Elekes and Szabó established non-trivial incidence bounds for binary algebraic relations in characteristic 0, generalizing the Szemerédi-Trotter theorem for point-line-incidence. This was later generalized to binary relations defined in reducts of so-called distal structures in a result of Chernikov, Peterzil and Starchenko. For fields of positive characteristic, such bounds fail to hold in general. Bays and Martin apply the bounds for distal structures in the context of valued fields to derive incidence bounds in the sense of Szemerédi-Trotter in fields admitting valuations with finite residue field, such as F_p(t). We show that this result can be made uniform in the size of the finite residue field, by making precise in some sense the intuition that ACVF is distal relative to the residue field. In this talk, I will introduce the relevant notions from incidence combinatorics and distality, before outlining a proof of the uniform-in-p result.

What is the common theory of all finite fields equipped with an additive and/or multiplicative character? Hrushovski answered this question in the additive case working in (a mild version of) continuous logic. Motivated by natural number-theoretic examples we generalise his results to the case allowing for both (non-trivial) additive character and (sufficiently generic) multiplicative character. Apart from answering the above question we obtain a quantifier elimination result and a generalisation of the definability of the Chatzidakis-Macintyre-van den Dries counting measure to this context. The proof relies on classical results on bounds of character sums following from the work of Weil.

This talk focuses on the logic side of the following result: the non-definability of free independence in the theory of tracial von Neumann algebras and C*-probability spaces. I will introduce continuous model theory, which is suitable for the study of metric structures. Definability in the continuous setting differs slightly from that in the discrete case. I will introduce its definition, give examples of definable sets, and prove an equivalent ultrapower condition of it. A. Berenstein and C. W. Henson exposited model theory for probability spaces in 2023, which was done with continuous model theory. It makes it natural for us to consider the definability of the notion of free independence in probability spaces. I will explain our result, which gives an example of a non-definable set.

This is work with William Boulanger and Emma Harvey, supervised by Jenny Pi and Jakub Curda.