University of Manchester

A self-similar group is a group $G$ acting on a regular, infinite rooted tree by automorphisms in such a way that the self-similarity of the tree is reflected in the group. The most common examples are generated by the states of a finite automaton. Many famous groups, like Grigorchuk's 2-group of intermediate growth, are of this form. Nekrashevych associated $C^*$-algebras and algebras with coefficients in a field to self-similar groups. In the case $G$ is trivial, the algebra is the classical Leavitt algebra, a famous finitely presented simple algebra. Nekrashevych showed that the algebra associated to the Grigorchuk group is not simple in characteristic 2, but Clark, Exel, Pardo, Sims, and Starling showed its Nekrashevych algebra is simple over all other fields. Nekrashevych then showed that the algebra associated to the Grigorchuk-Erschler group is not simple over any field (the first such example). The Grigorchuk and Grigorchuk-Erschler groups are contracting self-similar groups. This important class of self-similar groups includes Gupta-Sidki p-groups and many iterated monodromy groups like the Basilica group. Nekrashevych proved algebras associated to contacting groups are finitely presented.

In this talk, we discuss a result of the speaker and Benjamin Steinberg characterizing simplicity of Nekrashevych algebras of contracting groups. In particular, we give an algorithm for deciding simplicity given an automaton generating the group. We apply our results to several families of contracting groups like GGS groups and Sunic's generalizations of Grigorchuk's group associated to polynomials over finite fields.

Despite their straightforward definition, the homeomorphism groups of surfaces are far from straightforward. Basic algebraic and dynamical problems are wide open for these groups, which is a far cry from the closely related and much better understood mapping class groups of surfaces. With Jonathan Bowden and Sebastian Hensel, we introduced the fine curve graph as a tool to study homeomorphism groups. Like its mapping class group counterpart, it is Gromov hyperbolic, and can shed light on algebraic properties such as scl, via geometric group theoretic techniques. This brings us to the enticing question of how much of Thurston's theory (e.g. Nielsen--Thurston classification, invariant foliations, etc.) for mapping class groups carries over to the homeomorphism groups. We will describe new phenomena which are not encountered in the mapping class group setting, and meet some new connections with topological dynamics, which is joint work with Bowden, Hensel, Kathryn Mann and Emmanuel Militon. I will survey what's known, describe some of the new and interesting problems that arise with this theory, and give an idea of what's next.

(jt work with David Cushing and George Stagg)

Prolog is a rather unusual programming language that was developed by Alain Colmerauer 50 years ago in one of the buildings on the way to the CIRM in Luminy. It is a declarative language that operates on a paradigm of first-order logic -- as distinct from imperative languages like C, GAP and Magma. Prolog operates by loading in a list of axioms as input, and then responds at the command line to queries that ask the language to achieve particular goals, given those axioms. It gained some notoriety through IBM’s implementation of ‘Watson’, which was a system designed to play the game show Jeopardy. Through a very efficiently implemented constraint logic programming module, it is also the worlds fastest sudoku solver. However, it has had barely any serious employment by pure mathematicians. So the aim of this talk is to advertise Prolog through an extended example: my co-authors and I used it to search for new simple Lie algebras over the field GF(2) and were able to classify a certain flavour of absolutely simple Lie algebra in dimensions 15 and 31, discovering a dozen or so new examples. With some further examples in dimension 63, we then extrapolated two previously undocumented infinite families of simple Lie algebras.

Assuming the Riemann Hypothesis, Soundararajan established almost sharp upper bounds for all positive moments of the Riemann zeta-function. This result was later improved by Harper, who proved upper bounds of the right order of magnitude. I will describe some of the ideas in their proofs, and then discuss recent joint work with Alexandra Florea, where we consider negative moments of the Riemann zeta-function. For example, we can obtain asymptotic formulas for negative moments when the shift in the zeta function is large enough, confirming a conjecture of Gonek.  We also obtain an upper bound for the average of the generalised Mobius function.

We outline Dirac cohomology for Lie algebras and Vogan’s conjecture. We then cover some basic material on Schur-Weyl duality and Arakawa-Suzuki functors. Finishing with current efforts and results on generalising Vogan’s conjecture to a Schur-Weyl duality setting. This would relate the centre of a Lie algebra with the centre of the relevant tantaliser algebra. We finish by considering a unitary module X and giving a bound on the action of the tantalizer algebra.

The infamous Ramanujan tau-function is the starting point for many mysterious conjectures and difficult open problems within the realm of modular forms. In this talk, I will discuss some of our recent results pertaining to odd values of the Ramanujan tau-function. We use a combination of tools which include the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue–Mahler equations due to Bugeaud and Gyory, and the modular approach via Galois representations of Frey-Hellegouarch elliptic curves. This is joint work with Mike Bennett (UBC), Adela Gherga (Warwick) and Samir Siksek (Warwick).

Spectral sequences are computational tools to find the (co-)homology of mathematical objects and are used across various fields. In this talk I will focus on the LHS spectral sequence, which we associate to an extension of groups to compute group cohomology. The first part of the talk will serve as introduction to both group cohomology and general spectral sequences, where I hope to provide and intuition and some reduced formalism. As main example, and core of this talk, we will look at the LHS spectral sequence associated to the group extension $(\mathbb{Z}/3\mathbb{Z})^3 \rightarrow S \rightarrow \mathbb{Z}/3\mathbb{Z}$, where $S$ is a Sylow-3-subgroup of $SL_2(\mathbb{Z}/9\mathbb{Z})$. In particular I will present arguments that all differentials on the $E^2$ page vanish.

The famous Birch & Swinnerton-Dyer conjecture predicts that the (algebraic) rank of an elliptic curve is equal to the so-called analytic rank, which is the order of vanishing of the associated L-functions at the central point. In this talk, we shall discuss the analytic rank of automorphic L-functions in an "alternate universe". This is joint work with Kyle Pratt and Alexandru Zaharescu.