SCIENTIFIC PROGRAMS AND ACTIVITIES

December 21, 2014
THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

Set Theory Seminar Series 2014-15
Fields Institute, Room 210
Friday 1:30 pm

Organizing Committee:
Miguel Angel Mota, Ilijas Farah, Juris Steprans, Paul Szeptycki
2014
Fridays

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

   
2014-15
Past Seminars
Speaker and Talk Title
December 16,
Tuesday
*CANCELLED*
BAHEN CENTRE, Room BA6183,
13:30-15:00, talk by Neil Hindman.
December 12, 2015
13:30-15:00
Dilip Raghavan.
Embedding $P(\omega)/FIN$ into the $P$-points

We show under $\mathfrak{p}=\mathfrak{c}$ that $P(\omega)/FIN$ can be embedded into the $P$-points under RK and Tukey reducibility.

December 5, 2015 no seminar scheduled due to Dow Conference.

November 28, 2014

12:30-3:00
Stewart Library

Antonio Avilés
A combinatorial lemma about cardinals $\aleph_n$ and its applications on Banach spa

The lemma mentioned in the title was used by Enflo and Rosenthal to show that the Banach space $L_p[0,1]^\Gamma$ does not have an unconditional basis when $|\Gamma|\geq \aleph_\omega$. In a joint work with Witold Marciszewski, we used some variation of it to show that there are no extension operators between balls of different radii in nonseparable Hilbert spaces.

Istvan Juhász
Lindelof spaces of small extent are $\omega$-resolvable

I intend to present the proof of the following result, joint with L. Soukup and Z. Szentmiklossy: Every regular space $X$ that satisfies $\Delta(X) > e(X)$ is $\omega$-resolvable, i.e. contains infinitely many pairwise disjoint dense subsets. Here $\Delta(X)$, the dispersion character of $X$, is the smallest size of a nonempty open set in $X$ and $e(X)$, the extent of $X$, is the supremum of the sizes of all closed-and-discrete subsets of $X$. In particular, regular Lindelof spaces of uncountable dispersion character are $\omega$-resolvable.

November 21, 2014

Miodrag Sokic
Functional classes

We consider the class of finite structures with functional symbols with respect to the Ramsey property.

November 14, 2014

Martino Lupini
Fraisse limits of operator spaces and the noncommutative Gurarij space

We realize the noncommutative Gurarij space introduced by Oikhberg as the Fraisse limit of the class of finite-dimensional 1-exact operator spaces. As a consequence we deduce that such a space is unique, homogeneous, universal among separable 1-exact operator spaces, and linearly isometric to the Gurarij Banach space.

November 7, 2014
Stewart Library

Juris Steprans
The descriptive set theoretic complexity of the weakly almost periodic functions in the dual of the group algebra

The almost periodic functions on a group G are those functions F from G to the complex number such that the uniform norm closure of all shifts of F is compact in the uniform norm. The weakly almost periodic functions are those for which the analogous statement holds for the weak topology. The family of sets whose characteristic functions are weakly almost periodic forms a Boolean algebra. The question of when this family is a complete $\Pi^1_1$ set will be examined.

October 31, 2014
Speaker 1 (from 12:30 to 13:30):
Vera Fischer
Definable Maximal Cofinitary Groups and Large Continuum
A cofinitary group is a subgroup of the group of all permutations of the natural numbers, all non-identity elements of which have only finitely many fixed points. A cofinitary group is maximal if it is not properly contained in any other cofinitary group. We will discuss the existence of nicely definable maximal cofinitary groups in the presence of large continuum and in particular, we will see the generic construction of a maximal cofinitary group with a $\Pi^1_2$ definable set of generators in the presence of $2^\omega=\aleph_2$.

Speaker 2 (from 13:30 to 15:00):
Menachem Magidor
On compactness for being $\lambda$ collectionwise hausdorff

A compactness property is the statement for a structure in a given class, if every smaller cardinality substructure has a certain property then the whole structure has this property. In this talk we shall deal with the compactness for the property of a topological space being collection wise Hausdorff. The space is X is said to be $\lambda$--collection wise Hausdorff ($\lambda$--cwH) if every closed discrete subset of X of cardinality less than $\lambda$ can be separated by a family of open sets. X is cwH if it is $\lambda$--cwH for every cardinal $\lambda$.

