Building on Leo’s talk last week, I will present the full Galois characterisation of henselianity and introduce some of the ‘explicit’ ingredients he referred to during his presentation. In particular, I will describe a Galois cohomology-inspired criterion for distinguishing between different characteristics. I will then outline the full proof of the Galois characterisation of p-adically closed fields, indicating how each of the ingredients enters the argument.

The absolute Galois group of ℚₚ determines its field structure: a field K is p-adically closed if and only if its absolute Galois group is isomorphic to that of ℚₚ. This Galois-theoretic characterisation was proved by Koenigsmann in 1995, building on previous work by Arason, Elman, Jacob, Ware, and Pop. Similar results were obtained by Efrat and further developed in his 2006 book.

Our project aims to provide an optimal proof of this characterisation, incorporating improvements and new developments. These include a revised proof strategy; Efrat's construction of valuations via multiplicative stratification; the Galois characterisation of henselianity; systematic use of the standard decomposition; and the function field analogy of Krasner-Kazhdan-Deligne type. Moreover, we replace arguments that use Galois cohomology with elementary ones.

In this talk, I will focus on two key components of the proof: the construction of valuations from rigid elements, and the role of the function field analogy as developed via the non-standard methods of Jahnke-Kartas.

This is joint work with Jochen Koenigsmann and Benedikt Stock.

We establish that a criterion based on ring-theoretic amenability is both necessary and sufficient for the abelian version of the Elekes-Szabó theorem to be sharp in the case of positive characteristic. Moreover, the criterion is always sufficient. We provide illustrative examples in the theories ACF_p and DCF_0.

I will outline some recent progress in identifying realistic models of particle physics in heterotic string theory, supported by several mathematical and computational advancements which include: analytic expressions for bundle valued cohomology dimensions on complex projective varieties, heuristic methods of discrete optimisation such as reinforcement learning and genetic algorithms, as well as efficient neural-network approaches for the computation of Ricci-flat metrics on Calabi-Yau manifolds, hermitian Yang-Mills connections on holomorphic vector bundles and bundle valued harmonic forms. I will present a proof of concept computation of quark masses in a string model that recovers the exact standard model spectrum and discuss several other models that can accommodate the entire range of flavour parameters observed in the standard model.