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 UK time, with a 15 minute coffee break before our hour-long seminar at 14:00 UK time — 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:

23rd June 2021
Speaker: Jinhe (Vincent) Ye, Institut de Mathématiques de Jussieu-Paris Rive Gauche
Title: TBA

16th June 2021
Speaker: Sylvy Anscombe, Institut de Mathématiques de Jussieu-Paris Rive Gauche
Title: Some existential theories of fields

Abstract

Building on previous work, I will discuss Turing reductions between various fragments of theories of fields. In particular, we exhibit several theories of fields Turing equivalent to the existential theory of the rational numbers. This is joint work with Arno Fehm.

9th June 2021
Speaker: Vahagn Aslanyan, University of East Anglia
Title: A geometric approach to some systems of exponential equations

Abstract

I will discuss three important conjectures on complex exponentiation, namely, Schanuel’s conjecture, Zilber’s Exponential Algebraic Closedness (EAC) conjecture and Zilber’s quasiminimality conjecture, and explain how those conjectures are related to each other and to the model theory of complex exponentiation. I will mainly focus on the EAC conjecture which states that certain systems of exponential equations have complex solutions. Then I will show how it can be verified for systems of exponential equations with dominant additive projection for abelian varieties. All the necessary concepts related to abelian varieties will be defined in the talk. The analogous problem for algebraic tori (i.e. for usual complex exponentiation) was solved earlier by Brownawell and Masser. If time permits, I will show how our method can be used to give a new proof of their result. This is joint work with Jonathan Kirby and Vincenzo Mantova.

2nd June 2021
Speaker: Jing Zhang, Bar-Ilan University
Title: When does compactness imply guessing?

Abstract

Large cardinal properties, or more generally compactness principles, usually give rise to certain guessing principles. For example, if kappa is measurable, then the diamond principle at kappa holds and if kappa is supercompact, then the Laver diamond principle holds. It is a long-standing open question whether weak compactness is consistent with the failure of diamond. In the 80’s, Woodin showed it is consistent that diamond fails at a greatly Mahlo cardinal, based on the analysis on Radin forcing. It turns out that this method cannot yield significant improvement to Woodin’s result. In particular, we show that in any Radin forcing extension with respect to a measure sequence on kappa, if kappa is weakly compact, then the diamond principle at kappa holds. Despite the negative result, there are still some positive results obtained by refining the analysis of Radin forcing, demonstrating that diamond can fail at a strongly inaccessible cardinal satisfying strong compactness properties. Joint work with Omer Ben-Neria.

26th May 2021 (will be at 16:45 BST)
Speaker: Nam Trang, University of California, Irvine
Title: Sealing of the Universally Baire sets
Slides

Abstract

A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. Sealing is a type of generic absoluteness condition introduced by H. W. Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by set forcings. The Largest Suslin Axiom (LSA) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. LSA-over-uB is the statement that in all (set) generic extensions there is a model of LSA whose Suslin, co-Suslin sets are the universally Baire sets.
The main result connecting these notions is: over some mild large cardinal theory, Sealing is equiconsistent with LSA-over-uB. As a consequence, we obtain that Sealing is weaker than the theory “ZFC+there is a Woodin cardinal which is a limit of Woodin cardinals”. This significantly improves upon the earlier consistency proof of Sealing by Woodin and shows that Sealing is not a strong consequence of supercompactness as suggested by Woodin’s result.
We discuss some history that leads up to these results as well as the role these notions and results play in recent developments in descriptive inner model theory, an emerging field in set theory that explores deep connections between descriptive set theory, in particular, the study of canonical models of determinacy and its HOD, and inner model theory, the study of canonical inner models of large cardinals. Time permitted, we will sketch proofs of some of the results.
This talk is based on joint work with G. Sargsyan.

19th May 2021
Speaker: Dorottya Sziráki, Alfréd Rényi Institute of Mathematics
Title: The open dihypergraph dichotomy and the Hurewicz dichotomy for generalized Baire spaces

Abstract

Generalized descriptive set theory studies analogues, associated to uncountable regular cardinals $\kappa$, of well known topological spaces such as the real line, the Cantor space and the Baire space. A canonical example is the generalized Baire space ${}^\kappa\kappa$ of functions $f:\kappa\to\kappa$ equipped with the ${<}\kappa$-support topology.
The open graph dichotomy for a given set $X$ of reals is a strengthening of the perfect set property for $X$, and it can also be viewed as the definable version of the open coloring axiom restricted to $X$. Raphaël Carroy, Benjamin Miller and Dániel Soukup have recently introduced an $\aleph_0$-dimensional generalization of the open graph dichotomy which implies several well-known dichotomy theorems for Polish spaces.
We show that in Solovay's model, this $\aleph_0$-dimensional open dihypergraph dichotomy holds for all sets of reals. In our main theorem, we obtain a version of this previous result for generalized Baire spaces ${}^\kappa\kappa$ for uncountable regular cardinals $\kappa$. As an application, we derive several versions of the Hurewicz dichotomy for definable subsets of ${}^\kappa\kappa$. This is joint work with Philipp Schlicht.

