May 17, 2024

Set Theory Seminar Series 2013-14
Fields Institute, Room 210

Organizing Committee:
Miguel Angel Mota, Ilijas Farah, Juris Steprans, Paul Szeptycki

Upcoming Seminars
at 1:30 p.m. in the Fields Institute, Room 210

June 27

Christopher Eagle
Model theory of abelian real rank zero C*-algebras

We consider algebras of the form $C(X)$, where $X$ is a $0$-dimensional compact Hausdorff space, from the point of view of continuous model theory. We characterize these algebras up to elementary equivalence in terms of invariants of the Boolean algebra $CL(X)$ of clopen subsets of $X$. We also describe several saturation properties that $C(X)$ may have, and relate these to topological properties of $X$ and saturation of $CL(X)$. We will discuss some consequences of saturation when we view $C(X)$ as a $C^*$-algebra. All the necessary background on continuous logic will be provided. This is joint work with Alessandro Vignati.

Past Seminars
Speaker and Talk Title
June 13 no seminar
June 6

no seminar

May 30

Asger Törnquist
Statements that are equivalent to CH and their Sigma-1-2 counterparts.

There is a large number of "peculiar" statements that have been shown over time to be equivalent to the Continuum Hypothesis, CH. For instance, a well-known theorem of Sierpinski says that CH is equivalent to the statement that the plane can be covered by countably many graphs of functions (countably many of which are functions of x, and countably many of which are functions of y.) What happens if we consider the natural Sigma-1-2 analogues of these statements (in the sense of descriptive set theory)? It turns out that then these statements are, in a surprising number of cases, equivalent to that all reals are constructible. In this talk I will give many examples of this phenomenon, and attempt to provide an explanation of why this occurs. This is joint work with William Weiss.
May 23

Martino Lupini.
The Lopez-Escobar theorem for metric structures and the topological Vaught conjecture.

I will present a generalization of the classical Lopez-Escobar theorem to the logic for metric structures. As an application I will provide a model-theoretic reformulation of the topological Vaught conjecture. This is joint work with Samuel Coskey.

May 16 no seminar
May 9

Frank Tall
What I am working on

It might be useful, especially to grad students just starting their research, if we had talks on the theme of "what I am working on", rather than waiting for all of the theorems to be proved and the presentation polished. In view of the fact that no one else wants to talk tomorrow, I am willing to give such a talk. My first topic grew outof Marion Scheepers' talk a few weeks ago, and concerns the question, due to Gruenhage and Ma, of whether, in the compact-open topology, the space of continuous real-valued functions on a locally compact normal space satisfies the Baire Category Theorem. I have several consistency results using PFA(S)[S], but am trying to settle the question in ZFC. The second topic also concerns PFA(S)[S] (which you do not have to know to understand my talk). I had characterizations under PFA(S)[S] of paracompactness in locally compact normal spaces that required the absence of perfect pre-images of omega_1; Together with Alan Dow, I have shown some of those characterizations can be improved to just require the absence of copies of omega_1, but others cannot. Some of this work requires an interesting but difficult PFA(S)[S] proof of Dow that I shall eventually present in the seminar.

May 2

Micheal Pawliuk
Packing Hedgehogs densely into l_2 to give a trivial G-compactification

In the early 80s Smirnov asked if every regular G-space admits an equivariant G-compactification. In 1988 Megrelishvili exhibited a G-space that does not (essentially the metrizable hedgehog with a nice group action). His example still leaves open the larger question of if a regular G-space can have a *trivial* G-compactification. In joint work with Pestov and Bartosova, we will give such an example by finding many copies of the metrizable hedgehog inside l_2.

April 25

Alessandro Vignati
An algebra whose subalgebras are characterized by density

A long-standing open problem is whether or not every amenable operator algebra is isomorphic to a C*-algebra. In a recent paper, Y. Choi, I. Farah and N. Ozawa provided a non separable counterexample. After an introduction, building on their work and using the full power of a Luzin gap, we provide an example of an amenable operator algebra A such that every amenable nonseparable subalgebra of A is not isomorphic to a C*-algebra, while some "reasonable" separable subalgebras are. In the end we describe some interesting property of the constructed object related to the Kadison-Kastler metric.

April 11

Marion Scheepers
Box powers of Baire spaces.

A topological space is a Baire space if any countable sequence of dense open subsets has a non empty intersection. In this talk we discuss an elegant (consistent module large cardinals) characterization of spaces that have the Baire property in all powers, considered in the box topology.

