Set Theory and Infinitary Combinatorics
Org: Stevo Todorcevic (Toronto)
- ILIJAS FARAH, York University, Toronto, ON
A topological group is extremely amenable if it does not have a
fixed-point free action on a compact Hausdorff space. I'll talk about
a new example of extremely amenable groups of functions with topology
of convergence with respect to a diffused submeasures. The main
result unifies earlier work of Christensen-Herer and Pestov. We deduce
our results from a Ramsey-type result and give an example showing
there is no concentration of measure in this context.
This is a joint work with Slawek Solecki.
- VALENTIN FERENCZI, Université Pierre et Marie Curie - Paris 6, Equipe
d'Analyse Fonctionnelle, Boîte 186, 4, place Jussieu,
75252 Paris Cedex 05, France
The class of separable reflexive Banach spaces is strongly
A class C of separable Banach spaces is said to be strongly bounded
if whenever A is an analytic subset of C, there exists a space in
C which contains an isomorphic copy of any X in A.
It is proved that the classes REFL of separable reflexive spaces and
SD of spaces with separable dual are strongly bounded. This gives
another proof of a result of E. Odell and T. Schlumprecht answering a
question of J. Bourgain: there exists a separable reflexive Banach
space which is universal for the class of separable uniformly convex
Joint work with Pandelis Dodos.
- VERA FISCHER, York University, 4700 Keele Street, Toronto, Ontario M3J 1P3
The Consistency of Arbitrarily Large Spread Between the
Bounding and the Splitting Numbers
In 1984, S. Shelah obtained the consistency of b = w1 < s = w2 using countable support iteration
of proper forcing notions. The method can not be further generalized
since subsequent iterations would collapse the continuum. However
finite support iteration of c.c.c. forcing notions does not have this
disadvantage and we succeed to extend the above result obtaining a
model of b = w1 < s = m for m
arbitrary regular cardinal. The c.c.c. forcing notions which we use
are closely related to the partial order used originally by S. Shelah.
- PIOTR KOSZMIDER, Universidade de São Paulo
Combinatorics of Boolean algebras applied in functional
We present some new constructions of Boolean algebras achieved by
combinatorial methods which have applications in the theory of Banach
- JORDI LOPEZ-ABAD, Université Paris 7
Partial unconditionality and Barriers
The purpose of this talk is to present a framework for studying
weakly-null sequence of Banach spaces using the Ramsey property of the
families of finite sets of integers called Barriers, introduced by
Nash-Williams. We focus on "partial unconditionality" properties of
weakly-null sequences of Banach spaces. Inspired by a recent work of
S. J. Dilworth, E. Odell, Th. Schlumprecht and A. Zsak, we give a
general notion of partial unconditionality that covers most of the
known cases, including the classical Elton's near unconditionality,
convex unconditionality or Schreier unconditionality, and some new
The method reduces the problem to the understanding of mappings
j: B ® FIN×c0, where FIN
denotes the family of finite sets, B Í FIN is a
barrier, and c0 is the Banach space of sequences of real numbers
converging to zero. We present several combinatorial results
concerning these mappings, starting with the simpler mappings j: B ® FIN. One of the main results here is
that every mapping j: B ® c0 has a
restriction which is, up to perturbation, what we call a L-mapping,
i.e., j has a precise Lipschitz property and
satisfies that the support suppj(s) of j(s) is
included in s for every s Î B. L-mappings allow to
define naturally a weakly-null sequence, called L-sequence, associated
Finally our approach shows that if for some notion of unconditionality
\mathfrak F there is a weakly-null sequence with no \mathfrakF-unconditional subsequence, then there must be an L-sequence with no
\mathfrak F-unconditional subsequence.
This is a joint work with S. Todorcevic.
- JUSTIN MOORE, Boise State University, Boise ID, 83725-1555, USA
On linear orders with no real or Aronszajn suborders
In this talk we will consider what can be said about a linear order which contains neither real nor Aronszajn suborders. It is easily seen that s-scattered orders fit this criterion. Baumgartner constructed an example which is not s-scattered and contains neither a real nor Aronszajn suborder. Baumgartner's example is necessarily not minimal with respect to not being s-scattered. We have shown that PFA implies any minimal non-s-scattered order of size À1 must either be a real or Aronszajn type. A version of this theorem for larger linear orders will also be discussed, along with the relevance to a theorem of Laver which asserts that the s-scattered orders are well quasi-ordered. The work presented is joint work with Tetsuya Ishiu.
- CRISTIAN ROSENDAL, University of Illinois at Urbana-Champaign
On the algebraic structure of the unitary group
We study the unitary group of separable infinite dimensional complex
Hilbert space as a discrete group and show that all of its actions by
isometries on a metric space have orbits of finite diameter. This
property is enough to ensure that the unitary group also satisfies
properties FH and FA of Serre.
This is a joint work with Eric Ricard of CNRS, Université
- SLAWOMIR SOLECKI, University of Illinois, Department of Mathematics,
1409 W. Green St., Urbana, IL 61801, USA
A Fubini theorem
Let I0 be the s-ideal of subsets of a Polish
group generated by Borel sets which have perfectly many pairwise
disjoint translates. I will present a Fubini-type theorem that holds
between I0 and the s-ideals of Haar measure zero
sets and of meager sets. I will show how to use this result to give a
simple proof of a generalization of a theorem of
Balcerzak-Roslanowski-Shelah stating that I0 on
2N strongly violates the countable chain condition.
- JURIS STEPRANS, York University
Localization properties of Silver forcing and countable
A partial order is said to have the n-localization property if every
real added by the partial order can be approximated by an
n-branching tree in the ground model. Well known arguments
establishing this type of property for variants of Sacks forcing do
not generalize (in the obvious fashion) to the case of Silver forcing.
It will be shown that, nevertheless, similar results do hold.
- FRANKLIN TALL, University of Toronto, 40 St. George St., Toronto, ON
PFA(S)[S]: mutually consistent consequences of PFA and V=L
Forcing with a coherent Souslin tree S, after iterating proper
partial orders preserving S, produces interesting models. One gets
"Souslin-type" consequences of PFA, as well as some combinatorics
normally obtained from V=L. The combination of these two kinds of
consequences leads to solutions of a number of previously intractable
problems. The applications so far have been to topology, but one can
expect to see other uses in the future. We shall discuss the method
(pioneered by Todorcevic) and some applications.
- WILLIAM WEISS, University of Toronto
Problems caused by singular cardinals of countable
In ZFC, two recursive topological constructions are obstucted at
singular cardinals of countable cofinality. We discuss recent work
which tries to avoid this obstacle. The remaining problems are to
exhibit a locally countable, countably compact space of large
cardinality and a Bernstein set for any metric space.