C3

TBA

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

TBA

https://woloszyn.org/

TBA

TBA

I will present a recent work with G. Kocharyan, where we show the undecidability of the following two problems: given a finitely generated subgroup G of GL(n,Q), a) determine whether G has a non-identity element whose (i,j) entry is equal to zero, and b) determine whether the stabilizer of a given vector in G is non-trivial. Undecidability of problem b) answers a question of Dixon from 1985. The proofs reduce to the undecidability of the word problem for finitely presented groups.

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.