N3.12

A variety of groups is an equationally defined class of groups, namely the class of groups in which each of a set of "laws" (or "identical relations") holds. Examples include the abelian groups (defined by the law $xy = yx$), the groups of exponent dividing $d$ (defined by the law $x^d$), the nilpotent groups of class at most some fixed integer, and the solvable groups of derived length at most some fixed integer. This talk will give an introduction to varieties of groups, and then conclude with recent work on determining for certain varieties whether, for fixed coprime $m$ and $n$, a group $G$ is in the variety if and only if the power subgroups $G^m$ and $G^n$ (generated by the $m$-th and $n$-th powers) are in the variety.

In this talk I will describe another strong link between the behaviour of a 3-manifold and the behaviour of its fundamental group- specifically the theorem that the group splits as a free product if and only if the 3-manifold may be divided into two parts using a 2-sphere inducing this splitting. This theorem is for some reason known as Kneser's conjecture despite having been proved half a century ago by Stallings.

This talk will be on general amalgamation theory, covering ground from the 1950s to original research, with applications and examples from many different areas of mathematics and ranging from classical results to open problems.

In recent years, Ardakov and Wadsley have been interested in extending the classical theory of Beilinson-Bernstein localisation to different contexts. The classical proof relies on fundamental geometric properties of the dual nilcone of a semisimple Lie algebra; in particular, finding a nice desingularisation of the nilcone and demonstrating that it is normal. I will attempt to explain the relationship between these properties and the proof, and discuss some areas of my own work, which focuses on proving analogues of these results in the case where the characteristic of the ground field K is bad.

In this talk, I wish to address the problem of evaluating an integral on an n-dimensional complex vector space whose n-form of integration has poles along a union of (affine) hyperplanes, following the work of Heckman and Opdam. Such situation arise often in the harmonic analysis of a reductive group and when that is the case, the singular hyperplane arrangement in question is dictated by the root system of the group. I will then try to explain how we can relate the intersection lattice of the hyperplane arrangement with nilpotent orbits of a complex Lie algebra related to the root system in question.

An expander is a family of finite graphs of uniformly bounded degree, increasing number of vertices and Cheeger constant bounded away from zero. They occur throughout mathematics and computer science; the most famous constructions of expanders rely on powerful results in geometric group theory and number theory, while expanders are used in everything from error-correcting codes, through disproving the strongest version of the Baum-Connes conjecture, to affine sieve theory and the twin prime, Mersenne prime and Hardy-Littlewood conjectures.

However, very little was known about how different the geometry of two expanders could be. This question was raised by Ostrovskii in 2013, and a year later Mendel and Naor gave the first example of two 'distinct' expanders.

In this talk I will construct a continuum of expanders which are, in a certain sense, geometrically incomparable. Once the existence of a single expander is accepted, the remainder of the proof is a heady mix of counting, addition, multiplication, and just for the experts, a little bit of division. Two very different - and very interesting - continuums of 'distinct' expanders have since been constructed by Khukhro-Valette and Das.