Models and Sets Seminar

Location: MALL
Title: Monadic second-order definability of classes of matroids
 

Matroids can be seen as abstractions of geometrical configurations. Every (finite) matroid consists of a (finite) set called the ground set, and a collection of distinguished subsets called the independent sets. A classic example arises when the ground set is a finite set of vectors from a vector space, and the independent subsets are exactly the subsets that are linearly independent. Any such matroid is said to be representable. We can think of a representable matroid as being a geometrical configuration where the points have been given coordinates from a field. Another important class arises when the points are given coordinates from a group. Such a class is said to be gain-graphic.Monadic second-order logic is a natural language for matroid applications. In this language we are able to quantify only over subsets of the ground set. The importance of monadic second-order logic comes from its connections to the theory of computation, as exemplified by Courcelle's Theorem. This theorem provides polynomial-time algorithms for recognising properties defined in monadic second-order logic (as long as we impose a bound on the structural complexity of the input objects).It is natural to ask which classes of matroids can be defined by sentences in monadic second-order logic. When the class consists of the matroids that are coordinatized by a field we have a complete answer to this question. When the class is coordinatized by a group the problem becomes much harder.This talk will contain a brief introduction to matroids.Based on work with Sapir Ben-Shahar, Matt Conder, Daryl Funk, Mike Newman, and Gabriel Verret.

Location: MALL
Title: On stable and distal expansions of $(\mathbb{Z}, +)$ and $(\mathbb{Z}, <, +)$ by a unary predicate
 

Proper expansions of $(\mathbb{Z}, +)$ (respectively $(\mathbb{Z}, <, +)$) by a unary predicate have proven to be a rich source of interesting stable (respectively distal) structures. Precisely which predicates give a stable (respectively distal) expansion remains an open question that, not only is of independent interest, but also has the potential to provide counterexamples to open problems on model-theoretic dividing lines. In this workshop, we survey some results that provide a partial answer to this question and discuss strategies to tackle this and adjacent open problems. In particular, we will discuss the existence of non-distal NIP expansions of $(\mathbb{N}, +)$, and we will consider a candidate counterexample to the conjecture that a structure has a distal expansion if and only if every formula has the strong Erdős-Hajnal property.

Location: MALL
Title: Almost free abelian groups and anti-compactness principles
 

An abelian group is almost free if any subgroup of smaller cardinality is free. We review some theorems mainly by Magidor and Shelah centered around the question of when almost free abelian groups are free. This also serves as an introduction to some anti-compactness principles such as squares.

Location: MALL
Title: Transfer of NIP in finitely ramified henselian valued fields
 

By a technique of Jahnke and Simon, itself building on work of Chernikov and Hils, we investigate the transfer of NIP from the residue field to the valued field, in the case of a finitely ramified henselian valued field of mixed characteristic. This relies on understanding the complete theories of such fields, on a certain Ax--Kochen/Ershov principle, and the stable embeddedness of the residue field. This talk will touch on joint work with Franziska Jahnke, and another project additionally with Philip Dittmann.

Location: MALL
Title: The space of types with a spectral topology
NOTE time change: 4 pm GMT

Influenced by results in real algebraic geometry, Pillay pointed out in 1988 that the space of types of an o-minimal expansion of a real closed field admits a spectral topology. With this topology, this space is quasi-compact and $T_0$, yet not Hausdorff. Nonetheless, the subspace of all closed points turns out to be quasi-compact and Hausdorff. The purpose of this talk is to present some results on the spectral space of types for o-minimal and more generally NIP theories. This is joint work with Elías Baro and José Fernando.

Location: MALL
Title: Small ordered theories with a maximum spectrum of countable models

For theories of totally ordered structures, we consider questions concerning the number of non-isomorphic countable models. The approach is to consider realisations of 1-types, and formulas acting on the set of all realisations of a type.

Location: MALL
Title: Large cardinals for independent families
 

A collection of subsets of the natural numbers is independent if all finite intersections of members of the collection with all finite intersections of complements of members of the intersection is never empty. These are interesting combinatorial families and it is easy enough to prove their existence. Furthermore, by Zorn's Lemma, one can always extend an independent family to be maximal (that is, contained in no further independent family). However, as soon as one tweaks these parameters, specifically the word 'finite', existence becomes much harder to work with. Kunen showed that even having a maximal $\sigma$-independent family (so we take countable intersections) implies that there is an inner model with a measurable cardinal, so this is already not provable from ZFC. Indeed, he also showed that this goes the other way: Beginning with a measurable cardinal, there is a forcing extension in which $2^{\omega_1}$ has a maximal $\sigma$-independent family. Moving up the ladder, one obtains from an $\aleph_1$-strongly compact cardinal a forcing extension in which there are maximal $\sigma$-independent families for a class of cardinals. The talk shall briefly introduce the basics of large cardinals and go over this method of converting large cardinals into maximal independent families.

Location: MALL
Title: Various contractions arising in Natural Ways
 
In the paper "Abelian Groups with Contractions", F.V Kuhlmann introduced the notion of contraction group. They consist of an ordered abelian group along with a unary map which collapses archimedean classes to a single point. The motivation behind them was to axiomatise the action of log on the value group of a non-standard model of $\mathbb{R}_{\text{exp}}$, however there are a few other natural ways in which contraction groups arise.
 
The first is the action of a hyper-logarithm (which can be thought of the composition of log $\omega$ many times) on the value group of a trans-exponential ordered field. The other is the action of the hyper-logaithmic derivative on the same structure. In this talk I'll go through how contraction groups arise in these circumstances, and state various model theoretic results concerning them.

Location: MALL
Title: Measuring sizes without the Axiom of Choice, and how bad can it possibly get?
 

Picture yourself in a boat, on a river, with sets that can't be well-ordered and marmalade skies. How can you decide which ones are big and which ones are bigger? Sure, we can compare them by injections, or by surjections, but these are not total and some sets won't be comparable. One way to get around that is to use the Hartogs and Lindenbaum numbers which help us measure how large a set is, although in different ways, and to an extent, the relationship between them also tells us how far a set is from being well-ordered.

In this talk we'll see the relevant definitions,  and see how to create sets which are both very large, very small, and very far from being well-ordered, and how, as always, the worst can happen if you try. This is a joint work with Calliope Ryan-Smith.

Location: Roger Stevens LT23 (8.23)
Title: O-minimality and finiteness of some atypical intersections for $Y(1)^n$
NOTE location change: Roger Stevens LT23 (8.23)

I will explain how o-minimality can be used to prove that, for a modular function f with Heegner divisor, there are only finitely many $n$-tuples of $f$-images of CM-points which are multiplicatively dependent. I will also discuss the relation between this result and the Zilber–Pink conjecture for atypical intersections.