This is the (temporary) home of the Leeds University Models and Sets seminar series organised by myself and Ibrahim Mohammed. We meet every Wednesday of term at 13:45 GMT, with a 15 minute coffee break before our hour-long seminar at 14:00 GMT — due to time zone differences, some talks will be scheduled at different times, so please check the details below.

The seminar is informal and friendly and audience members are encouraged to participate and ask questions. All are welcome – email me at b.adam-day ‘at’ leeds.ac.uk to be added to the mailing list and for access to the virtual meeting.

The speakers for this academic year are as follows:

**24 ^{th} March 2021**

Speaker: Silvia Barbina, The Open University

Title: Model theory of Steiner triple systems

Abstract: A Steiner triple system (STS) is a set together with a collection B of subsets of size 3 such that any two elements of the set belong to exactly one subset in B. Finite STSs are well known combinatorial objects for which the literature is extensive. Far fewer results have been obtained on their infinite counterparts, which are natural candidates for model-theoretic investigation. I shall review some constructions of infinite STSs, including the Fraïssé limit of the class of finite STSs. I will then give an axiomatisation of the theory of the Fraïssé limit and describe some of its properties. This is joint work with Enrique Casanovas.

**17 ^{th} March 2021** (will be at 14:45 GMT)

Speaker: Sonia Navarro Flores, Universidad Nacional Autónoma de México

Title: Ramsey spaces and Borel ideals

Abstract: It is known that the Ellentuck space, which is forcing equivalent to the Boolean algebra forces a selective ultrafilter. The Ellentuck space is the prototypical example of a Ramsey space. The connection between Ramsey spaces, ultrafilters, and ideals has been explored in different ways. Ramsey spaces theory has shown to be crucial to investigate Tukey order, Karetov order, and combinatorial properties. This is why we investigate which ideals are related to a Ramsey space in the same sense that the ideal is related to the Ellentuck space. In this talk, we present some results obtained.

**10 ^{th} March 2021**

Speaker: Dana Bartošová, University of Florida

Title: Universal minimal flows of group extensions

Abstract: Minimal flows of a topological group are often described as the building blocks of dynamical systems with the acting group . The universal minimal flow is the most complicated one, in the sense that it is minimal and admits a homomorphism onto any minimal flow. We will study how group extensions interact with universal minimal flows, in particular extensions of and by a compact group.

**3 ^{rd} March 2021**

Speaker: Marlene Koelbing, Universität Wien

Title: Distributivity spectrum of forcing notions

Slides

Abstract: In my talk, I will introduce two different notions of a spectrum of distributivity of forcings. The first one is the fresh function spectrum, which is the set of regular cardinals , such that the forcing adds a new function with domain all whose initial segments are in the ground model. I will provide several examples as well as general facts how to compute the fresh function spectrum, also discussing what sets are realizable as a fresh function spectrum of a forcing.

The second notion is the combinatorial distributivity spectrum, which is the set of possible regular heights of refining systems of maximal antichains without common refinement. We discuss the relation between the fresh function spectrum and the combinatorial distributivity spectrum. We consider the special case of (for which is the minimum of the spectrum), and use a forcing construction to show that it is consistent that the combinatorial distributivity spectrum of contains more than one element. This is joint work with Vera Fischer and Wolfgang Wohofsky.

**24 ^{th} February 2021** (will be at 16:45 GMT)

Speaker: Erin Carmody, Fordham College

Title: The relationships between measurable and strongly compact cardinals. (Part 2)

Abstract: This talk is about the ongoing investigation of the relationships between measurable and strongly compact cardinals. I will present some of the history of the theorems in this theme, including Magidor’s identity crisis, and give new results. The theorems presented are in particular about the relationships between strongly compact cardinals and measurable cardinals of different Mitchell orders. One of the main theorems is that there is a universe where and are the first and second strongly compact cardinals, respectively, and where is least with Mitchell order 1, and is the least with Mitchell order 2. Another main theorem is that there is a universe where and are the first and second strongly compact cardinals, respectively, with the least measurable cardinal such that and the least measurable cardinal above . This is a joint work in progress with Victoria Gitman and Arthur Apter.

**17 ^{th} February 2021** (will be at 16:45 GMT)

Speaker: Erin Carmody, Fordham College

Title: The relationships between measurable and strongly compact cardinals. (Part 1)

Abstract: This talk is about the ongoing investigation of the relationships between measurable and strongly compact cardinals. I will present some of the history of the theorems in this theme, including Magidor’s identity crisis, and give new results. The theorems presented are in particular about the relationships between strongly compact cardinals and measurable cardinals of different Mitchell orders. One of the main theorems is that there is a universe where and are the first and second strongly compact cardinals, respectively, and where is least with Mitchell order 1, and is the least with Mitchell order 2. Another main theorem is that there is a universe where and are the first and second strongly compact cardinals, respectively, with the least measurable cardinal such that and the least measurable cardinal above . This is a joint work in progress with Victoria Gitman and Arthur Apter.

