Logic group

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.