# SCIENTIFIC PROGRAMS AND ACTIVITIES

September  3, 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 2009-10
For a complete listing of seminars and abstracts: http://www.math.yorku.ca/~ifarah/seminar.html
 2010-11 Fridays Speaker and Talk Title June 24 Room 210 Peter Krautzberger (Michigan) Union ultrafilters are fascinating — a survey June 17 Room 210 Dilip Raghavan (Fields) Weak squares 2010-11 PAST SEMINARS June 10 Room 210 Paul Larson Choosing Ideals We show that, in certain inner models of determinacy, there is a definable procedure which, given a tall ideal $I$ on $\omega$ containing all finite sets, and a function from $I \setminus \mathrm{Fin}$ to a countable set, chooses a finite subset of the range of this function. In the case we are most interested in, $I$ is generated by a countable collection of pairwise orthogonal ideals. In this context, $I$ represents a $\mathcal{P}(\omega)/\mathrm{Fin}$-name for an element of a countable set. Our result then says that, whenever $M[U]$ is a $\mathcal{P}(\omega)/\mathrm{Fin}$-extension of a model $M$ of the type we consider here, if $F$ is a function in $M[U]$ with domain $X \in M$, and $G$ is a function in $M$ with domain $X$ such that, for all $x \in X$, $F(x) \in G(x)$ and $G(x)$ is countable, then there exists a function $G'$ in in $M$ with domain $X$ such that, for all $x \in X$, $G'(x)$ is a finite subset of $G(x)$. This gives another proof of a theorem of Di Prisco-Todorcevic that in $M[U]$ there is no function which selects a single member from each $E_{0}$-equivalence class, where $E_{0}$ is the relation of mod-finite agreement on the Baire space. June 3 Room 210 No Seminar May 27 --1:30 to 3pm Room 210 Jan van Mill (VU Amsterdam) Homeomorphism groups of homogeneous compacta need not be minimal We present an example of a homogeneous compact space, the homeomorphism group of which is not minimal. This answers a question of Stoyanov from about 1984. If time permits, we will also talk about unique homogeneity and raise some open problems. Friday, May 20 No seminar Friday, May 13 from 1:30 to 3pm Fields Institute, Room 210 Peng Yinhe (Toronto and Singapore) An L space with non-Lindelof square There is an L space with non-Lindelof square. Moreover, there is an L space with some stronger property that its square is the closure of a countable union of closed discrete subsets. May 6 1:30 p.m. Room 210 Vladimir Pestov (Ottawa) Some set-theoretic motives in statistical learning theory II We will discuss a few instances where the Vapnik-Chervonenkis theory of statistical learning has potentially interesting overlaps with set theory / model theory / extremal combinatorics. This will be illustrated on speaker's recent results about PAC learnability over non-atomic measures (a solution of a problem by Vidyasagar with the help of Martin's Axiom), as well as learnability over exchangeable data inputs, and a reputed 25-year old open problem concerning the so-called sample compression schemes. Apr. 29 1:30 p.m. Room 210 NO SEMINAR Apr. 15 1:30 p.m. Room 210 Harvey Friedman (Ohio State) Concrete mathematical incompleteness An unprovable theorem is a theorem about basic mathematical objects that can only be proved using more than the usual axioms for mathematics (ZFC = Zermelo Frankel set theory with the Axiom of Choice) - and that has been proved using standard extensions of ZFC generally adopted by the mathematical logic community. The highlight of the talk is the presentation of unprovable theorems stated in terms of self embeddings of maximal cliques in graphs. We first review some previous examples of unprovable theorems. 1-5 are unprovable in the weaker sense that any proof demonstrably requires some use of logical principles transcendental to the problem statement. 6 is BRT (Boolean Relation Theory). 1. Patterns in finite sequences from a finite alphabet.