April 4

David Fernandez
Two microcontributions to the theory of Strongly Summable Ultrafilters

Strongly Summable Ultrafilters are those generated by FS-sets (where FS(X) is the set of all possible sums of finitely many elements from X (you can only add each element once)). I will show two little results (with nice little neat proofs!) about these: first, that every strongly summable ultrafilter on the countable Boolean group is rapid. Second, that there is a model where strongly summable ultrafilters (on any abelian group really, but without loss of generality on the countable Boolean group) exist yet Martin's axiom for countable forcing notions fails (up until now, these ultrafilters were only known to exist under this hypothesis).

March 28 Konstantinos Tyros
Primitive recursive bounds for the finite version of Gowers' $c_0$ theorem, Talk 2

In this talk we will present proofs for the finite version of Gowers' $c_0$ theorem for both the positive and the general case providing primitive recursive bounds. Multidimensional versions of these result will be presented too.

March 21 Konstantinos Tyros
Primitive recursive bounds for the finite version of Gowers' $c_0$ theorem, Talk 1

In this talk we will present proofs for the finite version of Gowers' $c_0$ theorem for both the positive and the general case providing primitive recursive bounds. Multidimensional versions of these result will be presented too.

March 14

Tomasz Kania.
A chain condition for operators from C(K)-spaces

Building upon work of Pelczynski, we introduce a chain condition, defined for operators acting on C(K)-spaces, which is weaker than weak compactness. We prove that if K is extremely disconnected and X is a Banach space then an operator T : C(K) -> X is weakly compact if and only if it satisfies our condition and this is if and only if the representing vector measure of T satisfies an analogous chain condition on Borel sets of K. As a tool for proving the above-mentioned result, we derive a topological counterpart of Rosenthal's lemma. We exhibit classes of compact Hausdorff spaces K for which the identity operator on C(K) satisfies our condition, for instance every class of compact spaces that is preserved when taking closed subspaces and Hausdorff quotients, and which contains no non-metrisable linearly ordered space (like the classes of Eberlein spaces, Corson compact spaces etc.) serves as an example. Using a Ramsey-type theorem, due to Dushnik and Miller, we prove that the collection of operators on a C(K)-space satisfying our condition forms a closed left ideal of B(C(K)), however in general, it does not form a right ideal. This work is based on two papers (one joint with K. P. Hart and T. Kochanek and the second one joint with. R. Smith).

March 7

Juris Steprans
Non-trivial automorphisms of $P(\omega_1)/fin$

Just as in the case of automorphisms of $P(\omega)/fin$, an automorphism of $P(\omega_1)/fin$ will be called trivial if it is induced by a bijection between cofinite subsets of $\omega_1$. Since a non-trivial automorphism of $P(\omega)/fin$ can easily be extended to a non-trivial automorphism of $P(\omega_1)/fin$ there is little interest examining the existence of non-trivial automorphisms of $P(\omega_1)/fin$ without further restrictions. So, an automorphism of $P(\omega_1)/fin$ will be called really non-trivial if it is non-trivial, yet its restriction to any subalgebra of the form $P(X)/fin$ is trivial when $X$ is countable. It will be shown to be consistent with set theory that there is a really non-trivial automorphism of $P(\omega_1)/fin$.
This is joint work with Assaf Rinot.

February 28 Daniel Soukup
Davies-trees in infinite combinatorics

The aim of this talk is to introduce Davies-trees and present new applications to combinatorics. Davies-trees are special sequences of countable elementary submodels which played important roles in generalizing arguments using CH to pure ZFC proofs. My goal is to present two unrelated but fascinating results due to P. Komjáth: we prove that the plane is the union of n+2 "clouds" provided that the continuum is at most $\aleph_n$ and that every uncountably chromatic graph contains k-connected uncountably chromatic subgraphs for each finite k. We hopefully have time to review the most important open problems around the second theorem.

February 21
Room 230
*Please note room change

Mohammed Bekkali
An overview of Boolean Algebras over partially ordered sets

Being at crossroads between Algebra, Topology, Logic, Set Theory and the Theory of Order; the class of Boolean Algebras over partially ordered sets offers more flexibility in representing no zero elements and describing Stone spaces. Some constructions and their interconnections will be discussed, motivating along the way a list of open problems.

February 14 Stevo Todorcevic
A new partition theorem for tress and is applications (Part II)

In a recent joint work with Antonio Aviles, in order to classify k-tuples of analytic hereditary families of subsequences of a fixed sequence of objects ( vectors, points in a topological space,etc), we needded to come up with a new Ramsey theorem for trees. The lecture will concentrate on stating the result and, if time permits, on giving some ideas from the proof.

