An anomalous symmetry of an operator algebra A is a mapping from a group G into the automorphism group of A which is multiplicative up to inner automorphisms. To any anomalous symmetry, there is an associated cohomology invariant in H^3(G,T). In the case that A is the Hyperfinite II_1 factor R and G is amenable, the associated cohomology invariant is shown to be a complete invariant for anomalous actions on R by the work of Connes, Jones, and Ocneanu.

In this talk, I will introduce anomalous actions from the basics discussing examples and the history of their study in the literature. I will then discuss two obstructions to possible cohomology invariants of anomalous actions on simple C*-algebras which arise from considering K-theoretic invariants of the algebras. One of the obstructions will be of algebraic flavour and the other will be of topological flavour. Finally, I will discuss the classification question for certain classes of anomalous actions.

abstract-Oxford.pdf

For a reductive group defined over a p-adic field, the wavefront set is an invariant of an admissible representations which roughly speaking measures the direction of the singularities of the character near the identity. Studied first by Roger Howe in the 70s, the wavefront set has important connections to Arthur packets, and has been the subject of thorough investigation in the intervening years. One of main lines of inquiry is to determine the relation between the wavefront set and the L-parameter of a representation. In this talk we present new results answering this question for unipotent representations with real infinitesimal character. The results are joint with Dan Ciubotaru and Lucas Mason-Brown.

The irreducible smooth representations of p-adic reductive groups and the enhanced Langlands parameters of these latter can both be partitioned into series indexed by "cuspidal data". On the representation side, cuspidality refers to supercuspidal representations of Levi subgroups, while on the Galois side, it refers to "cuspidal unipotent pairs", as introduced by Lusztig, in certain subgroups of the Langlands dual groups.

In addition, on both sides, the elements in a given series are in bijection with the simple modules of a generalized affine Hecke algebra. 

The cuspidal data on one side are expected to be in bijection with the cuspidal data on the other side. We will formulate conditions on this bijection that will guarantee the existence of a bijection between the simple modules of the attached generalized affine Hecke algebras. For the exceptional group of type G_2 and for all pure inner forms of quasi-split classical groups, the Hecke algebras are actually isomorphic.

Let K be an algebraically closed field. Given three elements a Lie algebra over K, we say that these elements form an sl_2-triple if they generate a subalgebra which is a homomorphic image of sl_2(K). In characteristic 0, the Jacobson-Morozov theorem provides a bijection between the orbits of nilpotent elements of the Lie algebra and the orbits of sl_2-triples. In this talk I will discuss the progress made in extending this result to fields of characteristic p, and discuss results for both the classical and exceptional Lie algebras. 

Let G and H be real reductive groups. To any L-homomorphism e: H^L \to G^L one can associate a map e_* from virtual representations of H to virtual representations of G. This map was predicted by Langlands and defined (in the real case) by Adams, Barbasch, and Vogan. Without further restrictions on e, this map can be very poorly behaved. A special case in which e_* exhibits especially nice behavior is the case when H is an endoscopic group. In this talk, I will introduce a more general class of L-homomorphisms that exhibit similar behavior to the endoscopic case. I will explain how this more general notion of endoscopic lifting relates to the theory of cohomological induction. I will also explain how this generalized notion of endoscopic lifting can be used to prove the unitarity of many Arthur packets. This is based on joint work with Jeffrey Adams and David Vogan.

The pro-C completion of a free profinite group on an infinite set of generators is a profinite group of a greater rank. However, it is still not known whether it is a free profinite group too.  We will discuss this question, present a positive answer for some special varieties, and show partial results regarding the general case. In addition, we present the infinite tower of profinite completions, which leads to a generalisation for completions of higher orders. 

The Satake isomorphism is a fundamental result in p-adic groups, and the affine Grassmannian is the natural setting where this geometrizes to the Geometric Satake Correspondence. In fact, it suffices to work with affine Grassmannian slices, which retain all of the information.

Recently, Braverman, Finkelberg, and Nakajima showed that affine Grassmannian slices arise as Coulomb branches of certain quiver gauge theories. Remarkably, their construction works in Kac-Moody type as well. Their work opens the door to studying affine Grassmannians and Geometric Satake Correspondence for Kac-Moody groups. Unfortunately, it is difficult at present to do any explicit geometry with the Coulomb branch definition. For example, a basic feature is that affine Grassmannian slices embed into one another. However, this is not apparent from the Coulomb branch definition. In this talk, I will explain why these embeddings are necessarily subtle. Nonetheless, I will show a way to construct the embeddings using fundamental monopole operators.

This is joint work with Alex Weekes.

The plethysm product on symmetric functions corresponds to composition of polynomial representations of general linear groups. Decomposing a plethysm product into Schur functions, or equivalently, writing the corresponding composition of Schur functors as a direct sum of Schur functors, is one of the main open problems in algebraic combinatorics. I will give an introduction to these mathematical objects emphasising the beautiful interplay between representation theory and combinatorics. I will end with new results obtained in joint work with Rowena Paget (University of Kent) on stability on plethysm coefficients. No specialist background knowledge will be assumed.

Rational Cherednik algebra is a flat deformation of a skew product of the Weyl algebra and a Coxeter group W. I am going to discuss two interesting subalgebras of Cherednik algebras going back to the work of Hakobyan and the speaker from 2015. They are flat deformations of skew products of quotients of the universal enveloping algebras of gl_n and so_n, respectively, with W. They also have to do with particular nilpotent orbits and generalised Howe duality.  Their central quotients can be given as the algebra of global sections of sheaves of Cherednik algebras. The talk is partly based on a joint work with D. Thompson.