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: 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: 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: 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: Algebraic flows on tori: an application of model theory
Speaker's homepage
A complex torus $\mathbb{T}$ is a Lie group which is obtained as a quotient of a finite dimensional complex vector space, $\mathbb{C}^g$, by a lattice; so there is a canonical projection map $\pi$ from $\mathbb{C}^g$ into $\mathbb{T}$. If we consider an algebraic subvariety $V$ of $\mathbb{C}^g$, 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 $\pi(V)$ in the Euclidean topology is given by a finite union of translates of closed subgroups of $\mathbb{T}$, 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 $\pi(V)$ in terms of finitely many closed subgroups of $\mathbb{T}$. In this talk, I'll introduce the problem and describe the main ingredients of the Peterzil-Starchenko proof.
Title: Split Principles and Large Cardinals
Speaker's homepageSlides
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.
Location: MALL
Title: The potentialist multiverse of classes
Speaker's homepage
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 ω1can be collapsed is open, but we know that would require it to be singular.
Title: Dependence logic and its axiomatization problem
Speaker's homepage
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.
