Angus Matthews (Leeds)
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.
Models Seminar
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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Floris Vermeulen (KU Leuven)
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.
Andrew Brooke-Taylor (University of Leeds)
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.
Aris Papadopoulos (University of Leeds)
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.
Pablo Andujar Guerrero (University of Leeds)
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.
Dugald Macpherson (Leeds)
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.
Ben De Smet (Leeds)
Location: MALL
Title: The Whitney embedding theorems and o-minimality
The Whitney embedding theorems (95%) and o-minimality (5%).
Pietro Freni (Leeds)
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.