
THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
20th
ANNIVERSARY
YEAR

JulyDecember
2012 Thematic Program on Forcing and its Applications
October 2226, 2012
Workshop on Forcing Axioms and their Applications
Organizing Committee:
Jordi Lopez Abad, Justin Tatch Moore, Stevo Todorcevic


Mailing List : To receive
updates on the program please subscribe to our mailing list at www.fields.utoronto.ca/maillist
Overview
This workshop will bring together researchers working on combinatorial
analysis of Banach spaces and those specializing in forcing axioms.
Central to the discussion will be combinatorial consequences of
Martin's Maximum which are driven by applications and which are
readily accessible to those working in analysis and other fields.
The driving goal will be to progress our understanding in well known
open problems in the the theory of Banach space such as the metrization
problem for compact convex sets, the smooth bump problem, and the
separable quotient problem.
Daniel Soukup has compiled all the open problems raised in the
workshop: http://settheory.mathtalks.org/?p=681
Schedule
Monday, October 22 
9:30 
Justin Tatch Moore, Cornell University
Martin's Maximum, a tutorial 1
The purpose of these talks will be to survey the consequences
of Martin's Maximum. This will be done by isolating consequences
of MM  specifically the Pideal Dichotomy, the Open Coloring
Axiom, and the Strong Reflection Principle  which have
proved useful in applications. A focus will be placed on
showing how these consequences can be applied in practice.
If time permits, we will also examine how to build the forcing
notions needed to show that these consequences follow from
MM.


Coffee break 
11:00 
Alan Dow, UNC Charlotte
Martin's Axiom and initially W1compact spaces
(slides) 
12:002:00 
Lunch Break 
2:00 
Michael Hrusak, Universidad Nacional Autónoma
de México
Malykhin's problem
Answering a 1978 problem of Malykhin we show that it is
relatively consistent with ZFC that every separable FrechetUrysohn
topological group is metrizable. (Joint work with Ulises
Ariet Ramos Garcia)


Tea Break 
3:30 
Back
to Fields Colloquium,
Valentin Ferenczi, Universidade de São Paulo and
l'Université Pierre et Marie Curie  Paris 6
On Gowers' classification program in Banach space theory
and
Christian Rosendal, University of Illinois at Chicago,
On bounded representations and maximal (slides)
symmetry 

Reception cash bar 
Tuesday, October 23 
9:30 
Justin Tatch Moore, Cornell
University
Martin's Maximum, a tutorial 2 

Coffee break 
11:00 
Richard Haydon, Mathematical Institute,
Oxford 

Lunch Break 
2:00 
Problems 

Tea Break 
3:30 
Paul Larson, Miami University
Models of size $\aleph_1$ in abstract elementary classes
We will present some applications of iterated generic ultrapowers
to the study of models of cardinality $\aleph_1$ in various
abstract elementary classes. The impetus for this work was
the stillopen question of absoluteness of $\aleph_1$categoricity
for the class of models of a fixed sentence of $L_{\omega_1,\omega}$.
Most of our result apply to the class of analytically presented
AEC's,those whose restrictions to countable models are analytic.
This is joint work with Baldwin and Shelah.

4:40 
Stevo Todorcevic, University of Toronto
and C.N.R.S., Paris
TBA 
Wednesday, October 24 
9:30 
Juris Steprans,
York University
Nontrivial automorphisms from variants of small d (slides) 

Coffee break 
11:00 
Matteo Viale, University of Torino
Forcing with forcings
Let $A$ be a class of partial orders and $B$ be a class
of complete embeddings between elements of $A$ closed under
composition.Then $(A,B)$ is a category whose objects are
elements of $A$ and whose arrows are elements of $B$. Moreover
$(A\cap V_\delta, B\cap V_\delta)$ is a partial order. Depending
on the nature of $A$ and $B$ this can be an interesting
or trivial partial order. If $A$ is the class of all posets
and $B$ is the class of all complete embeddings and $\delta$
is limit $(A\cap V_\delta, B\cap V_\delta)$ is a trivial
partial order since all elements of this partial order are
compatible.
We shall study the case in which $A$ is the class of stationary
set preserving (semiproper, proper) posets, and $B$ is the
class of complete embeddings between elements of $A$ with
a stationary set preserving (semiproper, proper) quotient.
We show that if $\delta$ has some degree of largeness which
depends on the choice of the category $(A,B)$, $(A\cap V_\delta,
B\cap V_\delta)$ is a STATIONARY SET PRESERVING partial
order which collapses $\delta$ to become $\aleph_2$ but
it should NEVER be a proper one and can be a semiproper
one only if $MM^{++}$ holds in the ground model.
Finally we briefly outline how these partial orders can
be of use to study absoluteness results for the theory of
the Chang model for sets of size $\aleph_1$. However this
will be the subject of a future talk.


Lunch Break 
2:00 
Hiroshi Sakai, Kobe University
Consequences of Martin's Maximum and Weak Square (slides)


Tea Break 
3:30 
Antonio Aviles, University of Murcia
A weak* separable C(K)* space whose ball is not weak* separable
We provide a ZFC example of a compact space K such that
C(K)* is w*separable but its closed unit ball is not w*separable.
All previous examples of such kind had been constructed
under CH. We also discuss the measurability of the supremum
norm on that C(K) equipped with its weak Baire sigmaalgebra.
(Joint work with Grzegorz Plebanek and Jose Rodriguez)

4:40 
Stevo Todorcevic, University of Toronto
and C.N.R.S., Paris 
Thursday, October 25 
9:30 
David Asperó i Herrando, Technische
Universitaet Wien
Iterated forcing with side conditions
I will present a technique for building finite support
forcing iterations with certain symmetric systems of structures
as side conditions. I will also give some applications of
the technique to the construction of models of versions
of Martin's Axiom for certain classes of $\aleph_2$c.c.
partial orders, and will say something about extensions
of these methods. Most of this is joint work with M.A. Mota.


