2018-01-11David Schrittesser (KGRC)On the Complexity of Maximal Cofinitary Groups
A maximal cofinitary group is a subgroup of the group of permutations of the set of natural numbers \(\mathbb{N}\) such that any group element has only finitely many fixed points, and no strictly larger group of permutations of \(\mathbb{N}\) has this property. Improving a result of Horowitz and Shelah, we show that there is a closed maximal cofinitary group.
2018-01-18Philipp Schlicht (Universität Bonn, Germany)Combinatorial variants of Lebesgue's density theorem
We introduce density points relative to an arbitrary sigma-ideal of Borel subsets of the Cantor or Baire space, and define the density property relative to a sigma-ideal as a variant of Lebesgue’s density property. For the ideal of null sets with respect to the uniform measure on the Cantor space, our density points coincide with the standard definition up to a null set. The main results show that many well-known ccc ideals have the density property and that this fails for many non-ccc ideals. In the talk, I will present the current status of this project.

This is joint work with Sandra Müller, David Schrittesser and Thilo Weinert.
2018-03-08Šárka Stejskalová (KGRC)The tree property and the continuum function
We will discuss the tree property, a compactness principle which can hold at successor cardinals such as \(\aleph_2\) or \(\aleph_3\). For a regular cardinal \(\kappa\), we say that \(\kappa\) has the tree property if there are no \(\kappa\)-Aronszajn trees. It is known that the tree property has the following non-trivial effect on the continuum function:

(*) If the tree property holds at \(\kappa^{++}\), then \(2^\kappa > \kappa^+\).

After defining the key notions, we will review some basic constructions related to the tree property and state some original results regarding the tree property which suggest that (*) is the only restriction which the tree property puts on the continuum function in addition to the usual restrictions provable in ZFC.
2018-03-15Thilo Weinert (KGRC)Cardinal Characteristics and Partition Properties
Many a partition relation has been proved assuming the Generalised Continuum Hypothesis. More precisely, many negative partition relations involving ordinals smaller than \(\omega_2\) have been proved assuming the Continuum Hypothesis. Some recent results in this vein for polarised partition relations came from Garti and Shelah. The talk will focus on classical partition relations. The relations \(\omega_1\omega \not\rightarrow (\omega_1\omega, 3)^2\) and \(\omega_1^2 \not\rightarrow (\omega_1\omega, 4)^2\) were both shown to follow from the Continuum Hypothesis, the former in 1971 by Erdős and Hajnal and the latter in 1987 by Baumgartner and Hajnal.

The former relation was shown to follow from both the dominating number and the stick number being \(\aleph_1\) in 1987 by Takahashi. In 1998 Jean Larson showed that simply the dominating number being \(\aleph_1\) suffices for this. It turns out that the unbounding number and the stick number both being \(\aleph_1\) yields the same result. Moreover, also the second relation follows both from the dominating number being \(\aleph_1\) and from both the unbounding number and the stick number being \(\aleph_1\) thus answering a question of Jean Larson.

This is both joint work with Chris Lambie-Hanson and with both William Chen and Shimon Garti.
2018-03-22Monroe Eskew (KGRC)Local saturation of the nonstationary ideals
It is consistent relative to a huge cardinal that for all successor cardinals \(\kappa\), there is a stationary \(S \subseteq \kappa\) such that the nonstationary ideal on \(\kappa\) restricted to \(S\) is \(\kappa^+\)-saturated. We will describe the construction of the model, focusing how to get this property on all \(\aleph_n\) simultaneously. Time permitting, we will also briefly discuss the Prikry-type forcing that extends this up to \(\aleph_{\omega+1}\).
2018-04-12Victoria Gitman (Graduate Center, City University of New York (CUNY), USA)Virtual large cardinal principles
Given a set-theoretic property \(\mathcal P\) characterized by the existence of elementary embeddings between some first-order structures, we say that \(\mathcal P\) holds virtually if the embeddings between structures from \(V\) characterizing \(\mathcal P\) exist somewhere in the generic multiverse. We showed with Schindler that virtual versions of supercompact, \(C^{(n)}\)-extendible, \(n\)-huge and rank-into-rank cardinals form a large cardinal hierarchy consistent with \(V=L\). Sitting atop the hierarchy are virtual versions of inconsistent large cardinal principles such as the existence of an elementary embedding \(j:V_\lambda\to V_\lambda\) for \(\lambda\) much larger than the supremum of the critical sequence. The Silver indiscernibles, under \(0^\sharp\), which have a number of large cardinal properties in \(L\), are also natural examples of virtual large cardinals. With Bagaria, Hamkins and Schindler, we investigated properties of the virtual version of Vopěnka's Principle, which is consistent with \(V=L\), and established some surprising differences from Vopěnka's Principle, stemming from the failure of Kunen's Inconsistency in the virtual setting. A recent new direction in the study of virtual large cardinal principles involves asking that the required embeddings exist in forcing extensions preserving a large segment of the cardinals. In the talk, I will discuss a mixture of results about the virtual large cardinal hierarchy and virtual Vopěnka's Principle. Time permitting, I will give an overview of Woodin's new results on virtual large cardinals in cardinal preserving extensions.
2018-04-19Diana Carolina Montoya Amaya (KGRC)On some ideals associated with independent families
The concept of independence was first introduced by Fichtenholz and Kantorovic to study the space of linear functionals on the unit interval. Since then, independent families have been an important object of study in the combinatorics of the real line. Particular interest has been given, for instance, to the study of their definability properties and to their possible sizes.