12th May 2021
Speaker: Ibrahim Mohammed, University of Leeds
Title: Hyperlogarithmic contraction groups

Abstract

Contraction groups are a model theoretic structure introduced by F.V Kuhlmann to help generalise the global behaviour of the logarithmic function on a non-archimedean field. They consist of an ordered abelian group augmented with a map called the contraction which collapses entire archimedean classes to a single point. Kuhlmann proved in his paper that the theory of a particular type of contraction group had quantifier elimination and was weakly o-minimal (so every definable set is the finite union of convex sets and points).
We can go further and ask how a hyperlogarithmic function behaves globally on a non-archimedean field. A hyper logarithm is the inverse of a trans exponential, which is any function that grows faster than all powers of exp. From an appropriate field equipped with a hyperlogarithm, we get a new type of structure with two contraction maps, which we will call 'Hyperlogarithmic contraction groups'. In this talk I will show how the proof for Q.E and weak o-minimality given by Kuhlmann can be adapted to show that Hyperlogrithmic contraction groups also have these properties.

5th May 2021 (will be at 16:45 BST)
Speaker: Natasha Dobrinen, University of Denver
Title:  Ramsey theory on infinite structures
Corrected paper

Abstract

The Infinite Ramsey Theorem says that for any positive integer $n$, given a coloring of all $n$-element subsets of the natural numbers into finitely many colors, there is an infinite set $M$ of natural numbers such that all n-element subsets of $M$ have the same color.  Infinite Structural Ramsey Theory is concerned with finding analogues of the Infinite Ramsey Theorem for Fraisse limits, and also more generally for universal structures.  In most cases, the exact analogue of Ramsey’s Theorem fails.  However, sometimes one can find bounds of the following sort:  Given a finite substructure $A$ of an infinite structure $S$, we let $T(A,S)$ denote the least number, if it exists, such that for any coloring of the copies of $A$ in $S$ into finitely many colors, there is a substructure $S’$ of $S$, isomorphic to $S$, such that the copies of $A$ in $S’$ take no more than $T(A,S)$ colors.  If for each finite substructure $A$ of $S$, this number $T(A,S)$ exists, then we say that $S$ has finite big Ramsey degrees.

In the past six years, there has been a resurgence of investigations into the existence and characterization of big Ramsey degrees for infinite structures, leading to many new and exciting results and methods.  We will present an overview of the area and some highlights of recent work by various author combinations from among Balko, Barbosa, Chodounsky, Coulson, Dobrinen, Hubicka, Konjecny, Masulovic, Nesetril, Patel, Vena, and Zucker.

28th April 2021 (will be at 14:45 UK time)
Speaker: Justine Falque, Université Paris-Sud
Title: Classification of oligomorphic groups with polynomial profiles, conjectures of Cameron and Macpherson.
Slides

Abstract

Let $G$ be a group of permutations of a denumerable set $E$. The profile of $G$ is the function $f$ which counts, for each $n$, the (possibly infinite) number $f(n)$ of orbits of $G$ acting on the $n$-subsets of $E$. When $f$ takes only finite values, $G$ is called oligomorphic.
Counting functions arising this way, and their associated generating series, form a rich yet apparently strongly constrained class. In particular, Cameron conjectured in the late seventies that, whenever the profile $f(n)$ is bounded by a polynomial (we say that $G$ is P-oligomorphic), it is asymptotically equivalent to a polynomial. In 1985, Macpherson further asked whether the orbit algebra of $G$ (a graded commutative algebra invented by Cameron and whose Hilbert function is $f$) was finitely generated.
After providing some context and definitions of the involved objects, this talk will outline the proof of a classification result of all (closed) $P$-oligomorphic groups, of which the conjectures of Cameron and Macpherson are corollaries.
The proof exploits classical notions from group theory (notably block systems and their lattice properties), commutative algebra, and invariant theory. This research was a joint work with Nicolas Thiéry.

24th 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.

17th March 2021
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 $P(\omega)/\operatorname{Fin}$ 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 $\operatorname{Fin}$ is related to the Ellentuck space. In this talk, we present some results obtained.

10th March 2021
Speaker: Dana Bartošová, University of Florida
Title: Universal minimal flows of group extensions

Abstract

Minimal flows of a topological group $G$ are often described as the building blocks of dynamical systems with the acting group $G$. 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.

