# SCIENTIFIC PROGRAMS AND ACTIVITIES

October 26, 2014
THE FIELDS INSTITUTE
FOR RESEARCH IN MATHEMATICAL SCIENCES
20th ANNIVERSARY YEAR
 Set Theory Seminar Series 2011-12 Fields Institute (map), Friday, 1:30 p.m. Organizing Committee: Organizers: Sam Coskey, Ilijas Farah, Juris Steprans, Paul Szeptycki
 Seminars held during 2008-09 Seminars held during 2009-10 Seminars held during 2010-11 York U Set Theory Page
 2011-12 Fridays Speaker and Talk Title For a list of previous talks, please see http://settheory.mathtalks.org/seminar/toronto/ PAST SEMINARS 2011-12 June 22 no seminar June 15 no seminar June 8 Franklin Tall (Toronto) Topological Problems for Set Theorists, II This is the second instalment of a talk I gave 25 years ago, listing a variety of topological problems with set-theoretic content. The material may be familiar to faculty, but grad students and postdocs may come across something worth chewing on. Posting: http://settheory.mathtalks.org/franklin-tall-topological-problems-for-set-theorists-ii/ slides available here June 1 no seminar May 25 Mike Pawliuk (Toronto) Set Theoretic attacks on Itzkowitz' problem In 1976, Gerald Itzkowitz asked if every functionally balanced group (one where every left uniformly, real-valued function is right uniformly continuous) is in fact balanced (the left uniformity and right uniformity coincide). The answer is "Yes" for topological groups that are even a little bit nice - locally compact or metrizable or locally connected. The question is still open in general. I will show how some large topological groups seem like good candidates for counterexamples. In particular we will look at isometry groups of large metric spaces like the Urysohn space. May 18 Assaf Rinot (Fields and UTM) On incompactness for chromatic number of graphs We shall discuss Shelah's paper #1006. The talk will be an expanded version of the following blog post. May 11 Jim McGarva (Toronto) Constructing a Taller Thin-Thick Space May 4 no seminar April 20 no seminar March 30 Franklin Tall, University of Toronto Lindelof spaces with small pseudocharacter, and an analog of Borel's Conjecture for subsets of uncountable products of [0,1] (With T. Usuba) We improve results of Shelah, myself, and Scheepers concerning the cardinality of Lindelof spaces with small pseudocharacter. We establish the consistency, modulo an inaccessible, of an equivalent of Borel's Conjecture "stepped up one cardinal". March 23 Miodrag Sokic, Caltech The Ramsey property for structures with an arbitrary linear ordering The class of finite ultrametric spaces with an arbitrary linear ordering is not a Ramsey class. Also the class of finite posets with an arbitrary linear ordering is not a Ramsey class. We will calculate the Ramsey degrees for these classes. March 16 Dimitrios Vlitas (Paris) An infinite self dual theorem Recall that the classical Ramsey theorem states that given any finite coloring of the set of all K elements subsets of ? there exists of an infnite subset A ? ? where the restriction of the coloring is constant. The dual form of Ramsey theorem, the Carlson-Simpson Theorem, states that given any finite Borel coloring of the set of all partitions of ? into K many classes, there exists a partition r of ? into ? many classes such that the set of all K partitions of ? resulting by identifying classes of r is monochromatic. There are also the corresponding ?nite versions of these results, the finite Ramsey Theorem, and the Graham-Rothschild theorem, respectively. S. Solecki recently proved a self dual theorem that implies simultaneously the finite version of the Ramsey theorem and the Graham-Rothschild theorem. He achieved that by introducing the notion of a connection, which roughly speaking is a labelled partition of L into K many classes, for K and L integers. He then proved that given any positive integers K, L and M there exists N such that for any L coloring of all labelled partitions of N into K many pieces, there exists a labelled partition of M into K pieces, such that the set of all labelled partitions of N into M composed with the particular labelled partition of M into K is monochromatic. The composition is defined in the most natural way by composing partitions, namely that partition N into M pieces and then M into K pieces, so we finally partition N into K.  