# SCIENTIFIC PROGRAMS AND ACTIVITIES

October  9, 2015

## Set Theory Seminar Series Fields Institute (map) Friday, 1:30 p.m.

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

 Seminars held during 2008-09 Associated Activities Math 6041, Forcing Course, Summer 2010 Seminars held during 2007-08 Directions to Fields
For a complete listing of seminars and abstracts: http://www.math.yorku.ca/~ifarah/seminar.html
 2009-10 PAST SEMINARS Friday, June 25, 1:30-3:00 (location to be announced). Adam Ostaszewski (London School of Economics and Political Science) Title: TBA June 19, 1:30-3:00 Room 230 Ilijas Farah (York University) All automorphisms of all Calkin algebra (continued) (Joint work with Paul McKenney and Ernest Schimmerling) I will prove that PFA implies that all automorphisms of the Calkin algebra associated with any infinite-dimensional Hilbert space are inner. June 11, 2010 Ilijas Farah (York University) TBA June 4, 2010 Dilip Raghavan (University of Toronto) Distinguishing two cardinal invariants of the continuum May 28, 2010 *** SEMINAR CANCELED *** May 14, 2010 *** SEMINAR CANCELED *** May 7, 2010 10:30 a.m. Room 230 Franklin D Tall (University of Toronto) Productively Lindelof spaces may all be D I will prove that the Continuum Hypothesis implies every productively Lindelof space is a D-space. Collateral damage includes that Borel's Conjecture implies every Rothberger space is Hurewicz. May 7, 2010 1:30 p.m. Room 210 Brice Minaud (Paris) Reflection principle implies the singular cardinal hypothesis, a simplified version of a proof by Shelah cont. Shelah proved in 2004 that Stationary Set Reflection implies SCH. We present a simpler version of the proof, involving a closed game. April 30, 2010 Teruyuki Yorioka (Shizuoka University) Aronszajn trees and weak fragments of Martin's Axiom I will talk a summary of the following papers: (1) Some Weak fragments of Martin's Axiom related to the rectangle refining property , Arch. Math. Logic 47 (2008), 79--90. (2) The inequality $\mathfrak{b}>\aleph_1$ can be considered as an analogue of Suslin's Hypothesis , Axiomatic Set Theory and Set-theoretic Topology (Kyoto 2007), S?rikaisekikenky?sho K?ky?roku No. 1595 (2008), 84--88. (3) A non-implication between fragments of Martin's Axiom related to some property which comes from Aronszajn trees , Ann. Pure Appl.Logic, 161 (2010), 469-487. (4) Uniformizing Ladder system colorings and the rectangle refining property , Proc. Amer. Math. Soc., to appear. (For (1), (2), (3), see Shizuoka University repository which is linked from my webpage: http://www.ipc.shizuoka.ac.jp/~styorio/ For (4), see Recently posted articles in PAMS website.) In these papers, some combinatorial properties of forcing notions are introduced. All of them originate from an analysis of Aronszajn trees, and are stronger than the countable chain condition. We argue some consistency results about fragments of MA_{aleph_1} restricted to forcing notions with their properties. In the talk, we prove that forcing notions with such properties add no random reals. April 23, 2010 Brice Minaud (Paris) Reflection principle implies the singular cardinal hypothesis, a simplified version of a proof by Shelah Shelah proved in 2004 that Stationary Set Reflection implies SCH. We present a simpler version of the proof, involving a closed game. April 16, 2010 Leandro Aurichi (Sao Paolo) Strongly D-spaces We will talk about a condition that is stronger than being D and that implies Lindelofness. April 9, 2010 Ralf Schindler (Universität Münster) Bounded forcing axioms and Pi-2 statements Many natural Pi-2 consequences of Martin's Maximum have been verified to follow from Bounded Martin's Maximum plus the statement that NS, the nonstationary ideal on \omega_1, be precipitous. The forcing whose variants are exploited here is semi-proper if and only if all stationary set preserving forcings are semi-proper. March 26, 2010 Carlos Azarel (University of Toronto) Gap Structure of Coherent Aronszajn Trees (continued) We give a detailed description of the gap structure of $({\cal C},\prec)$ (the class of coherent Aronszajn trees) under the assumption of $MA_{\omega_1}$. Our study on gaps shows that the class $MA_{\omega_1}$ is universal for all linear orders of cardinality at most $\aleph_2$, i.e.