In this talk we focus on two ideals which are naturally associated with independent families: The first of them is characterized by a diagonalization property, which allows us to add a maximal independent family along a finite support iteration of some ccc posets. The second ideal originates in Shelah's proof of the consistency of \(i<u\) (here \(i\) and \(u\) are the independence and ultrafilter numbers respectively). Additionally, we study the relationship between these two ideals for an arbitrary independent family \(A\), and define a class of maximal independent families — which we call densely independent — for which the ideals mentioned above coincide. Building upon the techniques of Shelah we (1) characterize Sacks indestructibility for such families in terms of properties of its associated diagonalization ideal, and (2) devise a countably closed poset which adjoins a Sacks indestructible densely maximal independent family.

This is joint work with Vera Fischer.
2018-04-26Filippo Calderoni (Università di Torino, Italy and Politecnico di Torino, Italy)The bi-embeddability relation for countable abelian groups
We analyze the Borel complexity of the bi‑embeddability relation for different classes of countable abelian groups. Most notably, we use the Ulm theory to prove that bi‑embeddability is incomparable with isomorphism in the case of p‑groups, and torsion groups. As I will explain, our result contrasts the arguable thesis that the bi‑embeddability relation on countable abelian p‑groups has strictly simpler complete invariants than isomorphism.

This is joint work with Simon Thomas.
2018-05-03Vincenzo Dimonte (University of Udine, Italy)The sensitive issue of iterability
In the momentous years when the community of set theorists was reaching the definite answer for the problem of the consistency of the Axiom of Determinacy, Martin wrote a small paper in the Proceedings of the International Congress of Mathematicians, 1978, in which he proved that the iterable version of I3, a very large cardinal, implied the determinacy of \(\Pi^1_2\) sets of reals. Later it was proved that AD had much lower consistency, and iterable I3 fell into oblivion. In the last decade interest on I3 re-emerged, but iterable I3 is still elusive, and the small paper by Martin is not helpful, as it is terse and full of gaps. Even the definition of iterable I3 is not convincing. In this seminar we will bring back to life this abandoned hypothesis, clean it up to modern standards, and reveal the existence of a new hierarchy of axioms that was previously overlooked.
2018-05-17Yair Hayut (Hebrew University of Jerusalem, Israel)Stationary reflection at $\aleph_{\omega+1}$
Stationary reflection is one of the basic prototypes of reflection phenomena, and its failure is related to many counterexamples for compactness properties (such as almost free non-free abelian groups, and more). In 1982, Magidor showed that it is consistent, relative to infinitely many supercomapct cardinals, that stationary reflections holds at \(\aleph_{\omega + 1}\). In this talk I'm going to present a new method for forcing stationary reflection at \(\aleph_{\omega+1}\), which allows to significantly reduce the upper bound for the consistency strength of the full stationary reflection at \(\aleph_{\omega+1}\) (below a single partially supercompact cardinal).

