Location: MALL
Title: Introduction to Graphons
Graphons ("graph"+"functions") are a fairly new piece of technology from combinatorics (they were introduced and developed by C. Borgs, J. T. Chayes, L. Lovász, V. T. Sós, B. Szegedy, and K. Vesztergombi about 15 years ago). The motivation behind graphons is to construct a space of objects which capture the concept of a "limit of a (convergent) sequence of graphs", in the same way that real numbers capture the concept of a limit of a (Cauchy) sequence of rationals. The aim of this (not exactly model theory) talk is to introduce some of the main terminology used in this area, construct the space of graphons and discuss some of the foundational theorems about this space. This talk essentially serves as an introduction to the recent preprint "Graphons arising from graphs definable over finite fields" by Džamonja and Tomašić.

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.

Title: Strongly NIP almost real closed fields
Speaker's homepageSlides

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.

Title: Combining logic and probability in the presence of symmetry

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.

Title: Arithmetic under negated induction
Speaker's homepageSlides

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.

Title: Semi-retractions and preservation of the Ramsey property
Speaker's homepage

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 $B$.

Title: Power-admissible sets and ill-founded omega-models
Speaker's homepageReferences

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.

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

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.

Title: Distributivity spectrum of forcing notions
Speaker's homepageSlides

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.

Title: Universal minimal flows of group extensions
Speaker's homepage

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.