We shall deal with the problem of when $\lambda$--cwH implies cwH, or just when does $\lambda$--cwH implies $\lambda^+$--cwH. A classical example of Bing provides for every cardinal $\lambda$ a space $X_\lambda$ which is $\lambda$-cwH but not $\lambda^+$--cwH. So if we hope to get any level of compactness for the the property of being cwH, we have to restrict the class of spaces we consider. A fruitful case is the case where we restrict the local cardinality of the space. A motivating result is the construction by Shelah (using supercompact cardinal) of a model of Set Theory in which a space which is locally countable and which is $\omega_2$--cwH is cwH.

Can the Shelah result be generalized to larger cardinals , e.g. can you get a model in which for spaces which are locally of cardinality $\leq \omega_1$ and which are $\omega_3$-cwH are cwH? In general for which pair of cardinals $(\lambda, \mu)$ we can have models in which a space which is locally of cardinality $< \mu$ and which is $\lambda$--cwH are $\lambda^+$--cwH? In this lecture we shall give few examples where we get some ZFC theorems showing that for some pairs $(\lambda, \mu)$ compactness necessarily fails, and cases of pairs for which one can consistently have compactness for the property of being cwH.

October 24, 2014

Jordi Lopez Abad
Ramsey properties of embeddings between finite dimensional normed spaces

Given d=m , let E m,n be the set of all m×d matrices (a i,j ) such that
(a) ? d j=1 |a i,j |=1 for every 1=i=m .
(b) max m i=1 |a i,j |=1 for every 1=j=d .

These matrices correspond to the linear isometric embeddings from the normed space l d 8 :=(R d ,?·? 8 ) into l d 8 , in their unit bases.
We will discuss and give (hints of) a proof of the following new approximate Ramsey result:
For every integers d , m and r and every e>0 there exists n such that for every coloring of E d,n into r -many colors there is A?E m,n and a color i<r such that A·E d,m ?(c -1 (i)) e . Its proof uses the Graham-Rothschild Theorem on partitions of finite sets. We extend this result, first for embeddings between \emph{polyhedral} normed spaces, and finally for arbitrary finite dimensional normed spaces to get the following:
For every finite dimensional normed spaces E and F , every ?>1 and e>0 , and every integer r , there is some n such that for every coloring of Emb ? 2 (F,l n 8 ) into r -many colors there is T?Emb ? (G,l n 8 ) and some color i<r such that T°Emb ? (F,G)?(c -1 (i)) ? 2 -1+e .
As a consequence, we obtain that the group of linear isometries of the Gurarij space is extremely amenable. A similar result for positive isometric embeddings gives that the universal minimal flow of the group of affine homeomorphisms of the Poulsen simplex is the Poulsen simplex itself.
This a joint work (in progress) with Dana Bartosova (University of Sao Paulo) and Brice Mbombo (University of Sao Paulo)

October 17, 2014

Speaker 1 (from 12:30 to 13:30):
Dana Bartosova
Finite Gowers' Theorem and the Lelek fan

The Lelek fan is a unique non-degenrate subcontinuum of the Cantor fan with a dense set of endpoints. We denote by $G$ the group of homeomorphisms of the Lelek fan with the compact-open topology. Studying the dynamics of $G$, we generalize finite Gowers' Theorem to a variety of operations and show how it applies to our original problem. This is joint work with Aleksandra Kwiatkowska.

Speaker 2 (from 13:30 to 15:00):
Assaf Rinot
Productivity of higher chain condition

We shall survey the history of the study of the productivity of the k-cc in partial orders, topological spaces, and Boolean algebras. We shall address a conjecture that tries to characterize such a productivity in Ramsey-type language. For this, a new oscillation function for successor cardinals, and a new characteristic function for walks on ordinals will be proposed and investigated.

October 10

Sheila Miller
Critical sequences of rank-to-rank embeddings and a tower of finite left distributive algebras

In the early 1990's Richard Laver discovered a deep and striking correspondence between critical sequences of rank-to-rank embeddings and finite left distributive algebras on integers. Each $A_n$ in the tower of finite algebras can be defined purely algebraically, with no reference to the elementary embeddings, and yet there are facts about the Laver tables that have only been proven from a large cardinal assumption. We present here some of Laver's foundational work on the algebra of critical sequences of rank-to-rank embeddings and some work of the author's, describe how the finite algebras arise from the large cardinal embeddings, and mention several related open problems.

October 3

Ilijas Farah
Omitting types in logic of metric structures is hard

One of the important tools for building models with prescribed second-order properties is the omitting types theorem. In logic of metric structures omitting types is much harder than in classical first-order logic (it is Pi-1-1 hard). Although the motivation for this work comes from C*-algebras, the talk will mostly be on descriptive set theory. The intended takeaway from the talk is "logic of metric structures blends with descriptive set theory beautifully." This is joint work with Menachem Magidor.