February 7

Stevo Todorcevic
A new partition theorem for tress and is applications (Part I)

In a recent joint work with Antonio Aviles, in order to classify k-tuples of analytic hereditary families of subsequences of a fixed sequence of objects ( vectors, points in a topological space,etc), we needded to come up with a new Ramsey theorem for trees. The lecture will concentrate on stating the result and, if time permits, on giving some ideas from the proof.


January 31

Dana Bartosova
Lelek fan from a projective Fraïssé limit

The Lelek fan is the unique subcontinuum of the Cantor fan whose set of endpoints is dense. The Cantor fan is the cone over the Cantor set, that is $C\times I/\sim,$ where $C$ is the Cantor set, $I$ is the closed unit interval and $(a,b)\sim (c,d)$ if and only if either $(a=c$ and $b=d)$ or $(b=d=0)$. We construct the Lelek fan as a
natural quotient of a projective Fra\"iss\'e limit and derive some properties of the Lelek fan and its homeomorphism group. This is joint with Aleksandra Kwiatkowska.

January 17

Miguel Angel Mota
Baumgartner's Conjecture and Bounded Forcing Axioms (Part I)

Using some variants of weak club guessing we separate some fragments of the proper forcing axiom: we show that for every two indecomposable ordinals $\alpha < \beta$, the forcing axiom for the class of all the $\beta$-proper posets does not imply the bounded forcing axiom for the class of all the $\alpha$-proper posets.

January 10

Rodrigo Hernandez
Wijsman hyperspaces of non-separable metric spaces

The hyperspace CL(X) of a topological space X (at least T1) is the set of all non-empty closed subsets of X. The usual choice for a topology in CL(X) is the Vietoris topology, which has been widely studied. However, in this talk we will consider the Wijsman topology on CL(X), which is defined when (X,d) is a metric space. The Wijsman topology is coarser than the Vietoris topology and in fact it depends on the metric d, not just on the topology. The problem we will address is that of normality of the Wijsman hyperspace. It is known since the 70s that the Vietoris hyperspace is normal if and only if X is compact. But a characterization of normality of the Wijsman hyperspace is still not known. It is conjectured that the Wijsman hyperspace if normal if and only if the space X is separable. Jointly with Paul Szeptycki, we have proved that if X is locally separable and of uncountable weight, then the Wijsman hyperspace is not normal.

December 13 Martino Lupini.
The descriptive set theory of Polish groupoids

I will present an overview of functorial classification within the framework of invariant descriptive set theory, based on the notion of Polish groupoid and Borel classifying functor. I will then explain how several results about Polish group actions admit natural generalizations to Polish groupoids, extending works of Becker-Kechris, Effros, Hjorth, and Ramsay.

Dec. 06 Konstantinos Tyros
An infinitary version of the Furstenberg-Weiss Theorem.
In 2003 H. Furstenberg and B. Weiss obtained a far reaching extension of the famous Szemer\'edi's theorem on arithmetic progressions. They establish the existence of finite strong subtrees of arbitrary height, having an arithmetic progression as a level set, inside subsets of positive measure of a homogeneous tree. In this talk an infinitary version of their result will be presented.

Nov. 29

Jan Pachl
One-point DTC sets for convolution semigroups

Every topological group G naturally embeds in the Banach algebra LUC(G)*. The topological centre of LUC(G)* is defined to be the set of its elements for which the left multiplication is w*--w*-continuous. Although the definition demands continuity on the whole algebra, for a large class of topological groups it is sufficient to test the continuity of the left multiplication at just one suitably chosen point; in other words, the algebra has a one-point DTC (Determining Topological Centre) set. More generally, the same result holds for many subsemigroups of LUC(G)*. In particular, for G in the same large class, the uniform compactification (the greatest ambit) of G has a one-point DTC set. These results, which generalize those previously known for locally compact groups, are from joint work with Stefano Ferri and Matthias Neufang.

Nov. 15

Revised Location:
Stewart Library

Piotr Koszmider (Talk 1 from 14:00 to 15:00)
Independent families in Boolean algebras with some separation properties

We prove that any Boolean algebra with the subsequential completeness property contains an independent family of size continuum. This improves a result of Argyros from the 80ties which asserted the existence of an uncountable independent family. In fact we prove it for a bigger class of Boolean algebras satisfying much weaker properties. It follows that the Stone spaces of all such Boolean algebras contains a copy of the Cech-Stone compactification of the integers and the Banach space of continuous functions on them has l-infinity as a quotient. Connections with the Grothendieck property in Banach spaces are discussed. The talk is based on the paper: Piotr Koszmider, Saharon Shelah; Independent families in Boolean algebras with some separation properties; Algebra Universalis 69 (2013), no. 4, 305 - 312.

