2008

Speaker and Talk
Title

Friday, June 27
1:303:00pm
U of T Bahen Centre,
Room 6183 
Ilijas Farah (York University)
The commutant of B(H) in its ultrapower and flat ultrafilters
A question of Eberhard Kirchberg has lead to an isolation
of a new type of ultrafilter on N and a consistency result.
This is a joint work with N. Christopher Phillips and Juris
Steprans.
PLEASE NOTE THE UNUSUAL LOCATION FOR THIS SEMINAR. THE BAHEN
CENTRE IS THE BUILDING HOUSING THE DEPARTMENT OF MATHEMATICS
BEHIND THE FIELDS INSTITUTE

May 9 
no seminar 
May 2 
no seminar 
Friday April 25
1:303:00 
Jan Pachl
Ambitable topological groups
A topological group G is called ambitable if every uniformly
bounded uniformly equicontinuous set of functions on G with
its right uniformity is contained in an ambit, within the Gflow
defined by right translations on uniformly continuous functions.
This notion is motivated by investigations of topological centres
in the semigroups that are of interest in abstract harmonic
analysis  for ambitable groups, topological centres have
an effective characterization. Simple sufficient conditions
for ambitability, expressed in terms of two cardinal functions,
yield a generalization of previously known characterizations
of topological centres.
The talk will follow the content of the paper with the same
name, available as arXiv:0803.3405.

Friday April 18
1:303:00 
Stevo Todorcevic (University
of Toronto and CNRS Paris)
Unconditional Basic Sequences in Spaces of High Density 
Friday April 11
1:303:00 
No Seminar 
Friday April 4
1:303:00 
Bohuslav Balcar (Czech Academy
of Sciences)
Refinement properties of countable sets
I will present a notion of almost disjoint refinement and its
application and relationships to ultrafilters on $\omega$, semiselective
coideals, Boolean algebras, and generic extensions of models
of set theory. 
Friday Mar. 28
1:303:00 
Pandelis Dodos (University
of Paris 6 and National Technical University of Athens)
Universality Problems and \Script L_\infty spaces 
Friday Mar. 14
1:303:00 
CANCELLED 
Friday Mar. 7
1:303:00 
Ernest Schimmerling (Carnegie
Mellon University)
Some open questions about L
There are indeed open questions about L. I'll state a few and
describe related results on two two topics: forcing axioms and
mutual stationarity. The talk will include an overview of Jensen's
fine structure theory and, as such, will be selfcontained. 
Friday Feb. 29
1:303:00 
Frank Tall (University of Toronto)
On a core concept of Arhangel'skii (Con't) 
Friday Feb. 22
1:303:00 
Frank Tall (University of Toronto)
On a core concept of Arhangel'skii
Arhangel'skii defines a locally compact space to have
a countable core if it is the union of countably many open
subspaces, each having the property that infinite subsets
have limit points in the whole space. Investigating the question
of when such spaces are sigmacompact as usual leads to the
consideration of additional settheoretic axioms.

Friday, Feb. 15,
1:30  3:00 
Natasha Dobrinen (University of Denver)
The consistency strength of the tree property at the double
successor of a measurable cardinal
This is joint work with SyDavid Friedman.
Koenig's Lemma is the wellknown theorem that any inifinite
finitely branching tree of height $\omega$ has an infinite path
through it. This theorem, however, fails for trees of height
$\omega_1$. More precisely, Aronszajn constructed a tree of
height $\omega_1$, all of whose levels are countable, with no
cofinal branch. Such a tree is naturally called an Aronszajn
tree.
For a regular uncountable cardinal $\kappa$, a "$\kappa$tree"
is a tree of height $\kappa$ all of whose levels have size less
than $\kappa$. A "$\kappa$Aronszajn" tree is a $\kappa$tree
with no cofinal branches. We say that the "tree property
at $\kappa$ holds" if there are no $\kappa$Aronszajn trees.
As soon as $\kappa$ is greater than $\aleph_1$, large cardinals
become necessary. Silver showed that if the tree property holds
at $\aleph_2$, then $\aleph_2$ must be weakly compact in $L$.
Mitchell showed that from a weakly compact cardinal one can
force the tree property at $\aleph_2$. Thus began the study
of the consistency strengths of the tree property at various
cardinals.
We show that the tree property at the double successor of a
measurable cardinal is equiconsistent with what we call a weakly
compact hypermeasurable cardinal.
Our methods us a reverse Easton iteration of iterated Sacks
forcings. The difficulty lies in showing that the weakly compact
hypermeasurable cardinal remains measurable after this forcing.

