Toronto Set Theory Seminar - Year 2007.

• Friday, December 7, 1:30-3:00pm in Fields Institue room 210
  Vera Fischer (York University
  The consistency of $b=\kappa
  Abstract: 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 November 30, 1:30-3:00pm in Fields Institute room 210
  Greg Hjorth (University of Melbourne, UCLA)
  Ends and percolation
  Abstract: 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 November 23, 1:30-3:00pm in Fields Institute room 210
  Marton Elekes (Hungarian Academy of Sciences)
  Partitioning multiple covers into many subcovers
  abstract
  
• Friday, November 16th, 1:30-3:00 in Fields room 210
  Magdalena Grzech (Politechnika Krakowska) Complemented subspaces of the Banach space $l_\infty / c_0$
  
• Friday and Saturday November 9-10.
  Conference in honour of the 60th Birthday of Professor Andreas R. Blass
  
• Friday, November 2, 1:30-3:00pm in Fields Institute room 210
  Bernhard Koenig (University of Toronto)
  Forcing axioms and two cardinal diamonds, Part 2.
  
• Friday, October 26, 1:30-3:00pm in Fields Institute room 210
  Bernhard Koenig (University of Toronto)
  Forcing axioms and two cardinal diamonds
  Abstract: 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 in Fields Institute room 210
  Arthur Fischer (University of Toronto)
  PID in PFA(S)[S].
  Abstract: 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 in Fields Institute room 210
  Istvan Juhasz (Hungarian Academy of Sciences)
  Discrete subspaces of compacta
  abstract
• Friday, October 5. NO SEMINAR
• Friday, September 28, 1:30-3:00 in Fields room 210
  Asger Tornquist (University of Toronto)
  Definable Davies' Theorem, Part II
  
• Friday, September 21, 1:30-3:00 in Fields room 210
  Asger Tornquist (University of Toronto)
  Definable Davies' Theorem
  Abstract: 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 September 14, 1:30-3:00 in the Fields Institute Room 210
  Frank Tall (University of Toronto)
  More Topological Applications of PFA(S)[S]
  Abstract: 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, September 7, 1:30-3:00 in Fields Institute Room 210
  Logan Hoehn (University of Toronto)
  A model theoretic approach in topology
  Abstract: 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, June 8, 1:30-3:00pm in Fields Institute Room 210
  Frank Tall (University of Toronto)
  On the metrizability of hereditarily normal manifolds of dimension > 1.
  Abstract: We show that a conjunction of four axioms, three of which are consequences of PFA and one of which is a consequence of V = L, implies such manifolds are metrizable. The consistency of such a result is a long-outstanding conjecture of Peter Nyikos. Two of the axioms are known to follow from PFA(S)[S]; I conjecture that the other two also follow.
  
• Friday, June 1, 1:30-3:00pm in Fields Institute Room 210
  Paul Szeptycki (York University)
  Normality in products with a countable factor.
  
• Friday May 25. NO SEMINAR
  
• Friday May 18, 1:30-3:00pm in Fields Institute Room 210
  Philip Kremer (Department of Philosophy, University of Toronto)
  Dynamic Topological Logic
  
• Friday May 11, 1:30-3:00 in Fields Institute Room 210
  Arthur Fischer (University of Toronto)
  'Balogh's Sigma' in models of PFA(S)[S] continued.
  
• Friday May 4, 1:30-3:00 in Fields Institute Room 210
  Arthur Fischer (University of Toronto)
  'Balogh's Sigma' in models of PFA(S)[S] continued.
  
