Talks

Gabriel Goldberg (UC Berkley)
Mini-course on choiceless large cardinals 

Taking the modern paradigm for formulating large cardinal hypotheses to its logical extreme, one inexorably arrives at the principle that there is a nontrivial elementary embedding from the universe of sets to itself. Reinhardt proposed this principle in the late 1960s and Kunen soon after refuted it using the Axiom of Choice. If one drops the Axiom of Choice, a hierarchy of very strong large cardinal hypotheses beyond the Kunen inconsistency emerges. It remains unclear whether these choiceless large cardinals really are consistent, but the past few years have seen some major advances on this problem, which I will describe in my first talk. In the later talks I will outline certain periodicity properties of the universe of sets under choiceless large cardinal axioms, structural features of the levels $V_\alpha$ of the cumulative hierarchy that vary depending on the parity of the ordinal $\alpha$.
 

Boban Velickovic (Université de Paris)

Mini-course on combinatorial L-forcing

Let us say we are given an $L_{\kappa,\omega}$-theory T (for some uncountable $\kappa$). We would like to know if we can add a model of T by some nice forcing (proper, semiproper, SSP, etc).
If you allow arbitrary forcing the answer is well-known:  player 2 has a winning strategy in the consistency game.
I would like to have an ‘algorithm’ that searches for a nice poset doing this, in particular without collapsing $\omega_1$. As you can guess this is inspired by the Aspero-Schindler result, but also most classical applications of forcing axioms can be seen as adding models for such theories.
The ultimate goal would be to merge forcing axioms with the study of determinacy.

Tom Benhamou  (Tel Aviv University)
The Galvin property at successors of singulars

We present a property of filters discovered by F. Galvin which he proved to hold for normal filters over strongly regular cardinals,
and gained renewed interest due to recent developments in set theory.
We will present some strengthening of his theorem and constructions of filters and ultrafilters without the Galvin property.
In the second part of the talk, we continue the work of U.Abraham and S.Shelah who produced a model where the club filter
fails to satisfy the Galvin property in a strong sense at $\kappa^+$, where $\kappa$ is a regular cardinal and $2^{\kappa}>\kappa^+$.
We will produce a model where the club filter fails to satisfy the Galvin property at $\kappa^+$, where $\kappa$ is singular and $2^{\kappa}>\kappa^+$.
We will obtain this model from the optimal large cardinal assumptions and explore the possibility of obtaining the stronger form of failure as in  the Abraham and Shelah model. 
This is a joint work with M. Gitik, S. Garti and A. Poveda.

James Cummings (Carnegie Mellon University)
Measures on a strong cardinal

We show that it is consistent for the Mitchell order to be linear at a strong cardinal where GCH fails. This is a joint work with Arthur Apter.

Mirna Džamonja (IRIF, CNRS & Université Paris-Cité)

Towards another vision of effectiveness at $\aleph_1$

The first uncountable cardinal does not easily mend itself to methods inherited from the countable. We know this through a whole list of failures of properties such as compactness and Ramsey theorems. Similarly, the descriptive set theory at this level is very different from the classical descriptive set theory and does not really seem to give as much of an idea of effectiveness. We shall propose to look at the effectiveness at $\aleph_1$ from the point of view of automata theory and generalized decidability. In so doing, we shall introduce new classes of automata and consider MSO of trees. 

Todd Eisworth (Ohio University)
On $\clubsuit_{AD}$ and its relatives

We will look at consistency results that complement a recent theorem of Rinot, Shalev, and Todorcevic that instances of the principle $\clubsuit_{AD}$ follow from the Continuum Hypothesis and even weaker assumptions.

Matthew Foreman (University of California, Irvine)
Welch games on larger cardinals

Weakly and Strongly compact and Supercompact cardinals were developed to study the compactness properties of infinitary languages. Welch proposed a game that turned out to have large cardinal properties intermediate between a weakly compact and a measurable cardinal.  This talk discusses the difficulties of generalizing the game to larger cardinals, and makes some conjectures.

