# SCIENTIFIC PROGRAMS AND ACTIVITIES

March 30, 2015

## Set Theory Seminar Series at the Fields Institute Friday, 1:30 p.m.

Organizers:
Ilijas Farah,
York University, Juris Steprans, York University.

 Seminars held during 2006-07 Seminars held during 2007-08 Directions to Fields Of interest: Winter School in Abstract Analysis - Section Topology 31st January — 7th February, 2009
For a complete listing of seminars and abstracts: http://www.math.yorku.ca/~ifarah/seminar.html
 2008/09 Fridays Speaker and Talk Title *Monday June 1* Fields Institute Room 210 10:30-12:00 Speaker KP Hart (Miami University, Ohio and Delft, Netherlands) Title: TBA 1:30-3:00 Speaker Marion Scheepers (Boise State) Title: TBA 3:30-5:00 Speaker: Paul Larson (Miami University, Ohio) Title: Universally measurable sets in generic extensions. Wednesday, May 27* 1:30-3:00 in BA6183 on 6th fl. Bahen Centre OF NOTE Speaker: Gary Gruenhage (Auburn) Title: SLIM DENSE SETS IN PRODUCTS *Thursday May 28* 1:30-3:00 in Bahen Centre 6183 OF NOTE Speaker: Justin Moore (Cornell) Title: Fast growth of the Folner function for Thompson's group F. May 22, 2009 1:30-3:00 Beatriz Zamora-Aviles (York University) Analytic P-ideals on $B(H)^+_{\leq 1}$ May 15, 2009 1:30-3:00 Tamas Matrai (University of Toronto) Infinite dimensional perfect set theorems We obtain infinite dimensional analogues of some classical perfect set theorems, and we indicate how these result can be applied to the study of analytic ideals. May 8, 2009 No talk scheduled Slides of Talk Simon Thomas (Rutgers University) Some Consequences of Martin's Conjecture Abstract: In this talk, I will explore some of the consequences of Martin's Conjecture on degree invariant Borel maps. These include the strongest conceivable ergodicity result for the Turing equivalence relation, as well as the statement that the complexity of a universal countable Borel equivalence relation always concentrates on a null set. April 24, 2009 1:30-3:00 lijas Farah (York University) Nonseparable UHF algebras Uniformly HyperFinite (UHF) algebras are those C* algebras in which every finite subset is `near' a finite-dimensional full matrix subalgebra. This can be formalized in three different ways, all three being equivalent in the separable case. Separable UHF algebras were classified in the 1960s by Glimm and Dixmier. Dixmier asked whether three definitions are equivalent in the nonseparable case. I will give a complete answer to this question as well as some remarks on extending the Glimm-Dixmier theorem to the nonseparable case. This is a joint work with Takeshi Katsura. April 17, 2009 1:30-3:00 Frank Tall (University of Toronto) Lindelof spaces and selection principles, II. We continue our investigation of Lindelof spaces and selection principles, focusing on the relationships among productive Lindelofness and the Menger and Hurewicz properties. March 27/09 1:30-3:00 Jordi Lopez Abad (Universite Denis-Diderot Paris7) Generic constructions of Banach spaces. The aim of this talk is to present a forcing construction "à la Cohen" of generic Banach spaces. These spaces are Gurarij spaces, and in the case of the non-separable context, they can be non-isomorphic. These constructions can also be used to distinguish the existence of different kind of uncountable biorthogonal-like sequences. This is a joint work with S. Todorcevic. March 20/09 1:30-3:00 Arnie Miller (University of Wisconsin, Madison) The hierarchy of $\omega_1$-Borel sets The family $\omega_1$-Borel sets is the smallest family of subsets of the real line which contains the family of open sets and is closed under complementation and $\omega_1$ unions. We show: Theorem 1. MA + not CH implies this hierarchy has length $\omega_2$. Theorem 2. In the Cohen real model it has length either $\omega_1+1$ or $omega_1+2. March 20/09 3:15 - 4:45 Natasha Dobrinen, (University of Denver) Tukey degrees of ultrafilters Let$U$and$V$be ultrafilters on$\omega$. We say that$V$is Tukey reducible to$U$($V\le_T U$) if there is a "Tukey map"$g: V\rightarrow U$, meaning that$g$maps unbounded subsets of$V$to unbounded subsets of$U$. Equivalently, there is a "cofinal" map$f: U\rightarrow V$which maps cofinal subsets of$U$to cofinal subsets of$V$. Tukey reducibility is a generalization of Rudin-Keisler reducibility. In general,$V\le_{RK} U$implies$V\le_T U$but not vice versa. However, it is still unknown if this is the case for p-points. We present some results on the structure of the Tukey degrees of ultrafilters on$\omega$, concentrating on p-points and ultrafilters with similar properties, along with many questions. This is joint work with Stevo Todorcevic. March 13 1:30-3:00 Alexander Pyshchev (Nipissing University) On nonstandard hull-like spaces. We investigate topological spaces obtained as quotients of internal sets in a nonstandard universe. March 13 3:15 - 4:45 Carlos DiPrisco, (IVIC, Venezuela) Chromatic numbers of analytic shift graphs February 27 Christopher Miller (Ohio State University) Tameness in expansions of the real field What should it mean for a first-order expansion of the field of real numbers to be tame, or well behaved? In recent years, much attention has been paid by model theorists and real-analytic geometers to the o-minimal setting: expansions of the real field in which every definable set has finitely many connected components. But there are expansions of the real field that are tame in some well-defined sense, yet define sets with infinitely many connected components. Moreover, there are