This is joint work with Michael Pinsker. A mixed identity for a group $G$ is a word $w(x_1, \dots, x_r, g_1,\dots, g_n)$ in the language of groups (with variables $x_1,\dots, x_r$ and constants $g_1, \dots, g_n\in G$) such that for any $h_1, \dots, h_r\in G$, $w(h_1, \dots, h_r, g_1,\dots, g_n)=1$. For example, in an Abelian group, $x y x^{-1} y^{-1}$ is a mixed identity (without constants). A mixed identity is singular if forgetting the constants and reducing the resulting resulting word, we get the identity. For example, $x g x^{-1}$ is singular, but $x g x$ is not. Recently, Bodirsky, Schneider, and Thom conjectured that if $G$ is the automorphism group of an $\omega$-categorical structure, then all of its mixed identities are singular. We prove that if $G$ has an action with no algebraicity, then all of its mixed identities are singular. Our result applies to the automorphism groups of a large class of $\omega$-categorical structures, including $(\mathbb{Q}, <)$, for which the aforementioned conjecture was open, but also to several other groups of interest to geometric group theory, such the Thompson group $F$, whose mixed identities were studied in works of Ivanov, Słanina, and Zarzycki, or the homeomorphism groups of manifolds. More generally, we prove that all mixed identities of a group $G$ are singular as long as $G$ admits an action for which algebraic closure forms a modular pregeometry and satisfies a certain higher dimensional variant of Neumann's lemma. This covers also infinite vector spaces over finite fields, whose mixed identities were studied by Bradford, Schneider, and Thom by different methods.
This talk is part of the Lancashire Yorkshire Model Theory Seminar.
Bounding Betti numbers of semialgebraic sets has a 70+ year history where questions are studied because they are interesting in their own right, and also because they have applications in many areas. Indeed, such bounds are crucial in incidence geometry and other closely related areas, especially in tools such as the seminal Polynomial Partitioning theorem proved by Guth and Katz. To push discrete geometry into settings where the sets are no longer semialgebraic, but are definable in any arbitrary o-minimal structure, one needs similar precise bounds on the Betti numbers of definable sets.
I will talk about some of my work along this line. In particular, I will talk about some of my work investigating if an analogue of the Polynomial Partitioning theorem can be established in the o-minimal world. I will then talk about our work in generalizing the polynomial partitioning theorem to settings that involve semi-Pfaffian sets. I will then discuss some avenues for future research that I think are particularly important.
This talk is part of the Lancashire Yorkshire Model Theory Seminar.
(Abelian) $ℓ$-groups are abelian groups equipped with a lattice order compatible with addition. A prototypical example is the additive group of continuous real-valued functions on a topological space.
As well as being an interesting variety of groups in their own right, $ℓ$-groups are also central objects in a wide array of modern mathematics, such as in functional analysis as reducts of rings of real-valued continuous functions on a topological space; and in valuation theory as the value groups of valued fields. In particular, a greater understanding of their model theory should hopefully advance our understanding of these areas also.
Work by Glass and Pierce in 1980 showed that the theory of $ℓ$-groups admits no model companion. In this talk, we will look at recent work of mine which seeks to rectify this problem. In particular, we define a multi-sorted extension for $ℓ$-groups, which behaves like the $ℓ$-group $C(X)$ equipped with the map $P : C(X) → Pow(X)$ sending $f$ to the set $\{x ∈ X | f(x) ⩾ 0\}$.
We will see that this theory admits well-behaved representations, generalising well-known results from the theory of $ℓ$-groups. Further, we will show that this theory is companionable, and this model companion is complete, with quantifier elimination in a small language extension. Time permitting, we will also briefly discuss other work in this area, including a similar result for the case of ordered abelian groups.
This talk is part of the Lancashire Yorkshire Model Theory Seminar.
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
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.
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.
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.
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.
This is a short series of talks on the extended manuscript `Oligomorphic groups and tensor categories’ by Nate Harman and Andrew Snowden 2204.04526. Motivated by representation theory of finite groups and Deligne interpolation, the authors construct concrete pre-Tannakian categories. The methods go via, for certain omega-categorical structures M, constructing measures on Aut(M). I will attempt to give an overview of the work.