Jordi Lopez Abad
(Talk 2 from 15:30 to 16:30)
Unconditional and subsymmetric sequences in Banach spaces of high density

We will discuss bounds and possible values for the minimal cardinal number $\kappa$ such that every Banach space of density $\kappa$ has an unconditional basic sequence, or the corresponding cardinal number for subsymmetric basic sequences.

Nov. 8

Ilijas Farah.
The other Kadison--Singer problem.

In their famous 1959 paper Kadison and Singer posed two problems. The famous one was recently solved by Marcus, Spielman and Srivastava, using work of Weaver. The other (much more set-theoretic) Kadison-Singer
problem was resolved using the Continuum Hypothesis by Akemann and Weaver in 2008. This assumption was weakened to Martin's Axiom by myself and Weaver, but the question remains whether the answer is independent from ZFC.

Nov. 1

no seminar

Oct. 18

Lionel Nguyen Van
Structural Ramsey theory and topological dynamics for automorphism groups of homogeneous structures

In 2005, Kechris, Pestov, and Todorcevic established a striking connection between structural Ramsey theory and the topological dynamics certain automorphism groups. The purpose of this talk will be to present this connection, together with recent related results.

Oct. 11

Eduardo Calderon
Asymptotic models and plegma families

We will discuss one of the usual ways in which Ramsey's theorem is applied to the study of Banach space geometry and then, by means of techniques closely following ones first developed by S. Argyros, V. Kanellopoulos, K. Tyros, we will introduce the concept of an asymptotic model of higher order of a Banach space and establish a relationship between these and higher order spreading models that extends their result of the impossibility of always finding a finite chain of spreading models reaching an $l_p$ space to the context of weakly generated asymptotic models.

Oct. 4

David Fernandez
Strongly Productive Ultrafilters

The concept of a Strongly Productive Ultrafilter on a semigroup (known as a "strongly summable ultrafilter" when the semigroup is additively denoted) constitute an important concept ever since Hindman defined it, while trying to prove the theorem that now bears his name. In a 1998 paper of Hindman, Protasov and Strauss, it shown that strongly productive ultrafilters on abelian groups are always idempotent, but no further generalization of this fact had been made afterwards. In this talk I will show (at least the main ideas, anyway) the proof that this result holds on a large class of semigroups, which includes all solvable groups and the free semigroup, among others. After that, I'll discuss a special class of strongly productive ultrafilters on the free semigroup (dubbed "very strongly productive ultrafilters" by N. Hindman and L. Jones), and show that they have the "trivial products property". This means that (thinking of the free semigroup S as a subset of the free group G) if p is a very strongly productive ultrafilter on S, and q,r are nonprincipal ultrafilters on G such that $qr=p$, then there must be an element x of G such that $q=px$ and $r=x^{-1}p$. This answers a question of Hindman and Jones. Joint work with Martino Lupini

Sept. 27

Stevo Todorcevic
A construction scheme on $\omega_{1}$

We describe a simple and general construction scheme for describing mathematical structures on domain $\omega_{1}$. Natural requirements on this scheme will reduce the nonseparable structural properties of the resulting mathematical object to some finite-dimensional problems that are easy to state and frequently also easy to solve. The construction scheme is in fact quite easy to use and we illustrate this by some application mainly towards compact convex spaces and normed spaces.

Sept. 20

Rodrigo Hernandez
Countable dense homogeneous spaces

A separable space X is countable dense homogeneous (CDH) if every time D and E are countable dense subsets of X, there exists a homeomorphism $h:X\to X$ such that $h[D]=E$. The first examples of CDH spaces were Polish spaces. So the natural open question was whether there exists a CDH metrizable space that is not Polish. By a characterization result by Hrusak and Zamora-Aviles, such a space must be non Borel. In this talk, we will focus on recent progress in this direction. In fact, we only know about two types of CDH non-Borel spaces: non-meager P-filters (with the Cantor set topology) and $\lambda$-sets. Moreover, by arguments similar to those used for the CDH $\lambda$-set, it has also been possible to construct a compact CDH space of uncountable weight.


Sept. 13

Daniel Soukup
Monochromatic partitions of edge-colored infinite graphs

Our goal is to find well behaved partitions of edge-colored infinite graphs following a long standing trend in finite combinatorics started by several authors including P. Erdos and R. Rado; in particular, we are interested in partitioning the vertices of complete or nearly complete graphs into monochromatic paths and powers of paths. One of our main results is that for every 2-edge-coloring of the complete graph on $\omega_1$ one can partition the vertices into two monochromatic paths of different colors. Our plan for the talk is to review some results from the literature (both on finite and infinite), sketch some of our results and the ideas involved and finally present the great deal of open problems we facing at the moment. This is a joint work with M. Elekes, L. Soukup and Z. Szentmiklóssy.