Moti Gitik (Tel Aviv University) On Cohens inside Prikry forcing and a question of Woodin

We show that it is possible to add $\kappa^{+}$-Cohen subsets to $\kappa$ with a Prikry forcing over $\kappa$. This answers a question from a paper by T. Benhamou, Y. Hayut and G.. In order to do this we introduce a strengthening of non-Galvin property of ultrafilters which is shown to be consistent using a single measurable cardinal. This improves a previous result by T. Benhamou, S. Garti and S. Shelah. A situation with extender-based Prikry forcings is examined.
Building on these ideas, we show that starting with a hypermeasurable cardinal it is possible to have the following:
there is a submodel $V’$ of a model $V$ such that $V \models cof(\kappa) = \omega + 2^\kappa > \kappa^{++} + GCH_{<\kappa}$,
$V’ \models (\kappa \text{ is a measurable cardinal and } 2^\kappa = (2^\kappa)^V)$. Such $\kappa$ may be $\aleph_{\omega}$ of $V$ .
This gives an affirmative answer to a question of H. Woodin from the early 90-th.
This is a joint work with Tom Benhamou.

Tanmay Inamdar (Bar Ilan University) From Sierpinski-type colourings to Ulam-type matrices

Ulam matrices were introduced by Ulam in his study of the measure problem. Ulam’s construction applies to all infinite successor cardinals, and later Hajnal extended the construction to apply to some limit cardinals as well. Our main result is that such matrices can also be constructed using walks on ordinals, assuming there is a non-trivial C-sequence. The inspiration is the onto mapping principle of Sierpinski. The results I present are joint work with Assaf Rinot.

Steve Jackson (University of North Texas)
New partition properties and applications to cardinalities under AD

Under AD many cardinals have infinite exponent partition relations, and these give rise to measures on the corresponding function spaces. We establish some new properties of these measures and give applications to cardinalities in AD models. In particular, we establish a monotonicity conjecture for these partition measures.

Asaf Karagila (University of Leeds)
How small can the first measurable cardinal be?

We will discuss the smallness of measurable cardinals in ZF. What does it mean to be small, and how small can we go with just a measurable cardinal on our back?

 

Menachem Kojman (Ben Gurion University)
Ramsey theory over partitions

I will present new results on the subject from a recent sequence of three papers with Assaf Rinot and Juris Steprans.

Péter Komjáth (Eötvös Loránd University)
Davies’s theorem and the coloring number of graphs

The theorem referred to in the title is the following: under CH,
if $F:R\times R\to R$ is an arbitrary function, then there exist
functions $g_n,h_n:R\to R$ ($n<\omega$) such that
for all $x$, $y$ one has $F(x,y)=\sum_n g_n(x)h_n(y)$.
A reasonable generalization to graphs is surprisingly connected

to the notion of coloring number.  

Chris Lambie-Hanson (Czech Academy of Sciences)
Two-cardinal combinatorics, guessing models, and cardinal arithmetic

Two-cardinal tree properties were first studied by Jech and Magidor in the 1970s and were used to characterize strongly compact and supercompact cardinals. In the 2000s, Weiss formulated the two-cardinal tree properties (I)TP and (I)SP, which capture many of the combinatorial consequences of strong compactness and supercompactness but also consistently hold at small cardinals. Subsequently, Viale and Weiss showed that at $\omega_2$, ISP, the strongest of these principles, is equivalent to the Guessing Model Property (GMP), which has proven to be a very fruitful set theoretic hypothesis. In this talk, we will discuss some recent results concerning these principles and their variants. We will begin by introducing weakenings of the GMP that are equivalent to SP or even further weakenings thereof but still have considerable combinatorial consequences, for instance implying global failures of square. Then, motivated by the question as to whether (I)TP at a cardinal $\kappa$ implies the Singular Cardinals Hypothesis (SCH) above $\kappa$, we will discuss the consequences of various related two-cardinal combinatorial principles, including generalized narrow system properties and the non-existence of certain strongly unbounded subadditive functions, on cardinal arithmetic, particularly on SCH and Shelah’s Strong Hypothesis. This is joint work with \v{S}\'{a}rka Stejskalov\'{a}. 

