# SCIENTIFIC PROGRAMS AND ACTIVITIES

July 28, 2016

## 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 2002-03 Seminars held during 2005-06 Seminars held during 2003-04 Seminars held during 2006-07 Seminars held during 2004-05 Directions to Fields
For a complete listing of seminars and abstracts: http://www.math.yorku.ca/~ifarah/seminar.html
 2008 Speaker and Talk Title Friday, June 27 1:30-3:00pm U of T Bahen Centre, Room 6183 Ilijas Farah (York University) The commutant of B(H) in its ultrapower and flat ultrafilters A question of Eberhard Kirchberg has lead to an isolation of a new type of ultrafilter on N and a consistency result. This is a joint work with N. Christopher Phillips and Juris Steprans. PLEASE NOTE THE UNUSUAL LOCATION FOR THIS SEMINAR. THE BAHEN CENTRE IS THE BUILDING HOUSING THE DEPARTMENT OF MATHEMATICS BEHIND THE FIELDS INSTITUTE May 9 no seminar May 2 no seminar Friday April 25 1:30-3:00 Jan Pachl Ambitable topological groups A topological group G is called ambitable if every uniformly bounded uniformly equicontinuous set of functions on G with its right uniformity is contained in an ambit, within the G-flow defined by right translations on uniformly continuous functions. This notion is motivated by investigations of topological centres in the semigroups that are of interest in abstract harmonic analysis --- for ambitable groups, topological centres have an effective characterization. Simple sufficient conditions for ambitability, expressed in terms of two cardinal functions, yield a generalization of previously known characterizations of topological centres. The talk will follow the content of the paper with the same name, available as arXiv:0803.3405. Friday April 18 1:30-3:00 Stevo Todorcevic (University of Toronto and CNRS Paris) Unconditional Basic Sequences in Spaces of High Density Friday April 11 1:30-3:00 No Seminar Friday April 4 1:30-3:00 Bohuslav Balcar (Czech Academy of Sciences) Refinement properties of countable sets I will present a notion of almost disjoint refinement and its application and relationships to ultrafilters on $\omega$, semiselective co-ideals, Boolean algebras, and generic extensions of models of set theory. Friday Mar. 28 1:30-3:00 Pandelis Dodos (University of Paris 6 and National Technical University of Athens) Universality Problems and \Script L_\infty -spaces Friday Mar. 14 1:30-3:00 CANCELLED Friday Mar. 7 1:30-3:00 Ernest Schimmerling (Carnegie Mellon University) Some open questions about L There are indeed open questions about L. I'll state a few and describe related results on two two topics: forcing axioms and mutual stationarity. The talk will include an overview of Jensen's fine structure theory and, as such, will be self-contained. Friday Feb. 29 1:30-3:00 Frank Tall (University of Toronto) On a core concept of Arhangel'skii (Con't) Friday Feb. 22 1:30-3:00 Frank Tall (University of Toronto) On a core concept of Arhangel'skii Arhangel'skii defines a locally compact space to have a countable core if it is the union of countably many open subspaces, each having the property that infinite subsets have limit points in the whole space. Investigating the question of when such spaces are sigma-compact as usual leads to the consideration of additional set-theoretic axioms. Friday, Feb. 15, 1:30 - 3:00 Natasha Dobrinen (University of Denver) The consistency strength of the tree property at the double successor of a measurable cardinal This is joint work with Sy-David Friedman. Koenig's Lemma is the well-known theorem that any inifinite finitely branching tree of height $\omega$ has an infinite path through it. This theorem, however, fails for trees of height $\omega_1$. More precisely, Aronszajn constructed a tree of height $\omega_1$, all of whose levels are countable, with no cofinal branch. Such a tree is naturally called an Aronszajn tree. For a regular uncountable cardinal $\kappa$, a "$\kappa$-tree" is a tree of height $\kappa$ all of whose levels have size less than $\kappa$. A "$\kappa$-Aronszajn" tree is a $\kappa$-tree with no cofinal branches. We say that the "tree property at $\kappa$ holds" if there are no $\kappa$-Aronszajn trees. As soon as $\kappa$ is greater than $\aleph_1$, large cardinals become necessary. Silver showed that if the tree property holds at $\aleph_2$, then $\aleph_2$ must be weakly compact in $L$. Mitchell showed that from a weakly compact cardinal one can force the tree property at $\aleph_2$. Thus began the study of the consistency strengths of the tree property at various cardinals. We show that the tree property at the double successor of a measurable cardinal is equiconsistent with what we call a weakly compact hypermeasurable cardinal. Our methods us a reverse Easton iteration of iterated Sacks forcings. The difficulty lies in showing that the weakly compact hypermeasurable cardinal remains measurable after this forcing. Friday, Feb. 8, 1:30 - 3:00 Bernhard Koenig (University of Toronto) Two ways to construct an omega_2-Suslin-tree from GCH plus additional assumptions Friday, Feb. 1, 1:30 - 3:00 (**talk cancelled due to weather) **Natasha Dobrinen (University of Denver) The consistency strength of the tree property at the double successor of a measurable cardinal Friday, Jan. 25, 1:30 - 3:00 Students will present problems, including: L. Hoehn: The chainable vs. span-zero problem for non-metrizable continua; B. Zamora: About Hadwin's conjecture; V. Fischer: Consistency results with large continuum; A. Fischer: PFA(S)[S] vs. PFA; C. Martinez: Some problems on Aronszajn lines; A. Brodsky: The sigma-finite chain condition and its use in absoluteness proofs. Friday, Jan. 18, 1:30 - 3:00 Udayan B. Darji (University of Louisville) Generating dense subgroups of some transformations groups Friday, Dec. 21, 1:30 - 3:00 Vera Fischer (York University) Towards the consistency of arbitrarily large spread between the bounding and the