In representation theory, the characters of induced representations are explicitly known in terms of the character of the inducing representation. This leads to the question of understanding the elliptic representation space, i.e., the space of representations modulo the properly (parabolically) induced characters. I will give an overview of the description of the elliptic space for finite Weyl groups, affine Weyl groups, affine Hecke algebras, and their connection with the geometry of the nilpotent cone of a semisimple complex Lie algebra. These results fit together in the representation theory of semisimple p-adic groups, where they lead to a new description of the elliptic space within the framework of the local Langlands parameterisation.

In this talk I will give a brief introduction to the field of post-quantum (PQ) cryptography, introducing a few of the most popular computational hardness assumptions. Second, I will give an overview of a recent work of mine on PQ electronic voting. I’ll finish by presenting a short selection of ‘exotic’ cryptographic constructions that I think are particularly hot at the moment (no, not blockchain). The talk will be definitionally light since I expect the area will be quite new to many and I hope this will make for a more engaging introduction.

For many spaces of interest to number theorists one can construct cycles which in some ways behave like the coefficients of modular forms. The aim of this talk is to give an introduction to this idea by focusing on examples coming from modular curves and Heegner points and the relevant work of Zagier, Gross-Kohnen-Zagier and Borcherds. If time permits I will discuss generalizations to other spaces.

For a complete theory T, Lascar associated with it a Galois group which we call the Lacsar group. We will talk about some of my work on recovering the Lascar group as the fundamental group of Mod(T) and some recent progress in understanding the higher homotopy groups.

This is a pre-seminar meeting for Margaret Bilu's talk "A motivic circle method", which takes place later in the day at 5PM in L3.

The local Kronecker-Weber theorem states that the maximal abelian extension of p-adic numbers Q_p is obtained from this field by adjoining all roots of unity. In 2018, Koenigsmann conjectured that the maximal abelian extension of Q_p is decidable. In my talk, we will discuss Koenigsmann's proposed axiomatisation. In contrast, the maximal unramified extension of Q_p is known to be decidable, admitting a complete axiomatisation by an informed but simple set of axioms (this is due to Kochen). We explain how the question of completeness can be reduced to an Ax-Kochen-Ershov result in residue characteristic 0 by the method of coarsening.