University of Oxford

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 Q_p^ab, the maximal extension of the p-adic numbers 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 Q_p) and Jahnke-Kartas, we give a uniform treatment of their underlying method. In particular, we highlight how this method can yield short, non-standard model-theoretic proofs of known results (e.g. finite extensions of perfectoid fields are perfectoid).

The celebrated Splitting Theorem by Cheeger-Gromoll states that a manifold with non-negative Ricci curvature which contains a line is isometric to a product, where one of the factors is the real line. A related result was later proved by Kasue. He showed that a manifold with non-negative Ricci curvature and two mean convex boundary components, one of which is compact, is also isometric to a product. In this talk, I will present a variant of Kasue’s result based on joint work with Andrea Mondino. We consider manifolds with non-negative Ricci curvature and disconnected mean convex boundary. We show that if one boundary component is parabolic and convex, then the manifold is a product, where one of the factors is an interval of the real line. The result is an application of recently developed tools in synthetic geometry and exploits the interplay between Ricci curvature and optimal transport.

Jochen Koenigsmann’s Habilitation introduced a classification of fields elementarily characterised by their absolute Galois groups, including two conjecturally empty families. The emptiness of one of these families would follow from a Galois cohomological conjecture concerning radically closed fields formulated by Koenigsmann. A promising approach to resolving this conjecture involves the use of local-global principles in Galois cohomology. This talk examines the conceptual foundations of this method, highlights its relevance to Koenigsmann’s classification, and evaluates existing local-global principles with regard to their applicability to this conjecture.

The study of polynomials whose coefficients lie in a given set $S$ (the most notable examples being $S=\{0,1\}$ or $\{-1,1\}$) has a long history leading to many interesting results and open problems. We begin with a brief general overview of this topic and then focus on the following old problem of Littlewood. Let $A$ be a set of positive integers, let $f_A(x)=\sum_{n\in A}\cos(nx)$ and define $Z(f_A)$ to be the number of zeros of $f_A$ in $[0,2\pi]$. The problem is to estimate the quantity $Z(N)$ which is defined to be the minimum of $Z(f_A)$ over all sets $A$ of size $N$. We discuss recent progress showing that $Z(N)\geqslant (\log \log N)^{1-o(1)}$ which provides an exponential improvement over the previous lower bound. 

A closely related question due to Borwein, Erd\'elyi and Littmann asks about the minimum number of zeros of a cosine polynomial with $\pm 1$-coefficients. Until recently it was unknown whether this even tends to infinity with the degree $N$. We also discuss work confirming this conjecture.

A number of results on classical problems in analytic number theory rely on bounds for multilinear forms of Kloosterman sums, which in turn use deep inputs from the spectral theory of automorphic forms. We’ll discuss our recent work available at arxiv.org/abs/2404.04239, which uses this interplay between counting problems, exponential sums, and automorphic forms to improve results on the greatest prime factor of $n^2+1$, and on the exponents of distribution of primes and smooth numbers in arithmetic progressions.
The key ingredient in this work are certain “large sieve inequalities” for exceptional Maass forms, which improve classical results of Deshouillers-Iwaniec in special settings. These act as on-average substitutes for Selberg’s eigenvalue conjecture, narrowing (and sometimes completely closing) the gap between previous conditional and unconditional results.

How much can the order type - the list of element orders (with multiplicities)—reveal about the structure of a finite group G? Can it tell us whether G is abelian, nilpotent? Can it always determine whether G is solvable? 

This last question was posed in 1987 by John G. Thompson and I answered it negatively this year. The search for a counterexample was quite a puzzle hunt! It involved turning the problem into linear algebra and solving an integer matrix equation Ax=b. This would be easy if not for the fact that the size of A was 100,000 by 10,000…

TBC