$({\cal C},\prec)$ contains an isomorphic copy of each linear ordered set of size less or equal than $\aleph_2$. March 19, 2010 Carlos Azarel (University of Toronto) Gap Structure of Coherent Aronszajn Trees We give a detailed description of the gap structure of $({\cal C},\prec)$ (the class of coherent Aronszajn trees) under the assumption of $MA_{\omega_1}$. Our study on gaps shows that the class $MA_{\omega_1}$ is universal for all linear orders of cardinality at most $\aleph_2$, i.e. $({\cal C},\prec)$ contains an isomorphic copy of each linear ordered set of size less or equal than $\aleph_2$. March 12, 2010 Benjamin Miller Defining non-empty small sets from families of infinite sets We consider circumstances under which non-empty small subsets of a space can be defined from families of infinite subsets of the space, in the process establishing generalizations of Mansfield's perfect set theorem and the Lusin-Novikov uniformization theorem. ***Special Day and Time*** March 10, 2010 3:30 p.m. Stewart Library Ryszard Frankiewicz (IM PAN, Warsaw) Remarks on covering by nowhere Ramsey sets March 5, 2010 Frank Tall (University of Toronto) Productively Lindelof Spaces A space is productively Lindelof if its product with every Lindelof space is Lindelof. We have a variety of new results about such spaces, obtained via assorted set-theoretic methods. February 19, 2010 Menacham Magidor (Hebrew University of Jerusalem) On maximal resolvability of monotonically normal spaces February 12, 2010 David Milovich (Texas A&M International) On the order theory of local bases The local Noetherian type of a point in a space is the least kappa such that that point has a local base that is kappa-like with respect to the containment ordering.Local Noetherian type is surprisingly connected to Van Douwen's Problem. GCH implies that the local Noetherian type of point in a homogeneous compactum cannot exceed the cellularity of that space. All known homogeneous compacta have cellularity at most $2^{\aleph_0}$ and local Noetherian type at most $\aleph_0$. Local Noetherian type is even more closely connected to Tukey reducibility. Indeed, it is at the heart of Isbell's Problem, which can be formulated as asking whether the Stone-Cech remainder of omega has a point with uncountable local Noetherian type (assuming only ZFC). For a connection to large cardinals, consider the local Noetherian type of an arbitrary point in the $G_\delta$ modification of $2^{\aleph_\omega}$. It is $\aleph_1$ assuming V=L; it is $\aleph_2$ assuming GCH and Chang's Conjecture at $\aleph_\omega$. My talk will (1) survey basic facts about local Noetherian types and connections mentioned above, (2) survey recent results about limit cardinals and product topologies, and (3) mention some results from the more difficult theories of (global) Noetherian type and Noetherian $\pi$-type. January 22 NO SEMINAR February 5 NO SEMINAR January 22 NO SEMINAR January 15 1:30-3:00 Stewart Library Frank Tall (University of Toronto) Set-theoretic problems concerning Lindelof spaces We survey a variety of classic problems involving Lindelof spaces that have partial set-theoretic solutions. These are good research problems for graduate students in set theory or set-theoretic topology. January 8 Talk Cancelled Dec. 18 1:30-3:00 Room 210 Márton Elekes (University of Toronto) Haar null sets and the consistent reflection of nonmeagerness in Cantor sets Dec. 11 1:30-3:00 Room 210 Ilijas Farah (York University) A dichotomy for the number of ultrapowers (continued). (joint work with S. Shelah.) The paper is now available at: http://www.math.yorku.ca/~ifarah/preprints.html Dec. 4, 1:30pm Room 210 Logan Hoehn (University of Toronto) A counterexample for Lelek's problem in continuum theory Nov. 27, 1:30pm Room 210 Ilijas Farah (York University) A dichotomy for the number of ultrapowers (joint work with S. Shelah.) Nov. 20, 1:30pm Room 210 David Fremlin (University of Essex) TBA Nov. 13, 1:30pm Room 210 No talk scheduled Nov. 6, 1:30pm Room 210 Miodrag Sokic (University of Toronto) Ramsey properties of finite posets, continued October 30 at 1:30pm Room 210 no talk scheduled October 23 at 1:30pm Room 