different kinds of tameness that can arise. The analysis of these structures tends to require a mixture of model-theoretic, analytic-geometric and descriptive set-theoretic techniques. Underlying all this is an idea that first-order definability, in combination with the field structure, can be used as a tool for determining how complicated given sets of real numbers are, in particular, this gives us a new way to think about projective sets. This will be a primarily expository survey talk, intended to be accessible to anyone with a background in basic logic. February 20 1:30-3:00 Frank Tall (University of Toronto) Selection principles and Lindelof spaces which are indestructible, productive, or D. February 13 2:00 - 3:30 Dilip Raghavan (University of Toronto) Suslin Lattices (Continued) February 6, 2009 2:00 - 3:30 Dilip Raghavan (University of Toronto) Suslin Lattices January 23/09 1:30-3:00 Leandro Aurichi (University of Sao Paolo) TBA January 16/09 1:30-3:00 Asger Tornquist (University of Vienna) Borel reducibility and von Neumann equivalence January 9/09 1:30-3:00 Lionel Nguyen Van Thé (University of Calgary) Problems and results around metric oscillation stability. In 1994, Odell et Schlumprecht built a uniformly continuous map from the unit sphere of the Hilbert space into the unit interval and which does not stabilize on any isometric copy of the sphere. This anti-Ramsey result allowed to show that the Hilbert space has a property known as 'distortion'. The purpose of this talk is to consider similar problems when the Hilbert space is replaced by the so-called Urysohn metric space. December 19, 1:30-3:00 Tamas Matrai (University of Toronto) Sigma-ideals of compact sets in the Tukey ordering, continued December 12 1:30-3:00 Tamas Matrai (University of Toronto) Sigma-ideals of compact sets in the Tukey ordering We introduce a construction scheme of G_delta sigma-ideals of compact sets and we try to find the place of the ideals obtained in the Tukey ordering. December 5 1:30-3:00 Matthew Foreman (University of California, Irvine) Rational Invariant Measures Global questions about classifications of ergodic measure preserving transformations are usually studied by adopting one or another universal model for the measure preserving transformations. In this lecture I describe a new universal model investigated in joint work with B. Weiss. The underlying space of this model is$\Sigma^Z$, where$\Sigma$is a countable set. The new part is that the invariant measures are required to give rational values to each cylindar set. November 28 1:30-3:00 Leandro Aurichi (University of Sao Palo, Brazil) D-spaces and games Some games related to D-spaces, Rothberger and Menger spaces November 21 1:30-3:00 (TALK HAS BEEN CANCELLED this week) Dilip Raghavan, University of Toronto The P-ideal Dichotomy and Lattices November 14 1:30-3:00 Bernhard Koenig (University of Toronto) Variations of Axiom (continued) It is known that Fleissner's Axiom R is basically a variation of the stationary reflection principle. We investigate the exact status of Axiom R within the realm of stationary reflection principles by presenting some implications but also independence results. November 7 1:30-3:00 Bernhard Koenig (University of Toronto) Variations of Axiom It is known that Fleissner's Axiom R is basically a variation of the stationary reflection principle. We investigate the exact status of Axiom R within the realm of stationary reflection principles by presenting some implications but also independence results. October 31 1:30-3:00 Carlos Azarel (University of Toronto) Well quasi-ordering Aronszjan lines II We will prove that under PFA the class of Aronszjan lines is well quasi-ordered. October 24, 1:30-3:00 Carlos Azarel (University of Toronto) Well quasi-ordering Aronszjan lines. We will prove that under PFA the class of Aronszjan lines is well quasi-ordered. October 17, 1:30-3:00 **Room change Stewart Library** Stevo Todorcevic (University of Toronto and CNRS Paris) Forcing with a coherent Souslin tree October 10 1:30-3:00pm Leandro Aurichi (University of Sao Paolo, Brazil) The Rothberger and Menger properties We show some applications of these properties, including some related to preservation by forcing. September 19 1:30-3:00pm Tamas Matrai (University of Toronto) Hurewicz testing September 12, 1:30-3:00 Tamas Matrai (University of Toronto) Introduction I would like to present three topics I plan to work on during my stay in Toronto. These are: 1) Tukey reducibility of ideals and$\sigma$-ideals 2) Borel reducibility among$\ell^{p}\$-like equivalence relations 3) Hurewicz testing September 5 1:30-3:00 Leandro Aurichi (University of Sao Paolo, Brazil) Trees with fine wedge and coarse wedge topologies. We present a technique for constructing some examples of spaces which answer some questions on discretely generated properties and spaces. August 29 1:30-3:00pm Asger Tornquist (University of Toronto) Essentially countable equivalence relations A Borel equivalence relation is essentially countable if it is Borel reducible to a Borel equivalence relation with countable classes. We will give a new and simpler proof of a Theorem of Greg Hjorth, which states that there is more essentially countable Borel equivalence relations than there are Borel equivalence relations with countable classes. August 22 1:30-3:00pm Dilip Raghavan (University of Toronto) Proof of a conjecture of Brendle July 11, 2008 room 210 Dilip Raghavan (University of Toronto) A Van Douwen MAD family in ZFC.