The centre of the universal enveloping algebra of a complex semisimple Lie algebra has been understood for a long time since the pioneering work of Harish-Chandra. In contrast, the centres of the equivalent notions in characteristic p are still yet to be computed explicitly. In this talk, Zhenyu Yang and Rick Chen will present an explicit basis for the centre of the restricted enveloping algebra of sl_2, constructed from explicit calculations combined with techniques from non-commutative rings and Morita equivalences. They will then explain how to generalise the argument to compute the centre of the distribution algebra of the second Frobenius kernel of the algebraic group SL_2. This work was part of their summer project under the supervision of Konstantin Ardakov.

Adam Jones will explore the relationship between the category of smooth representations of a semisimple p-adic Lie group G and the module category over its associated pro-p Iwahori-Hecke algebra via the canonical invariance adjunction. This relationship is well understood in characteristic 0, in fact it yields a category equivalence equivalence, but in characteristic p it is very mysterious and largely defies understanding. We will explore methods of constructing an appropriate subcategory of Hecke modules which is well behaved under the adjunction, and which can be shown to contain all parabolic inductions. He will give examples of this yielding results when G has rank 1, and more recently when G = SL_3 in certain cases.

Introduced by Lusztig in the eighties, character sheaves are the geometrical counterpart of irreducible representations of finite groups of Lie type. Defined over algebraic groups, they allow us to use geometrical tools to deduce information on the finite groups. In this talk, Marie Roth will give a definition of character sheaves before explaining how to compute their restriction to conjugacy classes (to some extent). This work was part of her PhD thesis under the supervision of Olivier Dudas and Gunter Malle.

In this talk, David Luo will use type theory to construct a family of depth $\frac{1}{N}$ minimax supercuspidal representations of $p$-adic $\text{GL}(2N, F)$ which we call \textit{middle supercuspidal representations}. These supercuspidals may be viewed as a natural generalization of simple supercuspidal representations, i.e. those supercuspidals of minimal positive depth. Via explicit computations of twisted gamma factors, David will show that middle supercuspidal representations may be uniquely determined through twisting by quasi-characters of $F^{\times}$ and simple supercuspidal representations of $\text{GL}(N, F)$. Furthermore, David poses a conjecture which refines the local converse theorem for general supercuspidal representations of $\text{GL}(n, F)$.