Models is a weekly seminar of model theorists in Leeds, that aims to foster collaboration and engagement in each other's research. Roughly twice a term, Models combines with Sets for a two-hour joint seminar. Please contact Mervyn Tong at mmhwmt (at) leeds.ac.uk if you have any questions.

Time and place: MALL 1, Wednesday 14.00 - 15.00

Current organiser: Mervyn Tong

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Results 1 to 10 of 10

Location: MALL

Title: Existentially closed nilpotent Lie algebras

I will present ongoing work joint with Müller, Ramsey and Siniora. A classical result of Macintyre and Saracino states that the theory of Lie algebras over a fixed field and of bounded nilpotency class does not admit a model-companion. We prove that by letting the field grow (i.e. with a separated sort for the field) the theory of Lie algebras of bounded nilpotency class admits a model-companion and that this theory relates asymptotically to the omega-categorical existentially closed c-nilpotent Lie algebra over a finite field F_p for c<p. We also prove that if the theory of the field is NSOP1 then the theory of the corresponding Lie algebra is NSOP4. We will explain how to get this result via a criterion for NSOP4 which does not use stationary independence relations.

Location: MALL

Title: Countable homogeneous ordered bipartite graphs

Note: this seminar will take place in the MAGIC Room (10.03).

The classification of the countable homogeneous bipartite graphs is rather straightforward; they are empty or complete, perfect matching or its complement, and generic. Chernikov and Kruckmann asked what happens if a linear order is imposed on the structure. This is relevant (and required) in structural Ramsey theory: the Nesetril-Rodl Theorem requires the universe of the structure to be linearly ordered. I present a solution to this, which while not so very complicated, does require some tricks, and a rather longer list of structures. One would really like to address the same question for multipartite graphs. I shall briefly recall the classification of the countable homogeneous multipartite graphs (with a fixed finite number of parts), in joint work with Jenkinson and Seidel. The extension to linearly ordered multipartite graphs would require further work.

Location: MALL

Title: The F-adjacency Graph, Part 2

In this talk, we will construct a locally finite, bounded exponent group with an infinite F-adjacency graph. This resolves an open question of Mayhew.

Location: MALL

Title: Parametrizations in valued fields

In the o-minimal setting, parametrizations of definable sets form a key component of the Pila-Wilkie counting theorem. A similar strategy based on parametrizations was developed by Cluckers-Comte-Loeser and Cluckers-Forey-Loeser to prove an analogue of the Pila-Wilkie theorem for subanalytic sets in $p$-adic fields. In ongoing work with R. Cluckers, P. Cubides-K. and I. Halupczok, we prove the existence of parametrizations for arbitrary definable sets in Hensel minimal fields, leading to a counting theorem in this general context.

Location: Roger Stevens LT 04 (8.04)

Title: "Categorifying" Borel reducibility

NOTE: this is a 2-hour seminar for both model and set theorists.

Please make note of the unusual venue!

Borel reducibility is a framework for comparing the complexities of different equivalence relations, and it has been used to great effect showing that various old classification programmes were impossible tasks. However, these days classification maps are generally expected to be functorial, which the classical Borel reducibility framework takes no account of. After going through preliminaries of the classical set-up, I will present a natural framework of Borel categories and functorial Borel reducibility that remedies this oversight. Notably, many examples of classes of structures that were known to be universal in a Borel reducibility sense - Borel complete - are not universal for our functorial version. I'll give many examples, including new ones for the old hands who've seen me talk about this stuff before. This is joint work with Filippo Calderoni.

Location: MALL

Title: Zarankiewicz’s Problem and Model Theory

NOTE: this is a 2-hour seminar for both model and set theorists.

A shower thought that anyone interested in graph theory must have had at some point in their lives is the following: `How “sparse" must a given graph be, if I know that it has no “dense” subgraphs?’. This curiosity definitely crossed the mind of Polish mathematician K. Zarankiewicz, who asked a version of this question formally in 1951. In the years that followed, many central figures in the development of extremal combinatorics contemplated this problem, giving various kinds of answers. Some of these will be surveyed in the first part of my talk.

So far so good, but this is a model (and set) theory seminar and the title does include the words “Model Theory"… In the second part of my talk, I will discuss how the celebrated Szemerédi-Trotter theorem gave a starting point to the study of Zarankiewicz’s problem in “geometric” contexts, and how the language of model theory has been able to capture exactly what these contexts are. I will then ramble about improvements to the classical answers to Zarankiewicz’s problem when we restrict our attention to one of: (a) semilinear/semibounded o-minimal structures; (b) Presburger arithmetic, and (c) various kinds of Hrushovski constructions. The second hour of the talk will essentially be devoted to proofs. Which of (a),(b), or (c) will occupy the second hour will depend on input from the audience.

The new results appearing in the talk were obtained jointly with Pantelis Eleftheriou.

Location: MALL

Title: Defining definable compactness

Can topological compactness be expressed as a first-order property within tame topology? Let's find out. In this talk I will present various attempts in the literature to capture this notion. We will go over the model theory behind them and present open questions.

Location: MALL

Title: Omega-categorical pseudofinite groups

I will discuss recent joint work with Katrin Tent on omega-categorical groups which are pseudofinite, i.e. satisfy every first order sentence which is true of all finite groups. It is fairly easy to show that they are nilpotent-by-finite, and we conjecture that they are finite-by-abelian-by-finite, and can reduce this to the nilpotent class 2 case. We show that certain class 2 groups constructed by amalgamation are NOT pseudofinite – in particular there is an example with supersimple theory which is not pseudofinite.

Location: MALL

Title: The Whitney embedding theorems and o-minimality

The Whitney embedding theorems (95%) and o-minimality (5%).

Location: MALL

Title: Weakly immediate types and T-convexity

For $T$ an o-minimal theory expanding RCF, a $T$-convex valuation ring on an o-minimal expansion of a RCF is a convex subring closed under continuous $T$-definable functions. This was first defined by Van Den Dries and Leweneberg who proved that the common theory $T_{\mathrm{convex}}$ of the expansions of models of $T$ by a non-trivial $T$-convex valuation ring is complete and weakly o-minimal. One of the key properties of the valuation theory of $T_{\mathrm{convex}}$ for power bounded $T$ is the so called residue-valuation property which can be restated as saying that every model of $T_{\mathrm{convex}}$ has a spherically complete maximal immediate extension. This is known to be false if $T$ defines an exponential. The goal of the talk will be to discuss potential analogues of the residue-valuation property in the exponential context.