**10 ^{th} February 2021**

Speaker: Adrian Mathias, Université de la Réunion

Title: Power-admissible sets and ill-founded omega-models

References

Abstract: In the 1960s admissible sets were introduced which are transitive sets modelling principles of set-recursion.

In 1971 Harvey Friedman introduced power-admissible sets, which are transitive sets modelling principles of , *roughly* recursion in the power-set function.

Several decades later I initiated the study of provident sets, which are

transitive sets modelling principles of rudimentary recursion. Over the last fifty-odd years several workers have found that ill-founded omega-models, the axiom of constructibility and techniques from proof theory bring unexpected insights into the structure of these

models of set-recursion.

In this talk I shall review these results and the methods of proof.

**3 ^{rd} February 2021** (will be at 14:45 GMT)

Speaker: Lynn Scow, California State University, San Bernardino

Title: Semi-retractions and preservation of the Ramsey property

Abstract: For structures and in possibly different languages we define what it means for to be a semi-retraction of . An injection is quantifier-free type respecting if tuples from that share the same quantifier-free type in are mapped by to tuples in that share the same quantifier-free type in . We say that *is a semi-retraction of* if there are quantifier-free type respecting injections and such that is an embedding.

We will talk about examples of semi-retractions and give conditions for when the Ramsey property for (the age of) is inherited by a semi-retraction of .

**27 ^{th} January 2021** (will be at 11:45 GMT)

Speaker: Tin Lok Wong, National University of Singapore

Title: Arithmetic under negated induction

Slides

Abstract: Arithmetic generally does not admit any non-trivial quantifier elimination. I will talk about one exception, where the negation of an induction axiom is included in the theory. Here the Weak Koenig Lemma from reverse mathematics arises as a model completion.

This work is joint with Marta Fiori-Carones, Leszek Aleksander Kolodziejczyk and Keita Yokoyama.

**20 ^{th} January 2021**

Speaker: Rehana Patel, African Institute for Mathematical Sciences Senegal

Title: Combining logic and probability in the presence of symmetry

Abstract: Among the many approaches to combining logic and probability, an important one has been to assign probabilities to formulas of a classical logic, instantiated from some fixed domain, in a manner that respects logical structure. A natural additional condition is to require that the distribution satisfy the symmetry property known as exchangeability. In this talk I will trace some of the history of this line of investigation, viewing exchangeability from a logical perspective. I will then report on the current status of a joint programme of Ackerman, Freer and myself on countable exchangeable structures, rounding out a story that has its beginnings in Leeds in 2011.

**13 ^{th} January 2021**

Speaker: Salma Kulhmann, University of Konstanz

Title: Strongly NIP almost real closed fields

Slides

Abstract: The following conjecture is due to Shelah: Any infinite strongly NIP field is either real closed, algebraically closed, or admits a non-trivial definable henselian valuation, in the language of rings. We specialise this conjecture to ordered fields in the language of ordered rings, which leads towards a systematic study of the class of strongly NIP almost real closed fields. As a result, we obtain a complete characterisation of this class. The talk is based on joint work with Lothar Sebastian Krapp and Gabriel Lehéricy, which is to appear in the Mathematical Logic Quarterly.

**16 ^{th} December 2020**

Speaker: Francesco Gallinaro

Title: Algebraic flows on tori: an application of model theory

Abstract: A complex torus is a Lie group which is obtained as a quotient of a finite dimensional complex vector space, , by a lattice; so there is a canonical projection map from into . If we consider an algebraic subvariety of , then we can ask what the image of under looks like: Ullmo and Yafaev proved that if has dimension 1, then the closure of in the Euclidean topology is given by a finite union of translates of closed subgroups of , and conjectured that this should hold in higher dimensions. Using model theoretic methods, Peterzil and Starchenko showed that this conjecture isn’t quite true, but that a similar, slightly more complicated statement holds, describing the closure of in terms of finitely many closed subgroups of . In this talk, I’ll introduce the problem and describe the main ingredients of the Peterzil-Starchenko proof.

**9 ^{th} December 2020**

Speaker: Kaethe Minden, Bard College at Simon’s Rock

Title: Split Principles and Large Cardinals

Slides

Abstract: The original split principle is an equivalent formulation of a cardinal failing to satisfy the combinatorial essence of weak compactness. Gunter Fuchs and I expanded the notion in order to characterize the negation of other large cardinal properties. These split principles give rise to seemingly new large cardinals. In this talk I plan to introduce split principles and potentially compare them with flipping properties, which are another way to characterize various large cardinal properties.

**2 ^{nd} December 2020**

Speaker: Ronnie Nagloo, Bronx Community College, City University of New York

Title: Geometric triviality in differentially closed fields

Slides

Abstract: In this talk we revisit the problem of describing the ‘finer’ structure of geometrically trivial strongly minimal sets in . In particular, I will explain how recent work joint with Guy Casale and James Freitag on Fuchsian groups (discrete subgroup of ) and automorphic functions, has lead to intriguing questions around the -categoricity conjecture of Daniel Lascar. This conjecture was disproved in its full generality by James Freitag and Tom Scanlon using the modular group and its automorphic uniformizer (the -function). I will explain how their counter-example fits into the larger context of arithmetic Fuchsian groups and has allowed us to ‘propose’ refinements to the original conjecture.

