Set-theoretic potentialism is the view that the universe of sets is never fully completed but is only given potentially. Tools from modal logic have been applied to understand the mathematics of potentialism. In recent work, Neil Barton and I extended this analysis to class-theoretic potentialism, the view that proper classes are given potentially (while the sets may or may not be fixed).

In this talk, I will survey some results from set-theoretic potentialism. After seeing how the tools apply in that context I will then discuss our work in the class-theoretic context

What does the countable chain condition mean without the axiom of choice? We will discuss several possible definitions, all equivalent in ZFC, none equivalent in ZF(+DC). We will also present two "external" definitions (due to Bukovský and to Mekler) and see how they fit into this picture.

We will show that a ccc forcing can collapse ω₁, and quite possibly be countably closed while doing so. On the other hand, with the "correct definition" of ccc, no cofinalities or cardinals are changed above ω₁. Whether or not ω₁ can be collapsed is open, but we know that would require it to be singular.

This is a joint work with Noah Schweber.

Dependence logic, introduced by Väänänen (2007), is a non-classical logic for reasoning about dependence and independence. The logic extends first-order logic with a new type of atomic formulas, called dependence atoms, to specify explicitly the dependence relation between variables. Dependence logic adopts an innovative semantics, called team semantics (Hodges 1997), in which formulas are evaluated on a model with respect to sets of assignments (called teams), instead of single assignments. Teams are essentially relations on the model. For this reason, dependence logic is equi-expressive with existential second-order logic, and thus not fully axiomatizable. In this talk, I will give a concise introduction to dependence logic, and I will also survey recent developments in finding partial axiomatizations for the logic.

The background for this work includes Freyd's Theorem, in which the unit interval is viewed as a final coalgebra of a certain endofunctor in the category of bipointed sets.  Leinster generalized this to a broad class of self-similar spaces in categories of sets, also characterizing them as topological spaces. Bhattacharya, Moss, Ratnayake, and Rose went in a different direction, working in categories of metric spaces, obtaining the unit interval and the Sierpinski Gasket as a final colagebras in the categories of

bipointed and tripointed metric spaces respectively.  To achieve this they used a Cauchy completion of an initial algebra to obtain the required final coalgebra. In their examples, the iterations of the fractals can be viewed as gluing together a finite number of scaled copies of some set at some finite set of points (e.g. corners of triangles). Here we will expand these ideas to apply to a broader class of fractals, in which copies of some set are glued along segments (e.g. sides of a square). We use the method of completing an initial algebra to obtain a final coalgebra which is Bilipschitz equivalent to the Sierpinski Carpet, and note that this requires substantially different machinery from previous results in order to handle the metric. Time permitting, we will expand on the Sierpinski Gasket results by considering different categories of metric spaces.

Joint work with Larry Moss.

Joint work with Martin Bays and Pierre Simon

Suppose that M is a model of an NIP theory, and X an externally definable subset: for some elementary extension N of M, and some c from N, X = {a\in M : phi(a,c) holds}.

How large should X be to contain an infinite M-definable subset? Chernikov and Simon asked whether aleph1 is enough. I will discuss this question and relate it to questions in model theory and infinite combinatorics.

In this talk, we will consider two aspects in the study of extremal sets of reals, sets like maximal families of eventually different functions, maximal cofinitary groups, or maximal independent families. On one side, we will discuss their spectrum, defined as the set of cardinalities of such families and on the other, the existence of witnesses of optimal projective complexity. We will emphasize recent developments in the area and indicate interesting remaining open questions.

This is work in progress, joint with Nick Ramsey (UCLA).

A conjecture, now disproved by Chernikov, Hrushovski, Kruckman,
Krupinski, Pillay and Ramsey, asked whether any group with a simple
theory is definably amenable.

It is well known that the counting measure on finite fields gives rise
to a non-standard counting measure on pseudo-finite fields (the infinite
models of the theory of finite fields). It was unknown whether other PAC
fields possessed a reasonable measure, and in this talk, we will show
that some of them do, although the measure we define does not have all
the nice properties of a counting measure when the field is not
pseudo-finite.

This result can be used to show that if G is a group
definable in an e-free perfect PAC field, then G is definably amenable.
It extends to groups definable in omega-free PAC fields.

I will also discuss possible extensions to wider classes of perfect PAC
fields.

It's been a couple of years since I've spoken about CW complexes in a Leeds seminar, so for the benefit of newcomers and with apologies to the old hands who've seen it all before, this talk will be about my result on products of CW complexes.  CW complexes are "nice" spaces that are often seen as good spaces to focus on for algebraic topology, avoiding many point-set-theoretic "pathologies".  However, the product (as topological spaces) of two CW complexes need not be a CW complex.  After giving all the necessary definitions I will go through my characterisation of exactly when the product is a CW complex; this characterisation involves the uncountable cardinal b.

Let M be a weakly o-minimal non-valuational structure, and N its canonical o-minimal extension (by Wencel). We prove that every group G definable in M is a dense subgroup of a group K definable in N. As an application, we obtain that G^{​​​00}​​​= G\cap K^{​​​00}​​​, and establish Pillay's Conjecture in this setting: G/G^{​​​00}​​​, equipped with the logic topology, is a compact Lie group, and if G has finitely satisfiable generics, then dim(G/G^{​​​00}​​​)= dim(G).