September 26

No seminar

September 19

Daniel Soukup
Trees, ladders and graphs

The chromatic number of a graph $G$ is the least (cardinal) number $\kappa$ such that the vertices of $G$ can be covered by $\kappa$ many independent sets. A fundamental problem of graph theory asks how large chromatic number affects structural properties of a graph and in particular, is it true that a graph with large chromatic number has certain obligatory subgraphs? The aim of this talk is to introduce a new and rather flexible method to construct uncountably chromatic graphs from non special trees and ladder systems. Answering a question of P. Erdos and A. Hajnal, we construct graphs of chromatic number $\omega_1$ without uncountable infinitely connected subgraphs.

September 12

Konstantinos Tyros
A disjoint union theorem for trees

In this talk we will present an infinitary disjoint union theorem for level products of trees. An easy consequence of the dual Ramsey theorem due to T.J. Carlson and S.G. Simpson is that for every Suslin measurable finite coloring of the power set of the natural numbers, there exists a sequence $(X_n)_{n\in\mathbb{N}}$ of disjoint non-empty subsets of $\mathbb{N}$ such that the set
\[\Big\{\bigcup_{n\in Y}X_n:\; Y\;\text{non-empty subset of }\mathbb{N}\Big\}\]
is monochromatic. The result that we will present is of this sort, where the underline structure is the level product of a finite sequence of uniquely rooted and finitely branching trees with no maximal nodes of height $\omega$ instead of the natural numbers.
As it is required by the proof of the above result, we develop an analogue of the infinite dimensional version of the Hales--Jewett Theorem for maps defined on a level product of trees, which we will also present, if time permits.

September 5

Diana Ojeda
Finite forms of Gowers' Theorem on the oscillation stability of $c_0$

We give a constructive proof of the finite version of Gowers' $FIN_k$ Theorem and analyze the corresponding upper bounds. The $FIN_k$ Theorem is closely related to the oscillation stability of $c_0$. The stabilization of Lipschitz functions on arbitrary finite dimensional Banach spaces was proved well before by V. Milman. We compare the finite $FIN_k$ Theorem with the Finite Stabilization Principle found by Milman in the case of spaces of the form $\ell_{\infty}^n$, $n\in N$, and establish a much slower growing upper bound for the finite stabilization principle in this particular case.

August 29

Seminar cancelled

 
August 22

Mike Pawliuk
Various types of products of Fraisse Classes, various types of amenability and various types of preservation results.
This is joint work with Miodrag Sokic.

In a recent paper of Jakub Jasinski, Claude Laflamme, Lionel Nguyen Van Thé and Robert Woodrow (Arxiv: 1310.6466) it was shown that certain Fraisse Classes are actually Ramsey classes. For many of those cases we have determined whether their automorphism groups are extremely amenable or not. Some of these spaces turn out to actually be a special type of product of Fraisse classes. We were able to prove that unique ergodicity (a type of amenability) is preserved under this type of product.

August 1

Martino Lupini
Functorial complexity of Polish and analytic groupoids

I will explain how one can generalize the theory of Borel complexity from analytic equivalence relations to groupoids by means of the notion of Borel classifying functor. This framework allows one to capture the complexity of classifying the objects of a category in a functorial way. I will then present the first results relating the functorial complexity of a groupoid and the complexity of its associated orbit equivalence relation, focusing on the case of Polish groupoids: For Polish groupoids with essentially treeable equivalence relations any Borel reduction between the orbit equivalence relations extends to a Borel classifying functor. On the other hand for any countable non-treeable equivalence relation E there are Polish groupoids of different functorial complexity both having E as associated orbit equivalence relation. The proof of these results involves a generalization of some fundamental results on the descriptive set theory of actions of Polish groups --such as the Becker-Kechris theorem on Polishability of Borel G-spaces-- to actions of Polish groupoids.

July 25

Saeed Ghasemi
An analogue of Feferman-Vaught theorem for reduced products of metric structures

I will give a metric version of the Feferman-Vaught theorem for reduced products of discrete spaces. We will use this to show that, under the continuum hypothesis, the reduced powers of any metric structure over atomless layered ideals are isomorphic. As another application, I will give an example of two reduced products of sequences of matrix algebras over Fin, which are elementarily equivalent, therefore isomorphic under the CH, with no trivial isomorphisms between them.

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.


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