Justin Moore (Cornell University)
Higher derived limits and partitions

Recently Bergfalk and Lambie-Hanson showed that in the weakly compact Hechler model, the higher derived limits ${\lim}^n \mathbb{A}$ vanish for all $n$, where $\mathbf{A}$ is a certain inverse system of abelian groups indexed by $\omega^\omega$.  We will show that this result can be seen to factor into a purely set theoretic Partition Hypothesis concerning finite powers of $\omega^\omega$ and

a lemma which is purely algebraic.  The Partition Hypothesis holds in the weakly compact Hechler model and the algebraic lemma is true in ZFC.  This is joint work with Bannister, Bergfalk, and Todorcevic.

Sandra Müller (TU Wien)
Inner Models, Determinacy, and Sealing

Inner model theory has been very successful in connecting determinacy axioms to the existence of inner models with large cardinals and other natural hypotheses. Recent results of Larson, Sargsyan, and Trang suggest that a Woodin limit of Woodin cardinals is a natural barrier for our current methods to prove these connections. One reason for this comes from Sealing, a generic absoluteness principle for the theory of the universally Baire sets of reals introduced by Woodin. Woodin showed in his famous Sealing Theorem that in the presence of a proper class of Woodin cardinals Sealing holds after collapsing a supercompact cardinal. I will outline the importance of Sealing and discuss a new and stationary-tower-free proof of Woodin’s Sealing Theorem that is based on Sargsyan’s and Trang’s proof of Sealing from iterability. This is joint work with Grigor Sargsyan and Bartosz Wcisło.  

Ralf Schindler (Universität Münster)
Partitioning $\mathbb{R}^3$ into circles of radius 1.

In ZFC, $\mathbb R^3$ can be partitioned into pairwise disjoint circles of radius 1. We show that the same is true in the Cohen-Halpern-Levy model. Questions about the transcendence degree of the reals in forcing extensions over various somewhat smaller fields come into play. This is joint work with my student Azul Fatalini.

Farmer Schlutzenberg (Universität Münster)
The extent of determinacy in $\omega$-small mice

Assuming large cardinals, $M_\omega$ denotes the canonical proper class inner model with infinitely many Woodin cardinals. Let $A=R\cap M_\omega$ be the set of reals in $M_\omega$. Then $L(A)$ does not model determinacy. There is an ordinal $\alpha$ such that $L_\alpha(A)$ can be $\Sigma_1$-elementarily embedded into $L(R)$ (hence $L_\alpha(A)$ models determinacy), and there is a wellorder of $A$ which is definable over $L_\alpha(A)$. Rudominer and Steel conjectured that a similar situation holds for many mice lower than $M_\omega$ in the hierarchy. They verified the conjecture or a weaker variant under extra assumptions, but the full conjecture has remained open. We will discuss progress toward a positive resolution of the conjecture. This is joint work with John Steel.

Assaf Shani (Harvard University)
Classifying invariants for $E_1$

We introduce a framework for studying “reasonable” classifying invariants, more permitting than “classification by countable structures”.

This framework respects the intuitions and results about classifications by countable structures, and allows for equivalence relations such as $E_1$ and $E_1^+$ to be “reasonably classifiable” as well. In this framework we show that $E_1$ has classifying invariants which are $\kappa$-sequences of $E_0$-classes for $\kappa=\mathfrak{b}$, and it does not have such classifying invariants if $\kappa<\mathbf{add}(\mathcal{B})$.

The result relies on analysing the tail intersection model $\bigcap_{n<\omega}V[c_n,c_{n+1},…]$, where $\left<c_0,c_1,…\right>$ is a generic sequence of Cohen reals.

Saharon Shelah (The Hebrew University of Jerusalem)
Corrected Iterations

It is known that if we consider FS iteration of ccc forcings,  each “nicely definable”, then restricting ourselves to  subsequence of the forcing gives us a complete sub-forcing.
 