2. Pointwise continuous embeddings between countable sets of reals (or more concretely, rationals).
3. Relations between f(n_1,...,n_k) and f(n_2,...,n_k+1).
4. Homeomorphic embeddings between finite trees.
5. Borel sets in the plane and graphs of one dimensional Borel functions.
6. Boolean relations between sets of integers and their images under integer functions. April 8 1:30 p.m. Room 210 Natasha May (York), Saeed Ghaseemi (York), Amit Gupta (Berkeley) Research Glimpses 2 On *-isomorphisms between the Calkin Algebra onto the tensor product of two copies of itself; On Conjectures of Rado and Galvin; and more April 1 1:30 p.m. Room 210 Jim McGarva, Francisco Kibedi, Christopher Eagle and Dana Bartosova Research glimpses 1. (Including topics: Maximal saturated linear orders, Thin-tall spaces and PCF structures, and more) Mar. 25 1:30 p.m. Room 210 J. Lopez-Abad (ICMAT, Madrid) Unconditional sequences in Banach spaces of high density It is well known that there are separable Banach spaces without unconditional basic sequences, and that every Banach space whose density is bigger than an $\omega$-Erdös cardinal contains an unconditional basic sequence. We prove that it is consistent that every Banach space of density bigger than $\aleph_\omega$ has an unconditional sequence. Consequently, it is consistent that every Banach space of such density has a separable quotient. The proof relies on a Ramsey-like combinatorial property that $\aleph_\omega$ may have. This is a joint work with P. Dodos (Athens) and S. Todorcevic (Toronto). Mar. 18 1:30 p.m. Room 210 J. Lopez-Abad (ICMAT, Madrid) Unconditional sequences in Banach spaces of high density It is well known that there are separable Banach spaces without unconditional basic sequences, and that every Banach space whose density is bigger than an $\omega$-Erdös cardinal contains an unconditional basic sequence. We prove that it is consistent that every Banach space of density bigger than $\aleph_\omega$ has an unconditional sequence. Consequently, it is consistent that every Banach space of such density has a separable quotient. The proof relies on a Ramsey-like combinatorial property that $\aleph_\omega$ may have. This is a joint work with P. Dodos (Athens) and S. Todorcevic (Toronto). Mar. 11 1:30 p.m. Stewart Library No Seminar Mar. 4 1:30 p.m. Room 210 Ilijas Farah (York) Order property of II_1 factors and its applications II Feb. 25 1:30 p.m. Stewart Library Ilijas Farah (York) Order property of II_1 factors and its applications Feb. 18 1:30 p.m. Room 210 Jocelyn Bell (Buffalo) The Uniform Box Product Problem An important unsolved problem in topology is the box product problem, which asks whether the product of compact spaces with the box topology is normal. Applying uniformities, we introduce a new topology on products which sits between the box and Tychonov products called the uniform box product. This new product is an extension of the sup metric to products of compact spaces. We will show, in ZFC, that the uniform box product of a certain non-metrizable compact space is normal. Feb. 11 1:30 p.m. Room 210 Paul Szeptycki (York) A Lindelof T_1 space that is not a D-space Feb. 4 1:30 p.m. Room 210 Stevo Todorcevic (Toronto) A higher-dimensional theory of gaps in P(N)/Fin, part III Jan. 28 1:30 p.m. Room 210 Stevo Todorcevic (Toronto) A higher-dimensional theory of gaps in P(N)/Fin, part II Jan. 21 1:30 p.m. Room 210 Stevo Todorcevic (Toronto) A higher-dimensional theory of gaps in P(N)/Fin Jan. 14 1:30 p.m. Room 210 Jakub Jasinski (Toronto) Boron Tree Structures Jan. 7 1:30 p.m. Room 210 Jakub Jasinski (Toronto) Finite subsets of finite dimensional Euclidean spaces Dec. 17 1:30 p.m. Room 210 **NO SEMINAR** Dec. 10 1:30 p.m. Room 210 Max Burke (PEI) Liftings and densities for derived algebras Dec. 3 1:30 p.m. Room 210 Samuel Coskey (Fyorks Universitute) Borel equivalence relations and models of arithmetic Nov. 26 1:30 p.m. Room 210 NO SEMINAR Nov. 19 1:30 p.m. Room 210 Vladimir Pestov (University of Ottawa) Some set-theoretic motives in statistical learning theory A set theorist will find interesting opportunities by turning to the Vapnik-Chervonenkis theory of statistical learning. To argue this point, we will discuss the basic concepts and results of the theory of probably approximately correct (PAC) learnability, and then proceed to some