splitting numbers We will suggest a countably closed forcing notion which satisfies the $\aleph_2$ chain condition and adds a centered family $C$ of Cohen names for pure conditions, which has the property that forcing with $Q(C)$ preserves a chosen subfamily of the Cohen reals unbounded and diagonalizes all of them. Friday, Dec. 14, 1:30 - 3:00 Vera Fischer (York University) The consistency of $b=\kappa + s=\kappa^+$, continued Friday, Dec. 7, 1:30 - 3:00 Vera Fischer (York University) The consistency of $b=\kappa + s=\kappa^+$ Using finite support iteration of c.c.c. partial orders we provide a model of b=\kappa < s=\kappa^+ for \kappa arbitrary regular, uncountable cardinal. Friday, Nov. 30, 1:30 - 3:00 Greg Hjorth (UCLA) Ends and percolation This is part of joint with Inessa Epstein, where we analyze countable Borel equivalence relations with many ends and the actions of groups with many ends. As a consequence of this work we obtain a result regarding the percolation on non-amenable groups that have infinite normal amenable subgroups. Friday, Nov. 23, 1:30 - 3:00 Marton Elekes, Renyi Institute, Budapest Partitioning multiple covers into many subcovers Motivated by a question of A. Hajnal originating from geometry we investi-gate the following set of problems. Let X be a set,  a cardinal number, and H a family that covers each x 2 X at least  times. Under what assumptions canwe decompose H into  many subcovers? Equivalently, under what assumptions can we colour H by  many colours so that for each x 2 X and each colour c there exists H 2 H of colour c containing x? The assumptions we make can be e.g. that H consists of open, closed, compact, convex sets, or polytopes in Rn, or intervals in a linearly ordered set, or we can make various restrictions on the cardinality of X, H or elements of H. Besides numerous positive and negative results many questions turn out tobe independent of the axioms of set theory. This is a joint work with T. M´atrai and L. Soukup. Friday, Nov. 16, 1:30 - 3:00 Magdalena Grzech, Cracow University of Technology Complemented subspaces of the Banach space $l_\infty / c_0$ Friday, Nov. 2, 1:30 - 3:00 Bernhard Koenig (University of Toronto), part 2 Forcing axioms and two cardinal diamonds I will present some known facts and new results concerning the consistency of forcing axioms with two cardinal diamond principles. Friday, October 26, 1:30 - 3:00 Bernhard Koenig (University of Toronto) Forcing axioms and two cardinal diamonds I will present some known facts and new results concerning the consistency of forcing axioms with two cardinal diamond principles. Friday, October 19, 1:30 - 3:00 Arthur Fischer (University of Toronto) PID in PFA(S)[S] We will demonstrate that the P-ideal dichotomy (PID) holds in models of the form PFA(S)[S]. Time permitting, we will also discuss how certain extensions of PID can be shown to hold in such models. Friday, October 12, 1:30 - 3:00 Istvan Juhasz (Hungarian Academy of Sciences) Discrete subspaces of compacta The following are the main results presented in this talk. Theorem 1. If X is a countably tight compactum such that every limit cardinal strictly below |X| is strong limit then |X| = |D| for some discrete subspace D of X. Theorem 2. If each point of a compactum X has character at leat  then X cannot be covered by fewer than 2-many discrete subspaces. Theorem 3. The !th power of any compactum has a -discrete dense subset. Theorem 4. If the character of a compactum is greater than  then there is a discrete subspace Y of X with |Y |  + and a point p whosecharacter in Y [ {p} is greater than . Theorems 1-3 are joint results with Z. Szentmikl´ossy. Several open problems will also be formulated. Friday, September 28, 1:30-3:00pm Asger Tornquist (University of Toronto) Definable Davies' Theorem, PART II A result due to Davies states that CH is equivalent to every real function on the plane being representable as a sum of square functions, i.e. functions of the form g(x)h(y). We give a definable version of this theorem: Every real is constructible precisely when every \Sigma^1_2 function allows a representation as a sum of \Sigma^1_2 squares. We also discuss the possibility of a stronger converse in this Theorem. Friday, Sept. 21, 1:30-3:00 Fields room 210 Asger Tornquist (University of Toronto) Definable Davies' Theorem, Part 1 A result due to Davies states that CH is equivalent to every real function on the plane being representable as a sum of square functions, i.e. functions of the form g(x)h(y). We give a definable version of this theorem: Every real is constructible precisely when every \Sigma^1_2 function allows a representation as a sum of \Sigma^1_2 squares. We also discuss the possibility of a stronger converse in this Theorem. Friday, Sept. 14, 1:30-3:00 Fields Room 210 Frank Tall (University of Toronto) More Topological Applications of PFA(S)[S] We continue our study of the paracompactness of locally compact normal spaces in models of PFA(S)[S]. Using P-ideal dichotomy, we are able to improve our previous results. The presentation should be understandable to regular seminar participants, even if they missed my lectures last year on paracompactness. Friday, Sept. 7, 1:30-3:00 Fields Room 210. Logan Hoehn (University of Toronto) A model theoretic approach in topology The Wallman representation theorem enables one to describe certain properties of compact Hausdorff spaces with sentences in a first order language, which makes them compatible with some model theoretic constructions. We state this theorem and discuss some of its potential applications and limitations. As a sample application, we show how a certain result about colorings of self-maps of compact finite-dimensional metric spaces can be extended to a broader class of spaces using this approach. Friday, Aug. 3, 1:30-3:00pm Fields, room 210 Bart Kastermans, University of Wisconsin, Madison Cofinitary groups