What about ($<\lambda$)-support iteration of (($<\lambda$)-complete forcing notion satisfying a suitable $\lambda^+$-cc, and allowing partial memory?  The answer is to change the iteration used to the so called corrected iteration.
 
A special case for $\lambda$ inaccessible and specific forcing 
is [Sh:1126], and the general case is Horowitz-Sh. [HoSh:1204]

Dima Sinapova (University of Illinois at Chicago)
Prikry sequences and square properties

We investigate the consequences of Prikry type forcing on adding square sequences. It is well known that if an inaccessible cardinal $\kappa$ is singularized to countable cofinality while preserving cardinals, then $\square_{\kappa,\omega}$ holds in the outer model. Moreover, this remains true even when relaxing the cardinal preservation assumption a bit. In this talk we focus on when Prikry forcing add weaker forms of square. We prove abstract theorems about when Prikry forcing with interleaved collapses to bring down the singularized cardinal to $\aleph_\omega$ will add a weak square sequence.
 

Sarka Stejskalova (Charles University)
The negation of the weak Kurepa hypothesis and guessing models

 

The weak Kurepa hypothesis at $\omega_1$, $\mathsf{wKH}(\omega_1)$, states that there exists a tree of size and height $\omega_1$ which has at least $\omega_2$ cofinal branches.

In the first part of the talk we will focus on $\neg\mathsf{wKH}(\omega_1)$ and its connection to the Guessing Model Property at $\omega_2$, $\mathsf{GMP}(\omega_2)$, which was introduced by Viale and Weiss. The first — more common — variant of $\mathsf{GMP}(\omega_2)$ states that there are stationarily many $\omega_1$-guessing models (in some $H(\theta)$), where $\omega_1$-guessing refers to approximations of size $<\!\omega_1$. By a result of Cox and Krueger, this variant of $\mathsf{GMP}(\omega_2)$ implies $\neg\mathsf{wKH}(\omega_1)$. If we weaken $\omega_1$-guessing to $\omega_2$-guessing, i.e.\ if we consider approximations of size $<\!\omega_2$, we get the second variant of $\mathsf{GMP}(\omega_2)$. We show that this variant is consistent with the existence of a Kurepa tree at $\omega_1$. This answers affirmatively the question of Viale who asked whether it is consistent that there are $\omega_2$-guessing models which are not $\omega_1$-guessing.

In the second part of the talk, we will discuss the effect of $\neg\mathsf{wKH}(\omega_1)$ on the cardinal arithmetic. We first review the result that $\neg\mathsf{wKH}(\omega_1) + 2^{\omega}<\aleph_{\omega_1}$ implies $2^\omega=2^{\omega_1}$. Next, we will consider a strengthening of $\neg\mathsf{wKH}(\omega_1)$ which requires that all trees of size $\omega_1$ are special (in the sense of Baumgartner) and thus can have at most $\omega_1$ cofinal branches. This principle is in a sense an “indestructible” version of $\neg\mathsf{wKH}(\omega_1)$ because the “non-weak-Kurepaness” is witnessed by a specialization function, so the tree cannot become a weak Kurepa tree unless cardinals are collapsed. We will show that this strengthening of $\neg\mathsf{wKH}(\omega_1)$ is consistent with $2^\omega=\aleph_{\omega_1}$. We will discuss this result in the context of the Indestructible Guessing Model Property at $\omega_2$, $\mathsf{IGMP}(\omega_2$), and the question whether $\mathsf{IGMP(\omega_2)}$ is consistent with $2^\omega$ having cofinality $\omega_1$.

This is joint work with Chris Lambie-Hanson.

Spencer Unger (University of Toronto) Stationary reflection at the successor of a singular

We compare and contrast recent results getting the consistency of stationary reflection at the successor of a singular cardinal.

Jindrich Zapletal (University of Florida) 
Games and chromatic numbers of graphs

Let G be an analytic graph on a Polish space. I provide a determined game characterizing countable list-chromatic number of G. A variation of this game then yields a variation of the countable list-chromatic number of G. It works for example for distance graphs on Euclidean spaces. This is a joint work with David Chodounsky.