Friday, Feb. 8,
1:30  3:00 
Bernhard Koenig (University
of Toronto)
Two ways to construct an omega_2Suslintree from GCH plus
additional assumptions 
Friday, Feb. 1,
1:30  3:00
(**talk cancelled due to weather) 
**Natasha
Dobrinen (University of Denver)
The consistency strength of the tree property at the double
successor of a measurable cardinal 
Friday, Jan. 25,
1:30  3:00 
Students will present problems, including:
L. Hoehn: The chainable vs. spanzero problem for
nonmetrizable continua;
B. Zamora: About Hadwin's conjecture;
V. Fischer: Consistency results with large continuum;
A. Fischer: PFA(S)[S] vs. PFA;
C. Martinez: Some problems on Aronszajn lines;
A. Brodsky: The sigmafinite chain condition and
its use in absoluteness proofs.

Friday, Jan. 18,
1:30  3:00 
Udayan B. Darji (University
of Louisville)
Generating dense subgroups of some transformations groups 
Friday, Dec. 21,
1:30  3:00 
Vera Fischer (York University)
Towards the consistency of arbitrarily large spread between
the bounding and the splitting numbers
We will suggest a countably closed forcing notion which
satisfies the $\aleph_2$ chain condition and adds a centered
family $C$ of Cohen names for pure conditions, which has the
property that forcing with $Q(C)$ preserves a chosen subfamily
of the Cohen reals unbounded and diagonalizes all of them. 
Friday, Dec. 14,
1:30  3:00 
Vera Fischer (York University)
The consistency of $b=\kappa + s=\kappa^+$, continued

Friday, Dec. 7,
1:30  3:00 
Vera Fischer (York University)
The consistency of $b=\kappa + s=\kappa^+$
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, Nov. 30,
1:30  3:00 
Greg Hjorth (UCLA)
Ends and percolation
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 nonamenable
groups that have infinite normal amenable subgroups. 
Friday, Nov. 23,
1:30  3:00 
Marton Elekes, Renyi Institute,
Budapest
Partitioning multiple covers into many subcovers
Motivated by a question of A. Hajnal originating from geometry
we investigate the following set of problems. Let X be a set,
a cardinal number, and H a family that covers each x 2 X at
least times. Under what assumptions canwe decompose H into
many subcovers? Equivalently, under what assumptions can we
colour H by many colours so that for each x 2 X and each colour
c there exists H 2 H of colour c containing x?
The assumptions we make can be e.g. that H consists of open,
closed, compact, convex sets, or polytopes in Rn, or intervals
in a linearly ordered set, or we can make various restrictions
on the cardinality of X, H or elements of H.
Besides numerous positive and negative results many questions
turn out tobe independent of the axioms of set theory. This
is a joint work with T. M´atrai and L. Soukup. 
Friday, Nov. 16,
1:30  3:00 
Magdalena Grzech, Cracow University
of Technology
Complemented subspaces of the Banach space $l_\infty / c_0$ 
Friday, Nov. 2,
1:30  3:00 
Bernhard Koenig (University
of Toronto), part 2
Forcing axioms and two cardinal diamonds
I will present some known facts and new results concerning the
consistency of forcing axioms with two cardinal diamond principles. 
Friday, October 26,
1:30  3:00 
Bernhard Koenig (University
of Toronto)
Forcing axioms and two cardinal diamonds
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 
Arthur Fischer (University
of Toronto)
PID in PFA(S)[S]
We will demonstrate that the Pideal 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 
Istvan Juhasz (Hungarian Academy
of Sciences)
Discrete subspaces of compacta
The following are the main results presented in this talk.
Theorem 1. If X is a countably tight compactum such that every
limit cardinal strictly below X is strong limit then X =
D for some discrete subspace D of X.
Theorem 2. If each point of a compactum X has character at leat
then X cannot be covered by fewer than 2many discrete subspaces.
Theorem 3. The !th power of any compactum has a discrete dense
subset.
Theorem 4. If the character of a compactum is greater than
then there is a discrete subspace Y of X with Y  + and
a point p whosecharacter in Y [ {p} is greater than .
Theorems 13 are joint results with Z. Szentmikl´ossy.
Several open problems will also be formulated. 
Friday, September 28,
1:303:00pm 
Asger Tornquist (University of Toronto)
Definable Davies' Theorem, PART II
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, Sept. 21,
1:303:00
Fields room 210 
Asger Tornquist (University
of Toronto)
Definable Davies' Theorem, Part 1
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, Sept. 14,
1:303:00
Fields Room 210 
Frank Tall (University of Toronto)
More Topological Applications of PFA(S)[S]
We continue our study of the paracompactness of locally
compact normal spaces in models of PFA(S)[S]. Using Pideal
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, Sept. 7,
1:303:00
Fields Room 210.

Logan Hoehn (University of Toronto)
A model theoretic approach in topology
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 selfmaps of compact finitedimensional
metric spaces can be extended to a broader class of spaces
using this approach.

Friday, Aug. 3,
1:303:00pm
Fields, room 210 
Bart Kastermans, University
of Wisconsin, Madison
Cofinitary groups