This is a joint work with Spencer Unger.
2018-05-24Borisha Kuzeljevic (Czech Academy of Sciences, Prague)P-ideal dichotomy and some versions of the Souslin Hypothesis
The talk will be about the relationship of PID with the statement that all Aronszajn trees are special. This is joint work with Stevo Todorcevic.
2018-06-07Andrea Medini (KGRC)Homogeneous spaces and Wadge theory
All spaces are assumed to be separable and metrizable. A space \(X\) is homogeneous if for all \(x,y\in X\) there exists a homeomorphism \(h:X\longrightarrow X\) such that \(h(x)=y\). A space \(X\) is strongly homogeneous if all non-empty clopen subspaces of \(X\) are homeomorphic to each other. We will show that, under the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (with the trivial exception of locally compact spaces). This extends results of van Engelen and complements a result of van Douwen. Our main tool will be Wadge theory, which provides an exhaustive analysis of the topological complexity of the subsets of \(2^\omega\).

This is joint work with Raphaël Carroy and Sandra Müller.
2018-06-14Zoltán Vidnyánszky (KGRC)Borel chromatic numbers: basis and antibasis results
We give a full description of the existence of a homomorphism basis for Borel graphs of given Borel chromatic number. In particular, we show that there is a Borel graph with Borel chromatic number 3 that admits a homomorphism to any Borel graph of Borel chromatic number at least 3. We also discuss the relation of these results to Hedetniemi's conjecture.
2018-10-04David Fernández-Bretón (KGRC)Finiteness classes arising from Ramsey-theoretic statements in set theory without choice
In the absence of the Axiom of Choice, there may be infinite sets for which certain Ramsey-theoretic statements – such as Ramsey's or (appropriately phrased) Hindman's theorem – fail. In this talk, we will analyse the existence of such sets, and their precise location within the hierarchy of infinite Dedekind-finite sets; independence proofs will be carried out using the Fränkel-Mostowski technique of permutation models.

This is joint work with Joshua Brot and Mengyang Cao.

''Note: A [ recording] of this talk is available on YouTube.''
2018-10-09Gerhard Jäger (Universität Bern, Switzerland)From fixed points in weak set theories to some open problems
Least fixed points of monotone operators are well-studied objects in many areas of mathematical logic. Typically, they are characterized as the intersection of all sets closed under the respective operator or as the result of its iteration from below.

In this talk I will start off from specific \(\Sigma_1\) operators in a Kripke-Platek environment and relate fixed point assertions to alternative set existence principles. By doing that, we are also led to some “largeness axioms” and to several open problems.
2018-10-11David Chodounský (Academy of Sciences of the Czech Republic)Silver forcing and P-points
I will give a full proof of a joint result with O. Guzman regarding a technique for destroying P-ultrafilters with Silver forcing. Time permitting, I will present several applications.

''Note: A [ recording] of this talk is available on YouTube.''
2018-10-18Monroe Eskew (KGRC)Rigid ideals
Using ideas from Foreman-Magidor-Shelah, one can force from a Wooden cardinal to show it is consistent that the nonstationary ideal on \(\omega_1\) is saturated while the quotient boolean algebra is rigid. The key is to apply Martin's Axiom to the almost-disjoint coding forcing to see how it interacts with a generic elementary embedding. This strategy requires the continuum hypothesis to fail. Towards showing the consistency of rigid ideals with GCH, the speaker investigated other coding strategies: stationary coding (with Brent Cody), a rigid version of the Levy collapse, and ladder-system coding (in recent work with Paul Larson). We have some equiconsistencies about rigid ideals on \(\omega_1\) and \(\omega_2\), as well as some global possibilities from very large cardinals. Some natural questions remain about \(\omega_1\) and successors of singulars.
2018-10-22David Asperó (University of East Anglia, Norwich, UK)Special $\aleph_2$-Aronszajn trees and GCH
In joint work with Mohammad Golshani, and assuming the existence of a weakly compact cardinal, we build a forcing extension in which GCH holds and every \(\aleph_2\)-Aronszajn tree is special. This answers a well-known question from the 1970’s. I will give the proof of this theorem, with as many details as possible.

''Note: Recordings of this talk are available on YouTube: [ part 1], [ part 2]''
2018-10-25David Schrittesser (KGRC)The Ramsey property, MAD families, and their multidimensional relatives
Suppose every set of real numbers has the Ramsey property and “uniformization on Ellentuck-comeager sets” as well as Dependent Choice hold (as is the case under the Axiom of Determinacy, but also in Solovay's model). Then there are no MAD families. As it turns out, there are also no (Fin x Fin)-MAD families, where Fin x Fin is the two-dimensional Fubini product of the ideal of finite sets. We also comment on higher dimensional products.

All results are joint work with Asger Törnquist.