Aug 30 no seminar
Aug 28
3:00 p.m.

Connor Meehan
Infinite Games and Analytic Sets

In the context of set theory, infinite games have been studied since the mid-20th century and have created an interesting web of connections, such as with measurable cardinals. Upon specifying a subset A of sequences of natural numbers, an infinite game G(A) involves two players alternately choosing natural numbers, with player 1 winning in the event that the resulting sequence x is in A. We will give proofs of Gale and Stewart's classic results that any open subset A of Baire space leads to the game G(A) being determined (i.e. one of the players has a winning strategy) and that the Axiom of Determinacy (stating that all games are determined) contradicts the Axiom of Choice. With the former we recreate Blackwell's groundbreaking proof of a classical result about co-analytic sets. A family U of subsets of Baire space is said to have the reduction property if for any B and C in U, there are respective disjoint subsets B* of B and C* of C in U with the same union as B and C; Blackwell proves that the co-analytic sets have the reduction property. Blackwell's new proof technique with this old result revitalized this area of descriptive set theory and began the development for a slew of new results.

Aug 23

Jack Wright
Nonstandard Analysis and an Application to Combinatorial Number Theory

Since nonstandard analysis was first formalized in the 60's it has given mathematicians a framework in which to do rigorous analysis with infinitesimals rather than epsilons and deltas. More importantly, it has also allowed for the application of powerful techniques from logic and model theory to analysis (and other areas of mathematics). This brief presentation will outline some of those tools and discuss one particular application of them.

I will briefly state the key techniques: the transfer principle, the internal definition principle, and the overflow principle. I will then give an indication of the usefulness of these techniques by showing how they have been used to garner some technical results that might be able to help solve the Erd\H{o}s' famous Conjecture on Arithmetic progressions.

Aug 9

Miguel Angel Mota
Instantiations of Club Guessing. Part I

We build a model where Weak Club Guessing fails, mho holds and the continuum is larger than the second uncountable cardinal. The dual of this result will be discussed in a future talks.

Aug 2 Carlos Uzcategui
Uniform Ramsey theoretic properties

The classical Ramsey theorem holds uniformly in the following sense. There is a Borel map that for a given coloring of pairs and an infinite set A, it selects an infinite homogeneous subset of A.
This fact sugests that the notions of a selective, Frechet, p+ and q+ ideal could also holds uniformly. We will discuss about some of those uniform Ramsey theoretic properties.

Jul 19

Todor Tsankov
On some generalizations of de Finetti's theorem

A permutation group G acting on a countable set M is called oligomorphic if the action of G on M^n has only finitely many orbits for each n. Those groups are well known to model-theorists as automorphism groups of omega-categorical structures. In this talk, I will consider the question of classifying all probability measures on [0, 1]^M invariant under the natural action of the group G. A number of classical results in probability theory due to de Finetti, Ryll-Nardzewski, Aldous, Hoover, Kallenberg, and others fit nicely into this framework. I will describe a couple of new results in the same spirit and a possible approach to carry out the classification in general.

Jul 12

Ari Brodsky
A theory of non-special trees, and a generalization of the Balanced Baumgartner-Hajnal-Todorcevic Theorem (slide presentation)

Building on early work by Stevo Todorcevic, we describe a theory of non-special trees of successor-cardinal height. We define the diagonal union of subsets of a tree, as well as normal ideals on a tree, and we characterize arbitrary subsets of a tree as being either stationary or non-stationary.
We then use this theory to prove a partition relation for trees:
Let $\nu$ and $\kappa$ be cardinals such that $\nu ^ {<\kappa} = \nu$, and let $T$ be a non-special tree of height $\nu^+$. Then for any ordinal $\xi$ such that $2^{\left|\xi\right|} < \kappa$, and finite $k$, we have $T \to (\kappa + \xi )^2_k$.
This is a generalization of the Balanced Baumgartner-Hajnal-Todorcevic Theorem, which is the special case of the above where the tree $T$ is replaced by the cardinal $\nu^+$.

Jul 5

Jose Iovino
Definability and Banach space geometry

A well known problem in Banach space theory, posed by Tim Gowers, is whether every Banach space that has an explicitly definable norm must contain one of the classical sequence spaces. I will discuss recent progress obtained jointly with Chris Eagle.

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