In my paper for the special issue on the occasion of the 75th birthday of Boris Zilber ('Model Theory' no. 2, vol. 3, 2024) I show that an analytic continuation property (ACP) for functions definable in a certain o-minimal structure implies his quasi-minimality conjecture, i.e. that every subset of the complex numbers definable in the complex exponential field is either countable or co-countable. In this lecture I shall explain this implication in more detail and point out that that the essential difficulty in establishing the ACP consists in tracking periodicity (in many variables) along unbounded paths in exponential-algebraic curves (everything else can be dealt with using o-minimal theory). This in turn necessitates the study of the range of o-minimally defined functions on subsequences of discrete paths in n-dimensional integer lattices. Unfortunately such subsequences might be too sparse for the usual o-minimal counting arguments to apply. But the situation is rescued by some subtle results in nonstandard analysis obtained in a series of papers by Renling Jin (see BSL no. 3, vol. 6, 2000 for a survey). He studies asymptotic density properties for sequences of integers via Loeb-measure preserving transformations and I hope to give an introduction to this rather neglected (at least by me) topic and to explain its relevance to Zilber's quasi-minimality problem.

Quantifier elimination gives an attractive description of definable in sets in model-theoretically tame fields such as the complex, real or p-adic numbers. However, one can also give a complete description of the much richer class of definable sets in less well-behaved rings, such as the integers, and polynomial rings over the integers, p-adics, or reals. These descriptions tend to involve either computability theory or descriptive set theory. I will discuss some these results, and also talk about the (a priori unrelated) problem of quasi-finite axiomatisability - that is, the problem of axiomatising such rings by single first-order sentences within a suitable class.

The results in this talk are largely due to other people, including my former PhD student Aaron Vollprecht.

There are compelling and long-established connections between automata theory and tame geometry, including Büchi automata and the additive group of real numbers. We say a subset X of the reals is "k-regular" if there is a Büchi automaton that accepts (one of) the base-k representations of every element of X, and rejects the base-k representations of each element in its complement. We say an expansion of the real additive group is k-regular if all of its definable sets are k-regular. In this talk we will discuss the hierarchy of Büchi automata in terms of their expressive power--how many different geometries can a k-regular expansion of the real additive group have? We will unpack the connection between this question and tools from both metric geometry and topological dynamics.

Distal structures are a class of NIP structures isolated by Simon (2013) as being 'purely unstable'. Chernikov and Simon (2015) showed that formulas in distal structures admit 'strong honest definitions', which give rise to a natural form of cell decompositions, leading to an abundance of combinatorial applications. In recent years, often with combinatorial motivations, higher-arity generalisations of NIP and its subclasses have been defined, with Walker (2023) doing so for distality. We discuss one definition of 2-distality and explain how to obtain a corresponding notion of strong honest definitions.

We say that a model complete theory T expanding the theory of fields of characteristic 0 admits generic derivations if T plus the axiom "δ is a derivation" has a model completion Tδg. An algebraically bounded theory T admits generic derivations: in that case, many model theoretic properties of T (NIP, simplicity, uniform finiteness, etc) are inherited by Tδg; a big open problem is to describe the imaginaries of Tδg in terms of the imaginaries of T. We can also characterize exactly which theories admit generic derivations and relate it to the possibility of doing algebraic geometry in a definable way inside T: however, it is not clear in this general situation how much the model theoretic properties of T are inherited by Tδg.

Joint work with Giuseppina Terzo.

We single out a class of definable regular bipartite graphs that admit definable, "close to perfect" matchings. We discuss some consequences for an associated Grothendieck group, as well as specific examples to which our results apply.