After a brief introduction to subject of spherical representations of hyperbolic groups, I will present a new construction motivated by a spectral formulation of the so-called Shalom conjecture.This a joint work with Dr Jan Spakula.

A parabolic representation of the free group  is one in which the images of both generators are parabolic elements of $PSL(2,\IC)$. The Riley slice is a closed subset ${\cal R}\subset \IC$ which is a model for the moduli space of parabolic, discrete and faithful representations. The complement of the Riley slice is a bounded Jordan domain within which there are isolated points, accumulating only at the boundary, corresponding to parabolic discrete and faithful representations of rigid subgroups of $PSL(2,\IC)$. Recent work of Aimi, Akiyoshi, Lee, Oshika, Parker, Lee, Sakai, Sakuma \& Yoshida, have topologically identified all these groups. Here we give the first  substantive properties of the nondiscrete representations using ergodic properties of the action of a polynomial semigroup and identifying the Riley slice as the ``Julia set’’ of this dynamical system. We prove a supergroup density theorem: given any irreducible parabolic representation of $F_2$ whatsoever, {\em any}  non-discrete parabolic representation has an arbitrarily small perturbation which contains that group as a conjugate.  Using these ideas we then show that there are nondiscrete parabolic representations with an arbitrarily large number of distinct Nielsen classes of parabolic generators.

To a group theorist ribbon groups look like knot groups, except  that we know everything about knot groups and next to nothing about ribbon groups.

I will talk about an old paper of mine with Peter Teichner where several questions on ribbon groups naturally arise.

Given a group G, one can seek to understand (some of) its subgroups. Centralisers of elements are easy to define, but maybe not so easy to understand: even in such well studied groups as Out(Fn) they are not yet understood in general. I'll discuss recent work with Armando Martino where we extend what is known in Out(Fn), involving a (surprising?) connection to free-by-cyclic groups and their automorphisms as well as working with actions on trees. The strategies seem like they should apply in many more cases, and if time allows I'll discuss ongoing work (with Gilbert Levitt and Armando Martino) exploring these possibilities.

This talk will be an invitation to the study of cubulated groups and their quotients via the tools of cubical small cancellation theory. Non-positively curved cube complexes are a class of cell-complexes whose geometry and combinatorial structure is closely related to the structure of the groups that act nicely on their universal covers. I will tell you a bit about what we know and don’t know about these groups and spaces, and about the tools we have to study their quotients. I will explain some applications of the study of these quotients to producing a large variety of examples of large-dimensional hyperbolic (and non-hyperbolic) groups.

Recent years have seen many new ideas appearing in the solution theories of singular stochastic partial differential equations. An exciting application of SPDEs that is beginning to emerge is to the construction and analysis of quantum field theories. In this talk, I will describe how stochastic quantisation of Parisi–Wu can be used to study QFTs, especially those arising from gauge theories, the rigorous construction of which, even in low dimensions, is largely open.

In their seminal 2012 paper, Elekes and Szabó found that a certain weak combinatorial non-expansion property of an algebraic relation suffices to trigger the group configuration theorem, showing that only (approximate subgroups of) algebraic groups can be responsible for it. I will discuss some more recent variations and elaborations on this result, focusing on the case of ternary relations on varieties of dimension >1.