Skew braces are a class of algebraic structures introduced by Guarnieri and Vendramin to study set-theoretic solutions of the Yang–Baxter equation. A skew brace consists of a set equipped with two group operations satisfying a compatibility condition known as skew-left distributivity. This framework bridges group theory, ring theory, and mathematical physics, and has attracted growing interest from an algebraic perspective in recent years. Notions such as solvability and nilpotency—particularly (strong) left nilpotency—have been developed and explored within this context.
In this talk, we investigate skew braces under the assumption of finite Morley rank, a model-theoretic concept that generalizes the idea of dimension from algebraic geometry. In particular, we present a complete classification of skew braces of Morley rank at most 3. Additionally, we prove that if both the additive and multiplicative groups of a skew brace are nilpotent, then the skew brace is strongly left nilpotent. Finally, under an assumption concerning the length of chains of left ideals, we show that if both the additive and multiplicative groups are solvable, then the skew brace is weakly solvable.
This talk is part of the Lancashire Yorkshire Model Theory Seminar
In classical set theory, Gödel's constructible universe 𝐿 enjoys strong absoluteness properties and remains unchanged in forcing extensions. However, Heyting-valued forcings portray a very different picture by introducing new non-classical ordinals (i.e. ordinals not linearly ordered by the membership relation), and thus new elements in 𝐿, in intuitionistic extensions that violate the law of excluded middle.
The method of incomparable codings is a family of approaches I developed in my PhD to use such ordinals to control (especially, enlarge) 𝐿. In arXiv:2601.23070 [math.LO], I proved the following theorem: for any set z in a ZFC universe, there is a Heyting-valued extension where its powerset 𝒫(ž) ∈ 𝐿. In this talk, we will provide a sneak peek of this mechanism by building the forcing extension needed for 𝒫(ω) ∈ 𝐿; if time allows, we will briefly talk about technicalities and new developments on how this extends to sets larger than ω.
We will discuss how Chow varieties can be used to control the large-scale behaviour of certain systems of exponential polynomials. This is a work-in-progress talk.
Henselian valued fields have a central place in model theory: they provide natural examples of tame theories in the sense of classification theory, while also offering a rich setting in which to study complexity phenomena in model theory. In this talk, I will survey some fundamental model-theoretic results on Henselian valued fields and present new results from joint work with Paul Wang, in which we generalize results from the algebraically closed valued field setting to the broader Henselian context.
NOTES: Unusual time 16:00. Part of workshop on "Tame Geometry and Combinatorics".
Large fields are an interesting class of fields first introduced by Pop in the 90's for Galois-theoretic reasons. They have subsequently been studied for a variety of reasons. As logicians we are interested in large fields because essentially all known logically tame fields are large. I will discuss recent work on the model theory of large fields, joint with Will Johnson, Chieu-Minh Tran, and Jinhe Ye. Only minimal background in algebra will be assumed.
Recent developments in neostability theory have witnessed the proliferation of versions of Kim's lemma used to characterise ever-wider classes of unstable first-order theories, including NTP_2, NSOP_1, NSOP_4, NBTP, and beyond. These results highlight some limitations of Adler's framework for the study of abstract independence relations and suggest that new tools are at play underneath this diversity. In this talk, I will extend Adler's framework to study Kim's lemma and its consequences in their full generality. After briefly reviewing the role of independence relations in developing the theory of several neostability-theoretic properties, I will reinterpret Kim's lemma as a binary relation between independence relations under minimal assumptions. I will exemplify how this generalises several results from the literature, offer some new results that this framework proves, and mention previously known theorems which can be obtained in a semantic fashion using this technology.
Ultraproducts of rings and their modules have been appearing in various places in representation theory and the theory of tensor categories. We consider how this is reflected in the model theory of modules over the rings and, in particular, we describe the effect on the lattices of pp formulas, definable subcategories, Ziegler spectra and the associated abelian categories of pp-imaginaries. We also look at what happens when we enrich the picture with a monoidal = tensor product structure.
Globally valued fields were introduced by Ben Yaacov and Hrushovski to capture the model theory of 'global fields with heights' in continuous logic. They subsequently proved that the algebraic closure of rational functions, with the usual projective heights, is existentially closed, while Szachniewicz proved the same for the field of algebraic numbers. I will give a minimal introduction to global fields and heights, keeping the presentation as close as possible to valuation theory, and to continuous logic, then explain what existential closedness means and how it relates to existing problems in Diophantine geometry. The talks are very loosely based on a small subsets of the 'foundations' paper by Ben Yaacov, Destic, Hrushovski, Szachniewicz and various other notes.
Globally valued fields were introduced by Ben Yaacov and Hrushovski to capture the model theory of 'global fields with heights' in continuous logic. They subsequently proved that the algebraic closure of rational functions, with the usual projective heights, is existentially closed, while Szachniewicz proved the same for the field of algebraic numbers. I will give a minimal introduction to global fields and heights, keeping the presentation as close as possible to valuation theory, and to continuous logic, then explain what existential closedness means and how it relates to existing problems in Diophantine geometry. The talks are very loosely based on a small subsets of the 'foundations' paper by Ben Yaacov, Destic, Hrushovski, Szachniewicz and various other notes.