Slides are available [ here], a video [ recording] of this talk is available on YouTube.
2018-11-08Thilo Weinert (KGRC)On Order-Types in Polarised Partition Relations
The history of the polarised partition relation goes back to the original seminal paper by Erdős and Rado from 1956. For the ordinary partition relation after some time one also investigated order-types. For the polarised partition relation however, I am only aware of three papers where order-type played a role and here nothing beyond well-orders ever seems to have been considered.

We are attempting to rectify this. We will present an analogue of a theorem of Jones and a potentially vacuous generalisation of a proposition of Garti and Shelah. Furthermore we will show limits to further generalisations and analogues and will exhibit some open problems.

This is joint work with Lukas Daniel Klausner.
2018-11-15Russell Miller (Queens College, City University of New York (CUNY), USA)Hilbert's Tenth Problem for Subrings of the Rational Numbers
When considering subrings of the field \(\mathbb Q\) of rational numbers, one can view Hilbert's Tenth Problem as an operator, mapping each set \(W\) of prime numbers to the set \(HTP(R_W)\) of polynomials in \(\mathbb Z[X_1,X_2,\ldots]\) with solutions in the ring \(R_W=\mathbb Z[W^{-1}]\). The set \(HTP(R_{\emptyset})\) is the original Hilbert’s Tenth Problem, known since 1970 to be undecidable. If \(W\) contains all primes, then one gets \(HTP(\mathbb Q)\), whose decidability status is open. In between lie the continuum-many other subrings of \(\mathbb Q\).

We will begin by discussing topological and measure-theoretic results on the space of all subrings of \(\mathbb Q\), which is homeomorphic to Cantor space. Then we will present a recent result by Ken Kramer and the speaker, showing that the HTP operator does not preserve Turing reducibility. Indeed, in some cases it reverses it: one can have \(V<_T W\), yet \(HTP(R_W) <_T HTP(R_V)\). Related techniques reveal that every Turing degree contains a set \(W\) which is HTP-complete, with \(W’\leq_1 HTP(R_W)\). On the other hand, the earlier results imply that very few sets \(W\) have this property: the collection of all HTP-complete sets is meager and has measure \(0\) in Cantor space.

There is a [ video] recording of this talk on YouTube.

Slides are [ available], too.
2018-11-22Raphaël Carroy (KGRC)A dichotomy for topological embeddability between continuous functions
We say a function \(f\) embeds (topologically) in a function \(g\) when there are two topological embeddings \(\sigma\) and \(\tau\) satisfying \(\tau \circ f = g \circ \sigma\). I will prove the following dichotomy: the quasi-order of topological embeddability between continuous functions on compact zero-dimensional Polish spaces is either an analytic complete quasi-order, or a well-quasi-order.

This is a joint work with Yann Pequignot and Zoltán Vidnyánszky.

A [ video recording] of this talk is available on YouTube.
2018-11-29Daniel Soukup (KGRC)New aspects of ladder system uniformization
After a brief overview of some classical results, we will survey new applications of ladder system uniformization. In particular, we constrast uniformizations defined on \(\omega_1\) and uniformizations on trees of height \(\omega_1\). The latter, introduced by J. Moore, played a critical role in understanding minimal uncountable linear orders under CH. One of our rather surprising new results is that whenever \(\diamondsuit^+\) holds, for any ladder system \(\mathbf C\) there is an Aronszajn tree \(T\) so that any monochromatic colouring of \(\mathbf C\) has a \(T\)-uniformization (cf. []).

A [ video recording] of this talk is available on YouTube.
2018-12-04 (Tuesday!)Stevo Todorcevic (University of Toronto, Canada)Ramsey degrees of topological spaces
This will be an overview of structural Ramsey theory when the objects are topological spaces. Open problems and directions for further research in this area will also be examined.
2018-12-13Daniel Soukup (KGRC)New aspects of ladder system uniformization II
We continue the [/2018/Talk_11-29_a.html previous lecture] and present proofs for some of the new results. We show that \(\diamondsuit\) implies that for any Aronszajn-tree \(T\), there is a ladder system with a 2-colouring with no \(T\)-uniformization. However, if \(\diamondsuit^+\) holds then for any ladder system \(\mathbf C\) there is an Aronszajn tree \(T\) so that any monochromatic colouring of \(\mathbf C\) has a \(T\)-uniformization.

A [ video recording] of this talk is available on YouTube.