Programme

Benjamin Siskind, Uniqueness of the core model

Assuming there is no inner model with a Woodin cardinal, Jensen and Steel identified their core model, a canonical inner model which is close to the full universe of sets. Earlier work of Steel identified the core model under the additional assumption that there is a measurable cardinal. In either context, the definition of the core model is very technical, relying on the full lexicon of inner model theory. We will give a simple definition which works in the presence of a proper class of measurable cardinals. (The other definitions are still essential for working with the core model, however!) Whether there is such a simple definition without the measurables is open. This is because some important properties of the core model have only been verified in the presence of a measurable cardinal. In ongoing work with Jan Kruschewski, we are re-establishing some of these properties. For example, we have shown that the core model is rigid, which was proven by Steel in the presence of a measurable cardinal.

Aleksandra Kwiatkowska, Universal homogeneous two-sorted ultrametric spaces

We view ultrametric spaces as two-sorted structures consisting of a set of points and of a linearly ordered set of distances. We call the appropriate notion of embeddings distance-carrying (dc- for short). Those are obtained by combining isometries and linear order embeddings. The class of all such finite spaces is Fraïssé. We study properties of its limit U, its Cauchy completion, as well as of their automorphism groups.

The space U is dc-universal for all countable ultrametric spaces, and its Cauchy completion is dc-universal for all separable ultrametric spaces, which is in contrast with the situation of classical ultrametric spaces and isometric embeddings, where no such universal space can exist.

We characterize the automorphism group of U as the semidirect product of a group of order preserving bijections and a group of isometries. We prove that Aut(U) has a comeager conjugacy class but no generic pair. Furthermore, we show universality of Aut(U) and identify its universal minimal flow.

This is joint work with Bartoš, Kubiś, and Malicki.

Hausdorff Medal Ceremony and Lecture Farmer Schlutzenberg, 

Tom Benhamou, The PCF theory of directed sets

The two main branches in the study of cofinality are the Tukey order and PCF theory, both of which have a wide range of applications. While the Tukey order has been studied in a general framework, PCF theory mostly revolves around products of cardinals below a singular cardinal. Recently, Gartside–Mamatelashvili, Gilton, and Gartside–Gilton studied certain relations between the Tukey type of a product of cardinals and PCF theory.

In this talk, we establish fundamental connections between PCF theory and Tukey order theory for arbitrary sets. In particular, we study the Tukey and depth spectra of a directed set—these are associated sets of regular cardinals, introduced by Tukey, Isbell, and Schmidt in the 1960s, which can be viewed as a measure of cofinal complexity. We show how PCF structures appear naturally in general directed sets and prove several closure properties of the point spectrum. We also provide some applications to ultrafilter theory. 

Concerning regular limit points, we will present recent and ongoing joint work with Hannes Jakob, we provide a counterexample to an old theorem of Isbell and improve several other results.

Siiri Kivimäki, On universality problems in non-elementary classes

A universal object of a class is an object into which any other member of the class embeds. We develop a way to force a universal object in some non-elementary classes consisting of uncountable objects, such as wide Aronsazjn trees, certain type of graphs and certain type of linear orders. This is joint work with Omer Ben-Neria, Menachem Magidor and Jouko Väänänen.

Early career Award Lecture Jan Grebik, From descriptive to distributed

In the past couple of years a rich connection has been found between the fields of descriptive set theory and distributed computing. Frequently, and less surprisingly, finitary algorithms can be adopted to the infinite setting, resulting in theorems about infinite, definable graphs. In this talk, I will discuss how results and ideas from descriptive set theory provide new insights and techniques to the theory of distributed computing. The talk is based on a survey paper written jointly with Zoltan Vidnyánszky.

Monroe Eskew, Some higher forcing axioms

Abstract: The effort to generalize iteration theorems and forcing axioms to higher cardinals faces significant barriers. For example, the axiom for meeting aleph_2 many dense subsets of any countably closed, aleph_2-cc forcing is inconsistent. Moreover, naive generalizations of classical iteration theorems provably fail at higher cardinals, with problems arising at low-confinality limit stages. In this talk I present results from two projects that address these issues in different ways. The first is joint work with Curial Gallart in which we introduce a fairly simple notion of “exact properness” that strengthens strong properness (for some class of models) and turns out to be iterable at higher cardinals. It includes many natural examples, but it doesn’t carry large cardinal strength.  The second is joint work with Rahman Mohammadpour that bypasses the trouble with iterations and instead introduces a “poset reflection principle” that has many consequences similar to those of PFA. 

