# Impromptu Leeds-McMaster model theory days

- Date
- Wednesday 24 - Friday 26 May, 2023
- Location
- MALL

An informal miniworkshop of early career model theorists in McMaster and (possibly formerly) Leeds. Carefully non-organised by Pantelis Eleftheriou and Vincenzo Mantova.

Online participants can join us on Zoom (registration).

In-person participants from outside Leeds: please drop an email to Pantelis and Vincenzo before Wednesday.

## Schedule:

## Wed 24^{th} 2.00pm: Adele Padgett (McMaster), *Regular solutions of systems of transexponential-polynomial equations*

It is unknown whether there are o-minimal fields that are transexponential, i.e., that define functions which eventually grow faster than any tower of exponential functions. In past work, I constructed a Hardy field closed under a transexponential function $E$ which satisfies $E(x+1) = \exp E(x)$. Since the germs at infinity of unary functions definable in an o-minimal structure form a Hardy field, this can be seen as evidence that the real field expanded by $E$ could be o-minimal. To prove o-minimality, a better understanding of definable functions in several variable is likely needed. I will discuss one approach using a criterion for o-minimality due to Lion. This ongoing work is joint with Vincent Bagayoko and Elliot Kaplan.

## Wed 24^{th} 3.00pm: Ibrahim Mohammed (Leeds), *Asymptotic couples and contraction groups*

## Wed 24^{th} 3.30pm: coffee break

## Wed 24^{th} 4.00pm: Elliot Kaplan (McMaster), *Hilbert polynomials for finitary matroids*

Eventual polynomial growth is a common theme in combinatorics and commutative algebra. The quintessential example of this phenomenon is the Hilbert polynomial, which eventually coincides with the linear dimension of the graded pieces of a finitely generated module over a polynomial ring. A later result of Kolchin shows that the transcendence degree of certain field extensions of a differential field is eventually polynomial. More recently, Khovanskii showed that for finite subsets $A$ and $B$ of a commutative semigroup, the size of the sumset $A+tB$ is eventually polynomial in $t$. I will present a common generalization of these three results in terms of finitary matroids (also called pregeometries). I’ll discuss other instances of eventual polynomial growth (like the Betti numbers of a simplicial complex) as well as some applications to bounding model-theoretic ranks. This is joint work with Antongiulio Fornasiero.

## Thu 25^{th} 2.30pm: Aris Papadopoulos (Leeds), *Monadic NIP in monotone classes of relational structures*

## Thu 25^{th} 3.00pm: Pietro Freni (Leeds), *What should Strong Vector Spaces be?*

Spaces of generalized power series have been important objects in asymptotic analysis and in the algebra and model theory of valued structures ever since the introduction of the first instances of them by Levi-Civita and Hahn.

A key feature in this sort of structures is a notion of formal summability and often "natural" linear maps built in this context (such as derivations) are required to preserve this stronger form of linearity, whence they are called strongly linear.

In the talk we will propose a framework for strong linearity: we will argue about a notion of reasonable category of strong vector spaces (r.c.s.v.) generalizing the usual setting for strong linearity and show that up to equivalence there is a universal locally small r.c.s.v. $\Sigma\mathrm{Vect}$.

We will then give a brief description of a monoidal closed structure for $\Sigma\mathrm{Vect}$ and of the relation $\Sigma\mathrm{Vect}$ has with another reasonable category of strong vector spaces given: the category of linearly topologized vector spaces that are topological unions of their linearly compact subspaces.

## Thu 25^{th} 3.30pm: coffee break

## Thu 25^{th} 4.00pm: Alexi Block-Gorman (McMaster), *Expansions of the group of reals by Büchi-automatic sets*

Büchi automata are the natural extension of finite automata to a model of computation that accepts infinite-length inputs. We say a subset $X$ of the reals is $k$-regular if there is a Büchi automaton that accepts (one of) the base-$k$ representations of every element of $X$, and rejects the base-$k$ representations of each element in its complement. These sets often exhibit fractal-like behavior–e.g., the Cantor set is $3$-regular. Let $V_k$ be a ternary predicate such that $V_k(x,u,d)$ holds if and only if $u$ is an integer power of $k$ and $d$ is the coefficient of the term $u$ in some base-$k$ expansion of $x$. For a fixed $k$ and for each natural number $n$, all of the $k$-regular subsets of Euclidean space definable in the expansion of the ordered additive group of reals by the predicate $V_k$. In this talk, we will discuss the significance of the ordered additive group of reals by $V_k$ (and its reducts) from the perspectives of tame geometry and neostability. We will also discuss ongoing work toward a characterization of the reducts of this structure in terms of definability, neostability, and fractal dimensions.

## Fri 26^{th} 1.30pm: Pablo Andújar-Guerrero (Leeds), *NIP compressible types*

## Fri 26^{th} 2.30pm: coffee break

## Fri 26^{th} 3.00pm: Rosario Mennuni (Pisa), *Some definable types in the wild*

While definable types are usually studied in "tame" contexts, their usefulness and amenability to model-theoretic investigation even "in the wild" is, historically, not a surprise: for instance, Lascar defined the tensor product of definable types by generalising the existing notion on ultrafilters, which may be viewed as (trivially) definable types in the richest possible language on a given set.

By pushing the tensor product forward along the addition, one shows that the usual sum of integers may be extended to the space of ultrafilters over $\mathbb{Z}$, yielding a compact right topological semigroup. The analogous construction also goes through for the product, and these facts had important applications in additive combinatorics and Ramsey theory.

Recently, B. Šobot introduced two (ternary) notions of congruence on the space above. I will talk about joint work with M. Di Nasso, L. Luperi Baglini, M. Pierobon and M. Ragosta, in which the study of these congruences led us to isolate a class of ultrafilters enjoying characterisations in terms of tensor products, directed sets, profinite groups, and more.

Pub and social dinner on Friday 26^{th}.