manifolds is in presence of amenable covers with controlled multiplicity. I will discuss some conditions under which this result can be extended to manifolds with boundary. To this end, I will follow Gromov's original approach via the theory of multicomplexes, whose foundations have been recently laid down by Frigerio and Moraschini.

The conjugacy growth function of a finitely generated group is a variation of the standard growth function, counting the number of conjugacy classes intersecting the n-ball in the Cayley graph. The asymptotic behaviour is not a commensurability invariant in general, but the conjugacy growth of finite extensions can be understood via the twisted conjugacy growth function, counting automorphism-twisted conjugacy classes. I will discuss what is known about the asymptotic and formal power series behaviour of (twisted) conjugacy growth, in particular some relatively recent results for certain groups of polynomial growth (i.e. virtually nilpotent groups).

The study of the rationality of the $L^2$-Betti numbers of a countable group has led to the development of a rich theory in $L^2$-homology with deep implications in structural properties of the groups. For decades almost nothing has been known about the general question of whether the strong Atiyah conjecture passes to free products of groups or not. In this talk, we will confirm that the strong and algebraic Atiyah conjectures are stable under the graph of groups construction provided that the edge groups are finite. Moreover, we shall see that in this case the $\ast$-regular closure of the group algebra is precisely a universal localization of the associated graph of rings

I will describe some recent developments in random walks on Gromov-hyperbolic spaces. I will focus in particular on the notions of Schottky sets and pivoting technique introduced respectively by Boulanger-Mathieu-S-Sisto and Gouëzel and mention some consequences. The talk will be introductory; I will not assume specialized knowledge in probability theory.