C3

TBA

https://woloszyn.org/

TBA

TBA

TBA

TBA

Motivated by work of Hrushovski on pseudofinite fields with an additive character we investigate the theory ACFA+ which is the model companion of the theory of difference fields with an additive character on the fixed field. Building on results by Hrushovski we can recover it as the characteristic 0-asymptotic theory of the algebraic closure of finite fields with the Frobenius-automorphism and the standard character on the fixed field. We characterise 3-amalgamation in ACFA+. As cosequences we obtain that ACFA+ is a simple theory, an explicit description of the connected component of the Kim-Pillay group and (weak) elimination of imaginaries. If time permits we present some results on higher amalgamation.

We will decide on speakers for Trinity term 2024.

I will talk about some model-theoretic properties of Booleanizations of theories, subdirect products of structures, and sheaves of structures. I will discuss a result of Macintyre from 1973 on model-completeness, and more recent results jointly with Ehud Hrushovski and with Angus Macintyre.

I will discuss aspects of some work in progress with Tingxiang Zou, in which we continue the investigation of pseudofinite sets coarsely respecting structures of algebraic geometry, focusing on algebraic group actions. Using a version of Balog-Szemerédi-Gowers-Tao for group actions, we find quite weak hypotheses which rule out non-abelian group actions, and we are applying this to obtain new Elekes-Szabó results in which the general position hypothesis is fully weakened in one co-ordinate.

This talk is based on a joint work with Vincent Jinhe Ye. I will define various classes of hyperbolic varieties (Broody hyperbolic, algebraically hyperbolic, bounded, groupless) and discuss existence of model companions of classes of fields that exclude them. This is related to moduli spaces of maps to hyperbolic varieties and to the (open) question whether the above mentioned hyperbolicity notions are in fact equivalent.