The composition of the label functions is done in the reverse order. We extend canonically his notion of connection to labeled partitions of ?, with finite or infinitely many classes and we prove the following: Theorem.  For any fnite Borel coloring of al l label led K-partitions of ? there is a fixed label led ?-partition of ? such that the set of al l of its reductions, i.e. label led K-partitions of ? which result from putting pieces of the ?xed partition together, is monochromatic. The proof is done by induction on K and the use of the left variable Hales-Jewett Theorem. In the ?nal section of the paper we extend this result by building the corresponding topological Ramsey space F?,? . March 9 Paul McKenney (CMU) Automorphisms of Calkin Algebras March 2 Santi Spadaro (York) Noetherian type and other topological cardinal invariants of an order-theoretic flavour Noetherian type is a cardinal function that was introduced by Peregudov in the 90s to capture some base properties studied by the Russian School in the 70s. It has a striking affinity to the Suslin Number and for this reason it has an interesting productive behavior. We will show an example of two spaces of uncountable Noetherian type whose product has countable Noetherian type and single out classes of spaces in which the Noetherian type cannot decrease by passing from a space to its square. Time permitting we will show some independence results regarding the Noetherian type of countably supported box products. This is joint work with Menachem Kojman and Dave Milovich. Feb. 24 Xianghui Shi (Beijing Normal University) A Posner-Robinson Theorem from Axiom I_0 Under a slightly stronger version of Axiom I_0: there is a *proper* elementary embedding j from L(V_{lambda+1}) to L(V_{lambda+1}) with critical point < lambda, we prove an analog of Perfect Set Theorem in the context of V_{lambda+1}. And as a collorary, we obtain a version of Posner-Robinson Theorem at V_{lambda+1}: for every A in V_{\lambda+1}, and for almost every B in V_{\lambda+1} (i.e. except a set of size lambda) that can compute A, there is a G in V_{lambda+1}\$ such that G joint B can compute the sharp of G. Here compute'' and joint'' are analogs of the notions in the structure of Turing degrees. This is a part of the study on the impact of large cardinal hypotheses on various generalized degree structures. Feb. 17 NO SEMINAR Feb. 10 Slawomir Solecki (UIUC) An abstract approach to Ramsey theory with applications to finite trees I will present an abstract approach to finite Ramsey theory. I will indicate how certain concrete Ramsey results for finite trees are obtained by applying the abstract result. Feb. 3 Assaf Rinot (Fields Institute and UTM) Generalizing Erd?s-Rado to singular cardinals One of the most famous implications of the infinite Ramsey theorem (1929) asserts that any infinite poset either contains an infinite antichain or an infinite chain. Ramsey's theorem has been generalized by Dushnik and Milner (1941), and subsequently by Erd?s to a theorem that implies that any poset of uncountable cardinality k either contains an antichain of size k, or an infinite chain. Is it possible to ask for a more sophisticated second alternative? More specifically, can the theorem be strengthened to yield the existence of an infinite chain *with a maximal element*? This question, restricted to uncountable regular cardinals, was answered by Erdos and Rado (1956). In this talk, we shall discuss the missing case - singular cardinals - and present a proof of a Theorem of Shelah (2009) in the positive direction. Our proof may be found in here: http://blog.assafrinot.com/?p=628 Jan. 20 Martino Lupini (York University) Logic for metric structures and the number of universal sofic groups Jan.13 Jan Pachl, (Fields) Measurable centres in convolution semigroups Every topological group G naturally embeds in larger spaces, algebraically and topologically. Two such convolution semigroups of particular interest in abstract harmonic analysis are the norm dual of the space of bounded right uniformly continuous functions on G, and the uniform compactification of G with its right uniformity. Our understanding of the structure of these spaces has been advanced by tractable characterizations of their topological centres, now available for "almost all" topological groups. In the seminar I will discuss a measurable analogue of the topological centre, for various notions of measurability. This