SCIENTIFIC PROGRAMS AND ACTIVITIES

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

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.

Posting: http://settheory.mathtalks.org/mike-pawliuk-set-theoretic-attacks-on-itzkowitz-problem/

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

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.

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.)

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

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

Assaf Rinot (Toronto)
Recent advances in the theory of strong colorings

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.)