• Friday April 27, 1:30-3:00, Fields Institute, Room 210
  Arthur Fischer (University of Toronto)
  'Balogh's Sigma' in models of the form PFA(S)[S]
  Abstract: Continuing the study of topological consequences of forcing with the coherent Souslin tree S in models of PFA(S), we will demonstrate that the following PFA result also holds: Every locally countable subspace Z of size À₁ in a compact sequential space X is s-closed discrete in some open neighbourhood of Z in X which "locally witnesses" the local countability of Z in X.
• Friday April 20, 1:30-3:00, Fields Institute, Room 210
  Paul Larson (Miami University, Ohio)
  The stationary set splitting game
  Abstract: The stationary set splitting game is a game of perfect information of length w₁ between two players, UNSPL and SPLS, in which UNSPLS chooses stationarily many countable ordinals and SPLS tries to continuously divide them into two stationary pieces. We show that it is possible in ZFC to force a winning strategy for either player, giving a new counterexample to S²₂ maximality with a predicate for the nonstationary ideal on w₁. We show that the determinacy of the game is consistent with Martin's Axiom but not Martin's Maximum. This is joint work with Saharon Shelah.
• Friday April 13, 1:30-3:00, Fields Institute, room 210
  Matthias Neufang (Carleton University)
  Decomposition of von Neumann algebras and the Mazur property
  Abstract
• Friday April 6. No seminar.
• Friday March 30, 1:30-3:00, Fields Institute, room 210
  Wieslaw Kubis (Akademia Swietokrzyska, Poland)
  Fraisse sequences and their limis - a category-theoretic approach.
  Abstract: Given a category, we define the notion of a Fraisse sequence: an inductive sequence which shares the properties of a typical chain of model-theoretic structures producing the Fraisse-Jonsson limit structure. We give a criterion for the existence of a Fraisse sequence of a given length and we shall present some sufficient conditions for its homogeneity, universality and uniqueness. For instance: a fixed category may have, up to isomorphism, at most one countable Fraisse sequence. We give an example of a certain category of countable trees with many non-isomorphic Fraisse sequences of length aleph one. Finally, we shall present some applications for constructing universal objects in certain natural categories of compact spaces, Banach spaces and linearly ordered sets.
• Friday March 23, 1:30-3:00 Fields Institute, room 210
  Slawomir Solecki (University of Illinois, USA)
  Polish Groups as Isometry Groups.
• Friday March 16, 1:30-3:00 Fields Institute, room 210
  Stevo Todorcevic
  Chain Conditions in Topology, III (continued)
• Friday March 9, 1:30-3:00 Fields Institute, room 210
  Stevo Todorcevic
  Chain Conditions in Topology, III
• Friday, February 23, 3:30-5:00 pm Fields Instute Room 230
  Andrew Toms
  Elliott's program and partially ordered Abelian groups
  Abstract: We give a (hopefully) not-too-technical introduction to Elliott's program to classify separable amenable C*-algebras via K-theoretic invariants. We will stress the role of partially ordered Abelian groups, and discuss how the decomposition of such groups into inductive limits of tractable building blocks gives attractive range results for K-theoretic invariants. Some time will also be spent on the question of "how many" such groups may arise as the K-groups of simple C*-algebras. Finally, if time permits, we will point out some recent results in C*-algebra theory which have a set theoretic flavour.
• Friday February 9, 1:30-3:00 Fields Institute, room 210
  Victoria Lubitch
  Note on quasi-Rosenthal compactum.
  Abstract: Let X be a Polish space. A compact subset K of R^X is called quasi-Rosenthal if the accumulation points of K are Baire-1 functions. S.A.Argyros, P.Dodos, VKanellopoulos asked whether every quasi-Rosenthal compactum is Frechet. I will present a counter-example.
• Friday February 2, 1:30-3:00 Fields Institute, room 210
  Wieslaw Kubis Kielce, Poland
  Universal Valdivia compact spaces
  Abstract: A space K is Valdivia compact if it is (homeomorphic to) a closed subset of a Tikhonov cube such that elements with countably many non-zero coordinates form a dense subset of K. I will describe the construction of a linearly ordered Valdivia compact which is a universal increasing pre-image for the class of all linearly ordered Valdivia compacta. The space has also some homogeneity properties which, together with universality, characterize it uniquely up to order isomorphism. If time permits, I shall also try to sketch the construction of a universal Valdivia compact for weight $\aleph_1$, under the Continuum Hypothesis. Again, the space is both universal and in some sense homogeneous, and these two properties describe it uniquely. Both constructions can be described as particular cases of a general theory of Fraisse sequences/limits, in the language of category theory.
• Friday January 26, 1:30-3:00 Fields Institute, Room 210
  Luis Pereira Universite Paris VII
  Cardinal estimates using a topological approach
  Slides from the talk
  Abstract: We will prove that Shelah's application of his estimates of the Galvin-Hajnal norm to cardinal arithmetic can be obtained via Shelah's PCF topology alone.
• Friday January 19, 1:30-3:00 Fields Institute, Room 210
  Vladimir Pestov University of Ottawa
  Subgroups of metric ultraproducts of unitary and of symmetric groups: hyperlinear and sofic groups.
  Slides from the talk
  Abstract: Relatively recently, two new classes of groups have been isolated: hyperlinear groups and sofic groups. They come from different corners of mathematics (operator algebras and symbolic dynamics, respectively), and were introduced independently from each other, but are closely related nevertheless. Hyperlinear groups are motivated by Connes' Embedding Conjecture about von Neumann factors of type II_1 and go back to Connes' work, while sofic groups, introduced by Gromov, are motivated by Gollschalk surjunctivity conjecture (can a shift A^G contain a proper isomorphic copy of itself, where A is a finite discrete space and G is a group?). Groups from both classes can be characterized as subgroups of metric ultraproducts of families of certain metric groups (formed in the same way as ultraproducts of Banach spaces): unitary groups of finite rank lead to hyperlinear groups, symmetric groups of finite rank - to sofic groups. We will survey results by Connes, Gromov, Benjy Weiss, Ozawa, Elek and Szabo, Radulescu, and Gordon, and also discuss open questions which are for the time being perhaps more numerous than the results.
• Friday January 12, 1:30-3:00 Fields Institute, Room 10
  Bernhard Koenig, University of Toronto
  Coherent trees, continued.
  
• Friday January 5, 1:30-3:00 Fields Institute, Room 210
  Bernhard Koenig, University of Toronto
  Coherent trees.