notion was investigated by Glasner (2009) for the compactification of a discrete group, using Borel measurability. The main result is that in convolution semigroups over locally compact groups the Borel-measurable centre coincides with the topological centre [arXiv:1107.3799]. It is an open question whether the same holds for all topological groups. One version of the similar statement in which universal measurability replaces Borel measurability is independent of ZFC. Dec. 16 Rodrigo R. Dias, (São Paulo) Indestructibility and selection principles In this talk we will explore the game-theoretic characterization of indestructibility of Lindelöf spaces. In particular, we will show that this property is not equivalent to the associated selection principle if CH is assumed. Dec. 2 Peter Burton, (Toronto) A quotient-like construction concerning elementary submodels, II No abstract provided Nov.25 Peter Burton, (Toronto) A quotient-like construction concerning elementary submodels No abstract provided Nov.18 Konstantinos Tyros (Toronto) Density theorems for strong subtrees In this talk we will present the main ingredients of the proof of the density version of Halpern Lauchli Theorem. We shall also discuss some of its applications. Nov.11 Natasha May (York) A Noetherian base for scattered linear orders A collection of sets is Noetherian if it contains no infinite ascending sequences. We show that every scattered LOTS of cardinality strictly less than the first strongly inaccessible cardinal has a Noetherian base.  I will also provide some motivation. Joint with Paul Szeptycki. Oct. 28 Set Theory and C*-algebras Seminar Stevo Todorcevic (Toronto) The unconditional basic sequence problem, revisited Oct. 26 11:00 am Set Theory and C*-algebras Seminar Oct. 21 David Milovich (Texas A&M International) On cofinal types in compacta: cubes, squares, and forbidden rectangles In every compactum, not every point's neighborhood filter has cofinal type omega times omega_2. (This is an instance of a more general theorem.) This can be interpreted as yet another partial result pointing toward the conjectures that homogeneous compacta cannot have cellularity greater than c (Van Douwen's Problem) nor an exponential gap between character and pi-character. There are compacta where every point's neighborhood filter has cofinal type omega times omega_1, but it is not known if there is a homogeneous compactum with this property. Continuing the theme of cofinal types of product orders, the Fubini cube and Fubini square of an arbitrary filter F on omega are cofinally equivalent to each other and to the direct product F^omega. (This generalizes to kappa-complete filters on regular kappa.) Friday, October 21 11:00 a.m. Stewart Library Set Theory and C* algebras Oct. 14 Daniel Soukup (Toronto) Variations on separability The aim of this talk is to review some recent results on variations of separability; we investigate spaces having sigma-discrete and meager dense sets and selective versions of these properties. Our results mostly determine the relations between these properties, as well as give some hint on the effect of various convergence properties on these weak types of separability. However, many questions are left open. This work was jointly done by D. Soukup, L. Soukup and S. Spadaro. Oct. 7 11:00 a.m. Room 210 A special seminar on Set Theory and C*-algebras. The first goal is to read the paper "Turbulence, orbit equivalene, and the classification of C*-algebras" by Farah/Toms/Törnquist. Oct. 7 Dilip Raghavan (Kobe) The Borel almost disjointness number Sept.30 Judy Roitman (Kansas) The Box Problem Sept. 16 Peter Burton (Toronto) Productive Lindelofness and a class of spaces considered by Z. Frolik July 22 Assaf Rinot (Toronto) Recent advances in the theory of strong colorings July 15 Franklin Tall (Toronto) Recent progress and problems concerning Lindelöf products and selection principles. Rodrigo Dias (Toronto) Some topological games and selection principles (Please note that Franklin Tall will also give two talks, on July 12 and 19 in Bahen 6180/3 at 11am in the student set theory seminar. The titles are PFA(S)[S]: topological applications of forcing with coherent Souslin trees, AND PFA(S)[S]: a method for proving a set of size aleph_1 is the union of countable many nice subsets.) July 8, 2011 Kostas Tyros (Toronto) Density Theorems for Trees