**25 ^{th} November 2020**

Speaker: Rob Sullivan, Imperial College London

Title: Type spaces, Hrushovski constructions and giraffes

Abstract: The KPT correspondence established a connection between extreme amenability of automorphism groups of first-order structures and Ramsey theory. In this talk, I will consider automorphism groups which fix points on type spaces via a natural action. We will explore a few examples from a combinatorial perspective, and building on work of Evans, Hubicka and Nesetril, we will see that there is an omega-categorical structure M which does not have any omega-categorical expansion with fixing points on type spaces.

**18 ^{th} November 2020**

Speaker: Vincenzo Mantova, University of Leeds

Title: Proving o-minimality of real exponentiation with restricted analytic functions (Part 2)

Slides

Abstract: The o-minimality of real exponentiation with restricted analytic functions is one of the most applied model theoretic results. I’ll discuss the key steps of the van den Dries-Macintyre-Marker proof, based on (1) quantifier elimination for restricted analytic functions, (2) the interplay between analytic functions and the Archimedean valuation, and (3) Hardy fields.

The talk will be self-contained, but it’s also meant to be a conclusion to our summer reading of van den Dries-Macintyre-Marker.

**11 ^{th} November 2020**

Speaker: Laura Fontanella, Université Paris-Est Créteil

Title: Realizability and the Axiom of Choice

Slides

Abstract: Realizability aims at extracting the computational content of mathematical proofs. Introduced in 1945 by Kleene as part of a broader program in constructive mathematics, realizability has later evolved to include classical logic and even set theory. Recent methods that generalize the technique of Forcing led to define realizability models for the theory ZF, but realizing the Axiom of Choice remains problematic. After a brief presentation of these methods, we will discuss the major obstacles for realizing the Axiom of Choice and I will present my recent joint work with Guillaume Geoffroy that led to realize weak versions of the Axiom of Choice.

**4 ^{th} November 2020 **

Speaker: Vincenzo Mantova, University of Leeds

Title: Proving o-minimality of real exponentiation with restricted analytic functions (Part 1)

Slides

Abstract: The o-minimality of real exponentiation with restricted analytic functions is one of the most applied model theoretic results. I’ll discuss the key steps of the van den Dries-Macintyre-Marker proof, based on (1) quantifier elimination for restricted analytic functions, (2) the interplay between analytic functions and the Archimedean valuation, and (3) Hardy fields.

The talk will be self-contained, but it’s also meant to be a conclusion to our summer reading of van den Dries-Macintyre-Marker.

**28 ^{th} October 2020**

Speaker: Zaniar Ghadernezhad, Imperial College London

Title: Building countable generic structures

Slides

Abstract: In this talk I will discuss a new method of building countable generic structures with the algebraic closure property. This method generalises the well-known methods of Fraïssé and Hrushovski pre-dimension construction. I will start with an overview of the construction method of Fraïssé-Hrushovski and then as an application of the new method I will construct a generic non-sparse graph that its automorphism group is not amenable. This method is particularly useful for constructing non-simple generic structures. Time permitting I will explain how to construct non-simple structures with and .

**21 ^{st} October 2020**

Speaker: Andrew Brooke-Taylor, University of Leeds

Title: An introduction to large cardinal axioms

Abstract: Large cardinal axioms are axioms that extend the standard ZFC axioms for set theory in a strong way – they allow you to prove the consistency of ZFC and the large cardinals that came below. I will give a brief survey of these axioms.

**14 ^{th} October 2020**

Speaker: Dugald Macpherson, University of Leeds

Title: Around stability theory (Part 2)

Abstract: Model-theoretic stability theory was developed in the 1970s, with Shelah in a lead role, as providing a notion of ‘tameness’ for first order theories. In particular, uncountably categorical theories are stable, and on the other hand a complete unstable theory over a countable language has nonisomorphic models of size for any uncountable cardinality . Stability can be characterised in many different ways, and stability provides a powerful notion of independence between subsets of a model.

I will give a very informal overview of stability theory, and of some of the generalisations of stability which have been developed more recently (in particular simplicity, and NIP).

**6 ^{th} October 2020**

Speaker: Dugald Macpherson, University of Leeds

Title: Around stability theory (Part 1)

Abstract: Model-theoretic stability theory was developed in the 1970s, with Shelah in a lead role, as providing a notion of ‘tameness’ for first order theories. In particular, uncountably categorical theories are stable, and on the other hand a complete unstable theory over a countable language has nonisomorphic models of size for any uncountable cardinality . Stability can be characterised in many different ways, and stability provides a powerful notion of independence between subsets of a model.

I will give a very informal overview of stability theory, and of some of the generalisations of stability which have been developed more recently (in particular simplicity, and NIP)