Programme and Speakers

Finite Steiner Triple Systems (STs) are well-known combinatorial objects for which the literature is extensive. An STS is a set $S$ together with a collection $B$ of subsets of $S$ of size $3$ such that any two elements of $S$ belong to exactly one element of $B$.

Recently, the infinite counterparts of these structures have proven interesting from a model-theoretic viewpoint. We will survey recent results in the area, in particular regarding the countable universal homogeneous STS and free constructions of STSs.

Let $\mathbb{R}_{\exp}$ be the real field with the exponential function. This structure satisfies the axioms of ordered fields, the axiom scheme of definable completeness (a first-order analogue of completeness), and an axiom expressing the fact that $\exp$ is a differentiable function satisfying the differential equation $\exp'(x) = \exp(x)$ and $\exp(0) = 1$. Assuming Schanuel’s conjecture and building on the work of Wilkie, we shall consider the problem whether this is a complete axiomatization (work in progress with Marcello Mamino). Related conjectures can be raised for the exponential function restricted to a bounded interval.

A conjecture, now disproved by Chernikov, Hrushovski, Kruckman, Krupinski, Pillay and Ramsey, asked whether any group with a simple theory is definably amenable.

It is well known that the counting measure on finite fields gives rise to a non-standard counting measure on pseudo-finite fields (the infinite models of the theory of finite fields). It was unknown whether other PAC fields possessed a reasonable measure, and in this talk, we will show that some of them do, although the measure we define does not have all the nice properties of a counting measure when the field is not pseudo-finite. This result can be used to show that if G is a group definable in an e-free or omega-free perfect PAC field, then G is definably amenable.

We hope to extend our results to a wider class of perfect PAC fields. I will also discuss the case of groups definable in certain bounded PRC fields.

This is work in progress, joint with Nick Ramsey (Notre Dame).

I will discuss some aspects of Hrushovski’s work on definability patterns (arXiv:1911.01129) and its connections to structural Ramsey theory and the Kechris-Pestov-Todorcevic correspondence. As well as Hrushovski’s paper, the lectures will be based on some notes of Pierre Simon and talks given in a joint London-Prague seminar during 2022.

Suppose $L$ is a first-order language and $T$ is a universal $L$-theory which is irreducible, meaning that its class of models has the Joint Embedding Property. If $V$ is a sort (or collection of sorts) associated to $T$, then Hrushovski shows that there is a canonical minimal Ramsey expansion $T^{Ram}_V$. We will explain what this means, talk about the connections to structural Ramsey theory and KPT and discuss the proof. The construction of $T^{Ram}_V$ depends on Hrushovski’s construction of the core of $T$ for free pattern types at $V$, which is a special case of a more general object. We will focus specifically on explaining what this means in the Ramsey context.