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: MALL
Title: The limit of Gödel's first incompleteness theorem
NOTE date and time change: Fri 4pm

In this talk, we discuss the limit of the first incompleteness theorem (G1). It is well known G1 can be extended to both extensions and weak sub-systems of PA. We examine the question: are there minimal theories for which G1 holds. The answer of this question depends on how we define the notion of minimality. We discuss different answers of this question based on varied notions of minimality.

The notion of interpretation provides us a general method to compare different theories in distinct languages. We examine the question: are there minimal theories for which G1 holds with respect to interpretability. It is known that G1 holds for essentially undecidable theories, and there are no minimal essentially undecidable theories with respect to interpretability. G1 holds for effectively inseparable (EI) theories and the notion of effective inseparability is much stronger than essential undecidability. A natural question is: are there minimal EI theories with respect to interpretability? We negatively answer this question and prove that there are no minimal effectively inseparable theories with respect to interpretability: for any EI theory T, we can effectively find a theory which is EI and strictly weaker than T with respect to interpretability. Moreover, we prove that there are no minimal finitely axiomatizable EI theories with respect to interpretability.

Finally, we give a summary of the similarities and differences between logical incompleteness and mathematical incompleteness based on technical evidences and philosophical reflections.

Location: Roger Stevens LT23 (8.23)
Title: Effective Pila–Wilkie bounds for Pfaffian sets with some diophantine applications
NOTE location change: Roger Stevens LT23 (8.23)

Following critical insights of Pila and Zannier, there are by now many applications of model theory to diophantine geometry arising from the celebrated counting theorem of Pila and Wilkie and its variants. The original Pila–Wilkie Theorem bounds the number of rational points of bounded numerator and denominator lying on (the transcendental parts of) sets definable in o-minimal expansions of the real field. However, the proof of this theorem (and that of its variants) does not provide an effective bound, which limits the precision of its applications. I will discuss some joint work with Gal Binyamini, Gareth O. Jones and Harry Schmidt in which we obtained effective forms of the Pila–Wilkie Theorem and its variants for sets definable in various structures described by Pfaffian functions (including an effective Yomdin–Gromov parameterization result for sets defined using restricted Pfaffian functions), and then used these effective estimates to derive several effective diophantine applications, including an effective, uniform Manin–Mumford statement for products of elliptic curves with complex multiplication.

Location: Roger Stevens LT23 (8.23)
Title: Algebraic minimality of automorphism groups of countable homogeneous structures
NOTE location change: Roger Stevens LT23 (8.23)

Permutation groups of a countable set are Hausdorff topological groups with the pointwise convergence topology. A Hausdorff topological group G is minimal if every bijective continuous homomorphism from G to another Hausdorff topological group is a homeomorphism. The Zariski topology is defined in a natural way for any group. However, a permutation group with the Zariski topology is not necessarily a topological group. When the Zariski topology is a topological group then it is minimal. In this talk we investigate the Zariski topology for the automorphism groups of some countable homogeneous structures. This is a joint work with Javier de la Nuez González.