In my talk, I will survey the main definitions and some important notions concerning these generalised indiscernibles and give some examples of characterising dividing lines by collapsing indiscernibles. Finally, if time permits, I will discuss an application of generalised indiscernibles to higher-arity independence properties, showing that IP_k can be witnessed by formulas in singleton variables if one allows parameters (from some model).

In the 1970s Saharon Shelah initiated a program to develop classification theory for non-elementary classes, and eventually settled on the setting of abstract elementary classes.  For over three decades, limited progress was made, most of which required additional set theoretic axioms. In 2001, Rami Grossberg and I introduced the model theoretic concept of tameness which opened the door for stability results in abstract elementary classes in ZFC.  During the following 20 years, tameness along with limit models have been used by several mathematicians to prove categoricity theorems and to develop non-first order analogs to forking calculus and stability theory, solving a very large number of problems posed by Shelah in ZFC. Recently, Marcus Mazari-Armida found applications to Abelian group theory and ring theory.  In this presentation I will highlight some of the more surprising results involving tameness and limit models.

Many natural set-theoretic structures satisfy the basic axioms of set theory, but not the powerset axiom. These include the collections $H_{\kappa^+}$ of sets whose transitive closure has size at most $\kappa$, forcing extensions of models of ${\rm ZFC}$ by pretame (but not tame) forcing, and first-order models that are morally equivalent to models of the second-order Kelley-Morse set theory (with class choice). It turns out that a reasonable set theory in the absence of the powerset axiom is not simply ${\rm ZFC}$ with the powerset axiom removed. Without the powerset axiom, the Replacement scheme is not equivalent to the Collection scheme, and the various forms of the Axiom of Choice are not equivalent. In this talk, I will give an overview of the properties of a robust set theory without powerset, ${\rm ZFC}^-$, whose axioms are ${\rm ZFC}$ without the powerset axiom, with the Collection scheme instead of the Replacement scheme and the Well-Ordering Principle instead of the Axiom of Choice. While a great deal of standard set theory can be carried out in ${\rm ZFC}^-$, for instance, forcing works mostly as it does in ${\rm ZFC}$, there are several important properties that are known to fail and some which we still don't know whether they hold. For example, the Intermediate Model Theorem fails for ${\rm ZFC}^-$, and so does ground model definability, and it is not known whether ${\rm HOD}$ is definable. I will also discuss a strengthening of ${\rm ZFC}^-$ obtained by adding the Dependent Choice Scheme, and some rather strange ${\rm ZFC}^-$-models.

The automorphism groups of many homogeneous structures (Riemannian symmetric spaces,  projective spaces, trees, algebraically closed fields, Urysohn space etc) are abstractly simple groups - or at least are simple after taking an obvious quotient.

We present criteria to prove simplicity for a broad range of structures based on the notion of stationary independence.

This talk will discuss recent work on Galvin's conjecture in Ramsey theory. I will review the background and discuss previous work on the two dimensional case before focusing on the recent work on dimensions greater than 2. This is joint work with Stevo Todorcevic.

We report on a joint work in progress with Rahman Mohammadpour in which we study the problem of  the possible existence of a universal tree under weak embeddings in the classes of  $\aleph_2$-Aronszajn and wide $\aleph_2$-Aronszajn trees. This problem is more complex than previously thought, in particular it seems not to be resolved under ShFA + CH using the technology of weakly Lipshitz trees. We show that under CH, for a given $\aleph_2$-Aronszajn tree T without a weak ascent path, there is an $\aleph_2$-cc countably closed forcing forcing which specialises T and adds an $\aleph_2$-Aronszajn tree which does not embed into T. One cannot however apply the ShFA to this forcing.

Further, we construct a model à la Laver-Shelah in which there are $\aleph_2$-Aronszajn trees, but none is universal. Work in progress is to obtain an analogue for universal wide $\aleph_2$-Aronszajn trees. We also comment on some negative ZFC results in the case that the embeddings are assumed to have a strong preservation property.

A branchwise-real tree order is a partial order tree in which every branch is isomorphic to a real interval. In this talk, I give several methods of constructing examples of these which are rigid (i.e. without non-trivial automorphisms), subject to increasing uniformity conditions. I show that there is a rigid branchwise-real tree order in which every branching point has the same degree, one in which every point is branching and of the same degree, and finally one in which every point is branching of the same degree and which admits no monotonic function into the reals. Trees are grown iteratively in stages, and a key technique is the construction (in ZFC) of a family of colourings of (0,infty) which is 'sufficiently generic', using these colourings to determine how to proceed with the construction.

For any field $K$, we introduce natural topologies on $K$-points of varieties over $K$, which is defined to be the weakest topology such that étale morphisms are open. This topology turns out to be natural in a lot of settings. For example, when $K$ is algebraically closed, it is easy to see that we have the Zariski topology, and the procedure picks up the valuation topology in many henselian valued fields. Moreover, many topological properties correspond to the algebraic properties of the field. As an application of this correspondence, we will show that large stable fields are separably closed. Joint work with Will Johnson, Chieu-Minh Tran, and Erik Walsberg.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 $B$.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 $NSOP1$.

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.

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).

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).