recent results by the speaker. Those include a solution of a problem by Vidyasagar about PAC learnability over non-atomic measures which use Martin's Axiom and a combinatorial parameter defined for Boolean algebras. We will also discuss some other results and open problems. Nov. 12 1:30 p.m. Room 210 Juris Steprans (York) Reflecting non-meagreness in the Erdos-Kakutani group (joint work with Marton Elekes) Nov. 5 1:30 p.m. Room 210 Todor Tsankov (Paris VI) Applications of Roelcke precompactness to representation theory I will discuss the notion of Roelcke precompactness and how it can be applied to the study of unitary representations. I plan to give a fairly complete proof of the classification theorem for the representations of automorphism groups of omega-categorical structures. Oct. 22 1:30 p.m. Room 210 Julien Melleray (Lyon) Polish topometric groups Oct. 29 1:30 p.m. Room 210 Lionel Nguyen Van Thé (Université Aix-Marseille 3, Paul Cézanne) Partition properties of the dense local order (joint with Claude Laflamme and Norbert Sauer, University of Calgary) In 1984, Lachlan classified the countable ultrahomogeneous tournaments (ie the countable directed graphs where every pair or points supports an arc, and where every isomorphism between finite subgraphs extends to an automorphism of the whole structure), and showed that there are only three such objects: the rationals, the countable random tournament, and the so-called dense local order. The purpose of this talk is to present the Ramsey properties of this latter object. Oct. 15 No seminar on Friday, October 15 due to:Workshop on the Concentration Phenomenon, Transformation Groups and Ramsey Theory October 12--15 Fields Institute Oct. 8 1:30 p.m. Rm 210 Peter Burton (UofT) CH implies the existence of stationary locally countable families; hence CH implies productively Lindelof spaces are powerfully Lindelof. (And, if time remains) Frank Tall (UofT) More applications of PFA(S)[S] Sep 24 1:30 p.m. Fields Rm 210 Peter Burton, Kevin Duanmu, Frank Tall (U of T) Lindelof products A problem in Przymusinski's Handbook survey on normality of products asks whether, if X x Y is Lindelof for every Lindelof Y, are all countable powers of X Lindelof. We show that the Continuum Hypothesis implies a positive answer. Sep 17 1:30 p.m. Fields Rm 210 Peter Burton, Kevin Duanmu, Frank Tall (U of T) Lindelof products A problem in Przymusinski's Handbook survey on normality of products asks whether, if X x Y is Lindelof for every Lindelof Y, are all countable powers of X Lindelof. We show that the Continuum Hypothesis implies a positive answer. Sep 10 1:30 p.m. Fields Rm 210 Franklin Tall, University of Toronto More applications of PFA(S)[S] Friday, Sept. 3 No Seminar Friday, August 27 Asger Törnquist ( KCRG Wien) Conjugacy, orbit equivalence and von Neumann equivalence are analytic. For a countably infinite discrete group G, there are three equivalence relations of major interest for its measure preserving ergodic actions: Conjugacy, orbit equivalence and von Neumann equivalence. These equivalence relations are prima facie analytic. Can they be Borel? For G amenable, only conjugacy is of interest. In the special case G=Z the question goes back to Halmos, but was only recently solved by Forman, Rudolph and Weiss, who showed that conjugacy is complete analytic (as a set.) In this talk I will show that for a class of non-amenable groups, including all free groups, that all three of the above equivalence relations are analytic and not Borel. Part of this work is joint with Inessa Epstein. Friday, August 20 Ilijas Farah (York) A very short introduction to Woodin's Pmax forcing Friday, August 13 Paul Larson (Miami University of Ohio) Fragments of Martin's Maximum in the Pmax extension Friday, August 6 Franklin Tall PFA(S)[S] and applications cont. We develop machinery for applying the method of forcing with a coherent Souslin tree S over models of PFA restricted to posets that preserve S. Friday, July 30 Franklin Tall PFA(S)[S] and applications We develop machinery for applying the method of forcing with a coherent Souslin tree S over models of PFA restricted to posets that preserve S. Friday, July 23 Dilip Raghavan (Fields) A Hausdorff real without dominating reals Friday, July 16 No Seminar Friday, July 9 Sam Coskey (CUNY) Classification and equivalence relations