Is it possible to tell the isomorphism type of an infinite group from its collection of finite quotients? This question, known as profinite rigidity, has deep roots in various areas of mathematics, ranging from arithmetic geometry to group theory. In this talk, I will introduce the question, its history and context. I will explain how profinite rigidity is studied using the machinery of profinite completions, including elementary proofs and counterexamples. Then I will outline some of the key results in the field, ranging from 1970 to the present day. Time permitting, I will elaborate on recent results of myself on the profinite rigidity of certain classes of solvable groups. 

Right-angled Artin groups (RAAGs) can be viewed as a generalisation of free groups. To what extent, then, do the techniques used to study automorphisms of free groups generalise to the setting of RAAGs? One significant advance in this direction is the construction of 'untwisted Outer space' for RAAGs, a generalisation of the influential Culler-Vogtmann Outer space for free groups. A consequence of this construction is an upper bound on the virtual cohomological dimension of the 'untwisted subgroup' of outer automorphisms of a RAAG. However, this bound is sometimes larger than one expects; I present work showing that, in fact, it can be arbitrarily so, by forming a new complex as a deformation retraction of the untwisted Outer space. In a different direction, another subgroup of interest is that consisting of symmetric automorphisms. Generalising work in the free groups setting from 1989, I present an Outer space for the symmetric automorphism group of a RAAG. A consequence of the proof is a strong finiteness property for many other subgroups of the outer automorphism group.

In the definition of the skein lasagna module of a $4$-manifold $X$, it is essential that the input TQFT be fully functorial for link cobordisms in $S^3 \times [0, 1]$. I will describe an approach to resolve existing sign ambiguities in Kronheimer and Mrowka's spectral sequence from Khovanov homology to singular instanton link homology. The goal is to obtain a theory that is fully functorial for link cobordisms in $S^3 \times [0,1]$, and where the $E_2$ page carries a canonical isomorphism to Khovanov-Rozansky $\mathfrak{gl}_2$ link homology. Possible applications include non-vanishing theorems for $4$-manifold Khovanov skein lasagna modules à la Ren-Willis.

Recently, much attention has been paid to the intersection between coarse geometry and graph theory, giving rise to the fresh, exciting new field aptly known as ‘coarse graph theory’. One aspect of this area is the study of so-called ‘fat minors’, a large-scale analogue of the usual idea of a graph minor.

In this talk, I will introduce this area and motivate some interesting questions and conjectures. I will then sketch a proof that a finitely presented group is either virtually planar or contains arbitrarily ‘fat’ copies of every finite graph.

No prior knowledge or passion for graph theory will be assumed in this talk.

In this talk, we give an accessible introduction to the theory of coarse cohomology of metric spaces in the sense of Margolis, which we present in direct analogy with group cohomology for discrete groups. We explain how this yields the robust notion of coarse cohomological dimension (due to Margolis), which is a genuine quasi-isometry invariant of metric spaces generalising the cohomological dimension of groups when the latter is finite. We then give applications to geometric properties of quasimorphisms and motivate how such considerations might be useful in the setting of non-positively curved groups. This is joint reading/work with Paula Heim.

Property (T) is a rigidity property for group representations. It is generally very difficult to determine whether an infinite group has property (T) or not. It has long been known that a discrete group with a finite symmetric generating set has property (T) if and only if the group Laplacian is a positive element in the maximal group C*-algebra. However, this characterization has not been useful in addressing the question for automorphism groups of (non-abelian) free groups. In his 2016 paper, Ozawa proved that the phenomenon of 'positivity' of the group Laplacian is observed in the real group algebra, meaning that the Laplacian can be decomposed into a 'sum of squares'. This result transformed checking property (T) into a finite-dimensional condition that can be performed with the assistance of computers. In this talk, we will introduce property (T) and discuss Ozawa's result in detail.