The assertion belongs to the representation theory of partially ordered sets, to Non-Hausdorff topology and to domain theory, but is (co-)motivated by model theoretic questions about the analysis of structures that can be seen as global sections of a sheaf (like a ring or like a generalized product in the Feferman-Vaught theorem). I will first explain my interest in the statement of the title and then construct the asserted space in a functorial way.

Local stability has been used in the recent years to treat problems in additive combinatorics. Whilst many of the techniques of geometric stability theory have been generalised to simple theories, there is no local treatment of simplicity. Kaplan and Shelah showed that the theory of the additive group of the integers together with a predicate for the prime integers is supersimple of rank 1, assuming Dickson’s conjecture. We will see how to use their result to deduce that all but finitely many integers belongs to infinitely many arithmetic progressions in the primes, which resonates with previous unconditional work (without assuming Dickson’s conjecture) of van der Corput and of Green. If times permits, we will discuss analogous results asymptotically for finite fields.

The classical sum-product phenomena refers to the fact that for any finite set of natural numbers, either its sum set or its product set is large. Erdös--Szemerédi conjectured a sharp lower bound for the maximum of the two. This conjecture is still open but various weaker versions have been shown. Bays--Breuillard generalized this phenomenon to algebraic groups. Further generalizations have been proved by Chernikov--Peterzil--Starchenko. Both of those groups used a mixture of model theory and incidence geometry. In joint work with Harrison and Mudgal we prove a Bourgain--Chang type result for complex algebraic groups of dimension 1. We use substantially different methods than the previous groups. Time permitting, I will also talk about applications of our methods to a question of Bremner.

We survey some results in the model theory of metric structures related to different generalisations of the classic Ehrenfeucht–Fraïssé game. Namely, we look at a game of length $\omega$ that is used to characterise separable structures up to different notions of approximate isomorphism (such as linear isomorphisms between Banach spaces) in a framework that resembles that of positive bounded formulas. Additionally, we look at the (finite-length) EF game for continuous first-order logic and its variant of Ehrenfeucht's theorem. Last, we mention recent work on game comonads for continuous logic.

Approximate subrings are subsets $A$ of a ring $R$ satisfying \[ A + A + AA \subset F + A \] for some finite $F \subset R$. They encode the failure of sum-product phenomena, much like approximate subgroups encode failure of growth in groups.

I will discuss how approximate subrings mirror approximate subgroups and how model-theoretic tools, such as a stabilizer lemma for approximate subrings due to Krupiński, lead to structural results implying a general, non-effective sum-product phenomenon in arbitrary rings: either sets grow rapidly under sum and product, or nilpotent ideals govern their structure. I will also outline related results for infinite approximate subrings and conjectures unifying known (effective) sum-product phenomena.

Based on joint work with Krzysztof Krupiński.