SCIENTIFIC PROGRAMS AND ACTIVITIES

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

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

No Seminar

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.

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.

Slides of talk
http://settheorytalks.files.wordpress.com/2011/05/pestov-6may2011.pdf

NO SEMINAR

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.<br>
2. Pointwise continuous embeddings between countable sets of reals (or more concretely, rationals).<br> 3. Relations between f(n_1,...,n_k) and f(n_2,...,n_k+1).<br> 4. Homeomorphic embeddings between finite trees.<br> 5. Borel sets in the plane and graphs of one dimensional Borel functions.<br> 6. Boolean relations between sets of integers and their images under integer functions.

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).

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).

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.

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.

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.

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.

No seminar on Friday, October 15
due to:Workshop on the Concentration Phenomenon, Transformation Groups and Ramsey Theory
October 12--15
Fields Institute

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]

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.

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.

Franklin Tall, University of Toronto
More applications of PFA(S)[S]

Dilip Raghavan (Fields)
A Hausdorff real without dominating reals

No Seminar

Sam Coskey (CUNY)
Classification and equivalence relations