210 David Fremlin (University of Essex) Various kinds of ultrafilter content: extracts from http://www.essex.ac.uk/maths/staff/fremlin/n09102.ps October 16 at 3:30pm Room 210 Frank Tall (University of Toronto) Applying PFA(S)[S] - some of the tools One of the aspects of Todorcevic's proof that PFA(S)[S] implies that "in compact countably tight spaces, locally countable subsets of size aleph_1 are sigma-discrete" is a general method for forcing certain collections to be 'sigma-small'. We will talk about this method, and work in progress applying it to problems concerning under what circumstances does normality imply collectionwise Hausdorffness. October 16 at 1:30pm Room 210 Miodrag Sokic (University of Toronto) Ramsey properties of finite posets Using result of Kechirs-Pestov-Todor?evi? we analyise Fraïssé classes of finite posets. In order to conduct such analysis we have to examine various classes of finite posets with linear orderings. There are two natural ways we can add linear orderings: to be arbitrary or to be linear extension of partial orderings. Our discussion contains examinantion of ordering property and Ramsey property of such classes where we use techinque of partite construction and concept of a-colored sets. On the end we give topological meaning of the previous obtained combinatorila results and give list of few extremely amenable groups as well as list of few universal minimal flows. October 9 at 1:30pm Room 210 Dilip Raghavan (University of Toronto) A model with no strongly completely separable MAD families, continued October 2 at 1:30pm Room 210 Boris Model (Ben Gurion University in the Negev) On the Theory of Infinite Stage Games of Search and Completion On the border of Games Theory and Set Theory there is a broad class of Infinite Stage Decision Making Processes that can be characterized by the main followingproperty: a Future development of the process depends on the process Present state and does not depend directly on the process Past [1]. As an example of such kind ofprocesses and some connected with them questions Infinite Stage Games of Search and Completion [2, 3], which are interesting by themselves also, are presented. For these games (of the same nature as for example chess and draughts have: a Future depends on the Present and does not depend directly on the Past) with the use ofcontinuum hypothesis much unexpected results can be proved, for example: The least guaranteed result of these simultaneously played but completely independent gamesturns out to be less than the sum of the least guaranteed results of constituent games (and not equal to this sum as it could be supposed and as it is in the case of chess or draughts!). References 1. B. I. Model’, The existences of an overall ?-optimal strategy and validity of Bellman’sfunctional equation in an extended class of dynamic processes. I; II, Engineering Cybernetics, No. 5, 1975, pp. 13 – 19; No. 6, 1975, pp. 12 – 19. 2. B. I. Model’, Games of search and completion, Journal of Mathematical Sciences, Vol. 80, No 2, 1996, pp. 1699 - 1744, Plenum Publishing Corporation, New York. 3. U. Abraham, R. Schipperus, Infinite Games on Finite Sets, Israel Journal of Mathematics, Vol. 159, 2007, pp. 205 -219. September 25 at 1:30pm ***SIDNEY SMITH 2118*** Gregory Chaitin (IBM Thomas J. Watson Research Centre) Leibniz, Complexity & Incompleteness September 18 at 1:30pm Room 210 Dilip Raghavan (University of Toronto) A model with no strongly completely separable MAD families August 28 at 1:30pm Room 210 Maxime Burke (University of Prince Edward Island) Approximating smooth functions by "generic" entire functions August 21 at 1:30pm Room 210 Dilip Raghavan (University of Toronto) Cofinal Types of Ultrafilters August 14 at 1:30pm Room 210 Asger Tornquist (University of Vienna) On the pointwise implementation of measure preserving actions, continued August 7 at 1:30pm Room 210 Asger Tornquist (University of Vienna) On the pointwise implementation of measure preserving actions Tuesday, July 14, at 3:30pm-5:00pm in BA2195, Of interest: Sean Saunders will give a presentation for his Master's summer project: Van der Waerden's Theorem and Hindman's Theorem: Topological Dynamics Proofs of Theorems in Number Theory.