3rd 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 $\lambda$, such that the forcing adds a new function with domain $\lambda$ 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 $P(\omega)/\operatorname{fin}$ (for which $h$ is the minimum of the spectrum), and use a forcing construction to show that it is consistent that the combinatorial distributivity spectrum of $P(\omega)/\operatorname{fin}$ contains more than one element. This is joint work with Vera Fischer and Wolfgang Wohofsky.

24th February 2021
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 $\kappa_1$ and $\kappa_2$ are the first and second strongly compact cardinals, respectively, and where $\kappa_1$ is least with Mitchell order 1, and $\kappa_2$ is the least with Mitchell order 2.  Another main theorem is that there is a universe where $\kappa_1$ and $\kappa_2$ are the first and second strongly compact cardinals, respectively, with $\kappa_1$ the least measurable cardinal such that $o(\kappa_1) = 2$ and $\kappa_2$ the least measurable cardinal above $\kappa_1$.  This is a joint work in progress with Victoria Gitman and Arthur Apter.

17th February 2021
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 $\kappa_1$ and $\kappa_2$ are the first and second strongly compact cardinals, respectively, and where $\kappa_1$ is least with Mitchell order 1, and $\kappa_2$ is the least with Mitchell order 2.  Another main theorem is that there is a universe where $\kappa_1$ and $\kappa_2$ are the first and second strongly compact cardinals, respectively, with $\kappa_1$ the least measurable cardinal such that $o(\kappa_1) = 2$ and $\kappa_2$ the least measurable cardinal above $\kappa_1$.  This is a joint work in progress with Victoria Gitman and Arthur Apter.

10th 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 $\Sigma_1$ set-recursion.
In 1971 Harvey Friedman introduced power-admissible sets, which are transitive sets modelling principles of $\Sigma_1^P$, roughly $\Sigma_1$ 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.

3rd February 2021
Speaker: Lynn Scow, California State University, San Bernardino
Title: Semi-retractions and preservation of the Ramsey property

Abstract

For structures $A$ and $B$ in possibly different languages we define what it means for $A$ to be a semi-retraction of $B$. An injection $f:A \rightarrow B$ is quantifier-free type respecting if tuples from $A$ that share the same quantifier-free type in $A$ are mapped by $f$ to tuples in $B$ that share the same quantifier-free type in $B$. We say that $A$ is a semi-retraction of $B$ if there are quantifier-free type respecting injections $g: A \rightarrow B$ and $f: B \rightarrow A$ such that $f \circ g : A \rightarrow A$ is an embedding.
We will talk about examples of semi-retractions and give conditions for when the Ramsey property for (the age of) $B$ is inherited by a semi-retraction $A$ of $B$.

27th January 2021
Speaker: Tin Lok (Lawrence) 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.

20th 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.

13th 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.

16th December 2020
Speaker: Francesco Gallinaro
Title: Algebraic flows on tori: an application of model theory

Abstract

A complex torus $T$ is a Lie group which is obtained as a quotient of a finite dimensional complex vector space, $C^g$, by a lattice; so there is a canonical projection map $p$ from $C^g$ into $T$ . If we consider an algebraic subvariety $V$ of $C^g$, then we can ask what the image of $V$ under $T$ looks like: Ullmo and Yafaev proved that if $V$ has dimension 1, then the closure of $p(V)$ in the Euclidean topology is given by a finite union of translates of closed subgroups of $T$, 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 $p(V)$ in terms of finitely many closed subgroups of $T$. In this talk, I'll introduce the problem and describe the main ingredients of the Peterzil-Starchenko proof.

9th 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.

2nd 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 $DCF_0$. In particular, I will explain how recent work joint with Guy Casale and James Freitag on Fuchsian groups (discrete subgroup of $SL_2(\mathbb{R})$) and automorphic functions, has lead to intriguing questions around the $\omega$-categoricity conjecture of Daniel Lascar. This conjecture was disproved in its full generality by James Freitag and Tom Scanlon using the modular group $SL_2(\mathbb{Z})$ and its automorphic uniformizer (the $j$-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.

25th 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 $\operatorname{Aut}(M)$ which fix points on type spaces $S_n(M)$ 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 $M'$ with $\operatorname{Aut}(M')$ fixing points on type spaces.

18th 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.

11th 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.

4th 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.

28th October 2020
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 $TP_2$ and $NSOP1$.

21st 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.

14th 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 $2^\kappa$ nonisomorphic models of size $\kappa$ for any uncountable cardinality $\kappa$. 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).

6th 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 $2^\kappa$ nonisomorphic models of size $\kappa$ for any uncountable cardinality $\kappa$. 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).