The large sieve inequality for Dirichlet characters is a central result in analytic number theory, which encodes a strong orthogonality property between primitive characters of varying conductors. This can be viewed as a statement about $\mathrm{GL}_1$ automorphic representations, and it is a key open problem to prove similar results in the higher $\mathrm{GL}_m$ setting; for $m \ge 2$, our best bounds are far from optimal. We'll outline two approaches to such results (sketching them first in the elementary case of Dirichlet characters), and discuss work-in-progress of Thorner and the author on an improved $\mathrm{GL}_m$ large sieve. No prior knowledge of automorphic representations will be assumed.

A generalisation of Wiles’ famous modularity theorem, the Fontaine-Mazur conjecture, predicts that two dimensional representations of the absolute Galois group of the rationals, with a few specific properties, exactly correspond to those representations coming from classical modular forms. Under some mild hypotheses, this is now a theorem of Kisin. In this talk, I will explain how one can p-adically interpolate the objects on both sides of this correspondence to construct an eigensurface and “trianguline” Galois deformation space, as well as outline a new approach to proving a theorem of Emerton, that these spaces are often isomorphic.

The class of henselian valued fields with non-discrete value group is not well-understood. In 2018, Koenigsmann conjectured that a list of seven natural axioms describes a complete axiomatisation of $\mathbb{Q}_p^{ab}$, the maximal extension of the $p$-adic numbers $\mathbb{Q}_p$ with abelian Galois group, which is an example of such a valued field. Informed by the recent work of Jahnke-Kartas on the model theory of perfectoid fields, we formulate an eighth axiom (the discriminant property) that is not a consequence of the other seven. Revisiting work by Koenigsmann (the Galois characterisation of $\mathbb{Q}_p$) and Jahnke-Kartas, we give a uniform treatment of their underlying method. In particular, we highlight how this method yields short, non-standard model-theoretic proofs of known results (e.g. finite extensions of perfectoid fields are perfectoid).

Numerically solving PDEs typically requires compressing infinite information into a finite system of algebraic equations. Pragmatically, we usually follow a recipe: “Assume solutions of form X; substitute into PDE Y; discard terms by rule Z.” In contrast, Lanczos’s pioneering “tau method” prescribes modifying the PDE to form an exact finite system. Crucially, any recipe-based method can be viewed as adding a small equation correction, enabling us to compare multiple schemes independently of the solver. 

This talk also addresses a paradox: PDEs often admit infinitely many solutions, but finite systems produce only a finite set. When we include a “small” correction, the missing solutions are effectively hidden. I will discuss how tau methods frame this perspective and outline proposals for systematically studying and optimising various residuals.