Piotr Borodulin-Nadzieja, Generalizations of ultrafilters

I will overview some recent results concerning objects which may be considered as generalizations of ultrafilters: measures and Boolean homomorphisms into complete Boolean algebras. In particular, I will discuss how different properties of ultrafilters may be interpreted in the measure context. Also, I will show that those generalizations may help us to handle ultrafilters in forcing extensions.

Jenna Zomback, Chaining in measurable dynamics

We introduce a strengthening of weak mixing, which we call $k$-chaining, for measure class preserving (mcp) Borel actions of countable groups. In this talk, we will define $k$-chaining and place this notion in the hierarchy of properties related to weak mixing.  

For measure preserving (pmp) actions, we show that weak mixing is equivalent to 1-chaining. For general mcp actions, k-chaining, for k≥1, is strictly between double ergodicity and metric ergodicity. As an application, we show that weak mixing is equivalent to 2k-chaining for boundary actions of free groups on k generators with respect to Markov measures. This is joint work with Anush Tserunyan.

William Chan, The ABCD Hypothesis

Among the very few relations between cardinal exponentiations that can be proved using basic set construction principles, one can show that if A, B, C, and D are cardinals, A \leq C, and B \leq D, then then exponentiation {}^A B injects into the exponentiation {}^C D. The ABCD hypothesis essentially states that this is the only possible relationship between infinite cardinal exponentation: If \omega \leq A \leq B  and \omega \leq C \leq D, then {}^A B injects into {}^C D if and only if A \leq C and B \leq D. The ABCD hypothesis yields a complete classification of the cardinality relation between any pair of infinite cardinal exponentiations. The ABCD Conjecture is the assertion that Woodin's extension AD^+ of the axiom of determinacy proves the ABCD hypothesis below Theta. This talk will survey the descriptive set theoretic initial segment of highly regular choiceless universes, sketch the solution to the ABCD Conjecture, and outline other internal structures of this initial segment. 

Jensen Memorial Lecture Ralf Schindler
 

Jonathan Cancino Manriquez, The Halpern-Läuchli theorem and the rationals

The Halpern-Läuchli theorem states that given any finite sequence of finitely branching trees with no leaves, <T_i:i<n>, if we partition their product $\Pi_{i<n} T_i$ into two pieces, then we can find strong subtrees whose level product on some infinite set is contained in one of the pieces of the partition. We will talk about one version of the Halpern-Lauchli theorem which involves the ideal of nowhere dense subsets of the rationals, and investigate the question of whether the set of homogeneous levels can be obtained to be somewhere dense. 

This is joint work with Osvaldo Guzmán.

Andrea Vaccaro, Hyper-u-amenability and hyperfiniteness of treeable equivalence relations

I will present the notion of hyper-u-amenability for countable Borel equivalence relations, a property that implies amenability and which is automatic for orbit equivalence relations of continuous amenable actions on sigma-compact Polish spaces, and for orbit equivalence relations of Borel actions of amenable groups. I will then show that hyper-u-amenable, treeable countable Borel equivalence relations are hyperfinite. As corollaries, I will show that, for orbit equivalence relations of free continuous actions of free groups on sigma-compact spaces, measure-hyperfiniteness implies hyperfiniteness, and that the orbit equivalence relation of a Borel action by an amenable group is hyperfinite, if treeable. The material presented is part of a joint work with Petr Naryshkin.

The concept of definability in Set Theory is highly mutable. A striking example of this mutability is the fact that the properties of the
inner model of hereditarily ordinal definable sets can be easily changed by forcing. So if we look for a more robust theory of definability we have to restrict the range of acceptable definitions. Generalized logics give a family of such restrictions of definability. Each such logic L gives rise to an inner model C(L) which is constructed similarly to the constructible universe L , but replacing first order definability by definability by the logic L.

In this tutorial we shall give a survey of the structure of C(L) for some natural generalized logics , L. In particular we shall talk about
the relationship of such models to the canonical inner models for large cardinals. A special attention will be given C(L^aa ) , where L^aa is stationary logic. Namely the logic in which one can express , for a structure A , the property ”There is a club of countable subsets of the model A such that...”

Through we shall have to assume some familiarity with the standard canonical inner models, and techniques like Woodin stationary towers, we shall try to make the talks more accessible, by referring to these techniques as black boxes.
Most of the results we shall talk about are Joint work with Kennedy and Vaananen, as well as Goldberg, Larson, Rajala, Schindler, Steel, Wilson, and Ya’ar.

In these tutorial lectures, we will discuss three classical iterated forcing constructions — creature posets, template iterations, and coherent systems of iterations. The appearance and continued development of these constructions are closely related to key advances in set theory of the reals, providing a finer understanding of the combinatorial structure of the reals.  Three special aspects of the combinatorial cardinal characteristics and the associated combinatorial sets of reals play a central role in these investigations: constellations, strong witnesses, and spectra.