Finite W-algebras are certain associative algebras determined from a nilpotent element in a complex reductive Lie algebra. The representation theory of finite W-algebras is closely connected to the theory of primitive ideals of universal enveloping algebras. Furthermore, the representation theory of modular versions of finite W-algebras has major applications to the representation theory of modular Lie algebras. In this talk we will give an overview of this theory and some of the many applications of the representation theory of W-algebras.

We will start by explaining what categorification is, and how it has been beneficial in representation theory. We will then focus on the categorification of Hecke algebras via Soergel bimodules and give a classification of simple 2-representations of the resulting 2-categories for finite Coxeter systems in characteristic 0.

We review recent results on global dynamics of radial self-gravitating compressible Euler flows, which arise in the mathematical description of stars. We will describe classes of smooth initial data that lead to the formation of imploding finite-time singularities as well as expanding star solutions. Our main focus is on the decisive role of scaling invariances and their interaction with the nonlinearities.

C*-algebras are norm closed self-adjoint algebras of bounded operators on Hilbert space, and so arise naturally from unitary representations of groups and group actions, amongst other constructions. Large scale work of many researchers over decades has recently culminated in a definitive classification theorem for simple amenable C*-algebras of finite topological dimension. I’ll start out by introducing C*-algebras and then the rest of the talk will be devoted to describing this theorem: which C*-algebras are classified, and by what data? I’ll discuss how this relates to the structure and classification theorems for von Neumann algebras from the 70s and 80s. The talk will be illustrated by examples coming from group actions.

After recalling a general framework to prove existence of Einstein and constant scalar curvature metrics on algebraic varieties, we apply a recent analytic breakthrough by Chen-Cheng to study the existence problem of such metrics on Galois covers, connecting the analytic theory to various geometrical constructions in the cyclic (classical), abelian and non-abelian cases. Joint work with Yalong Shi (Nanjing) and Alberto Della Vedova (Milano Bicocca).

Symplectic manifolds encompass both the phase spaces of dynamical systems and smooth complex algebraic varieties.  A fundamental problem is to understand the group of Hamiltonian diffeomorphisms of a symplectic manifold, both as an algebraic object and as a topological object. I will describe some recent progress, based on novel ways of approaching well-known enumerative invariants which count holomorphic curves.  

Some of the most important problems in combinatorial number theory ask for the size of the largest subset of the integers in an interval lacking points in a fixed arithmetically defined pattern. One example of such a problem is to prove the best possible bounds in Szemerédi's theorem on arithmetic progressions, i.e., to determine the size of the largest subset of {1,...,N} with no nontrivial k-term arithmetic progression x, x+y,..., x+(k-1)y. Gowers initiated the study of higher order Fourier analysis while seeking to answer this question, and used it to give the first reasonable upper bounds for arbitrary k. In this talk, I'll discuss recent progress on quantitative polynomial, multidimensional, and nonabelian variants of Szemerédi's theorem and on related problems in harmonic analysis and ergodic theory.

The Cohen-Lenstra-Martinet conjectures have been verified in only two cases. Davenport and Heilbronn compute the average size of the 3-torsion subgroups in the class group of quadratic fields and Bhargava computes the average size of the 2-torsion subgroups in the class groups of cubic fields. The values computed in the above two results are remarkably stable. In particular, work of Bhargava--Varma shows that they do not change if one instead averages over the family of quadratic or cubic fields satisfying any finite set of splitting conditions.

However for certain "thin" families of number fields, the story is very different. In this talk, I will discuss two results in this direction. First, I will talk about joint work with Bhargava and Hanke in which we consider families of monogenic cubic fields, i.e, fields whose rings of integers are generated by a single element. Second, I will talk about joint work with Setayesh, Siad, and Swaminathan in which we consider the family of fields that have a unit monogenizer, i.e., fields whose rings of integers are generated by a unit. The average values of the 2-torsion in the class groups  differ from the Cohen-Lenstra-Matrinet heuristics in both these cases.

Exploration via random walks is a key tool in the analysis of large directed networks. For instance, the walk's stationary distribution plays a prominent role in ranking systems and search algorithms. In this lecture, we present some recent progress in the analysis of random walks for a class of sparse directed networks generated by the so-called configuration model. We discuss various properties of the stationary distribution, including bulk behaviour and extremal values. We also consider the mixing time, that is the time needed to reach stationarity, and show that the walk typically displays a cutoff behaviour. Finally, we analyse the convergence to stationarity when the walk experiences regeneration events such as teleportation, as in the PageRank algorithm, or resampling of the underlying graph, as in a dynamically evolving network.

We study the biased random walk for dynamical percolation on Z^d . We establish a law of large numbers and an invariance principle for the random walk using regeneration times. Moreover, we verify that the Einstein relation holds, and we investigate the speed of the walk as a function of the bias. While for d = 1 the speed is monotone increasing, we show that this fails in general dimension d, and that some new phenomena occur. (Based on joint work with Sebastian Andres, Perla Sousi and Dominik Schmid.)

Mathematics is the most popular A-Level in the UK with around 45% getting an A or A*. Yet recruitment to mathematics degrees remains comparatively low. And while student numbers have been steady for a decade some mathematics departments in HE have grown enormously and others have contracted accordingly.

Mathematics teachers are in short supply and retention is poor but the Prime Minister wants all pupils to study mathematics until the age of 18. Studying mathematics has been a central part of the school curriculum for many years yet many pupils leave school believing that maths has no relevance to their lives, are unaware of its importance or utility and, worse, are happy to boast that they aren’t good at it.

In this talk I will discuss these and other problems such as sexism, racism, extra-curricula activities, and Further Maths. I’ll try to offer some ideas as to what mathematical communities in the four regions of the UK can do to fix these problems.

Mathematicians get involved in outreach for many different reasons, ranging from the desire to teach people some beautiful mathematics, to the need to communicate important implications of their research to policy-makers, to the joy of performing on stage. The type of mathematics that we can explain depends on both the intended audience and on the format: a book can go on a much longer journey than a lecture. The panellists will discuss which generally comes first for them: content, audience or format, and how their answer to this first question shapes their answers for the other two. 

Panellists: Colva Roney-Dougal (Chair), Chris Budd, Colin Wright, Marianne Freiberger (Plus Magazine)