Coffee break 
11:00 
Jordi Lopez
Abad, Consejo Superior de Investigaciones Cientificas (CSIC)
Geometry and operators of some generic Banach spaces
We will present some properties, with hints of the proofs,
of our recent examples of generic Banach spaces. In particular,
we will talk about operators on such spaces. This is a joint
work with S. Todorcevic.


Lunch Break 
2:00 
Problems 

Tea Break 
3:30 
Piotr Koszmider, Polish Academy of Sciences
Universality in classes of Banach spaces and compact spaces (slides)
In the context of classical associations between classes
of Banach spaces and classes of compact Hausdorff spaces
we survey known results and open questions concerning the
existence and nonexistence of universal Banach spaces and
of universal compact spaces in various classes. This gives
quite a complex network of interrelations which often can
only be decided using additional settheoretic assumptions
or forcing arguments.

4:40 
Carlos MartinezRanero,
UNAM
Invariance properties of almost disjoint families
(Well quasiordering Aronszajn lines) (slides)
We will consider two kinds of closely related mathematical
structures almost disjoint families and cofinitary groups.
We shall present some constructions of cofinitary groups
with some special topological properties and we will combine
these techniques to construct a MAD family which is maximal
in the Katetov ordering.

Friday, October 26 
9:30 
Wieslaw Kubis, Academy of
Sciences of the Czech Republic
A strong Gurarii space of density aleph one
A strong Gurarii space is a Banach space containing isometric
copies of all finitedimensional spaces which is additionally
homogeneous with respect to finitedimensional subspaces.
The latter means that every linear isometry between its
finitedimensional subspaces extends to a bijective isometry
of the entire space. It is wellknown (already noticed by
Gurarii) that no separable Banach space can be a strong
Gurarii space. On the other hand, there exist strong Gurarii
spaces of density at least the continuum. This leads to
a natural settheoretic question whether consistently one
can have a strong Gurarii space of a smaller density. We
show that a strong Gurarii space of density aleph actually
exists in ZFC .
This is a joint work with Antonio Aviles.


Coffee break 
11:00 
Itay Neeman, University of California, Los Angeles
Higher analogs of the proper forcing axiom
We will present a higher analogue of the proper forcing
axiom, and discuss some of its applications. The higher
analogue we present is an axiom that allows meeting collections
of $\aleph_2$ maximal antichains, in specific classes of
posets that preserve both $\aleph_1$ and $\aleph_2$.


Lunch Break 
2:00 
Christina Brech,
University of São Paulo
Biorthogonal systems and the cardinal b under the PID
We show that under the assumptions of the Pideal dichotomy
and that the bounding number b is larger than $\omega_1$,
every Banach space of density $\omega_1$ with weak* sequentially
compact dual ball has a quotient of density $\omega_1$ with
a Schauder basis. Together with an example of Todorcevic,
it follows that under the PID, b=$\omega_1$ is equivalent
to the existence of a nonseparable Asplund space with no
uncountable biorthogonal systems.
This is a joint work with S. Todorcevic.


Tea Break 
3:30 
Maryanthe Malliaris, University of Chicago
Cofinality spectrum problems in model theory, set theory
and general topology
Recent work of Malliaris and Shelah on modeltheoretic
questions around saturation of regular ultrapowers has led
also to theorems in set theory and general topology, notably
the result that p = t. The talk will outline our general
program and some features of this recent proof.



Participants as of October 17
Full Name 
University/Affiliation 
Asperó i Herrando, David 
Technische Universitaet Wien 
Audrito, Giorgio 
Università degli Studi di Torino 
Avilés, Antonio 
University of Murcia 
Bartosova, Dana 
University of Toronto 
Bice, Tristan 
The Fields Institute 
Blackmon, Michael 
University of North Carolina Charlotte 
Blass, Andreas 
University of Michigan 
BorodulinNadzieja, Piotr 
University of Wroclaw 
Brech, Christina 
University of São Paulo 
Brendle, Joerg 
Kobe University 
Brenken, Berndt 
University of Calgary 
Brodsky, Ari 
University of Toronto 
CancinoManríquez, Jonathan 
UNAMUMSNH 
Chodounsky, David 
The Fields Institute 
Cody, Brent 
The Fields Institute 
Cox, Sean 
The Fields Institute 
Di Prisco, Carlos 
Instituto Venezoland de Investigaciones Cientifigas 
Dobrinen, Natasha 
University of Denver 
Dow, Alan 
UNC Charlotte 
Drucker, Ohad 
The Hebrew University 
Eagle, Christopher 
University of Toronto 
Eskew, Monroe 
UC Irvine 
Ferenczi, Valentin 
Universidade de São Paulo 
Fernández Bretón, David 
York University 
Garcia Balan, Sergio Atayan 
FCFM  BUAP 
Gaspar Arreola, Miguel Angel 
UNAMUMSNH 
Ghasemi, Saeed 
York University 
Ghasemloo, Kaveh 
University of Toronto 
Guzman, Osvaldo 
Centro de Ciencias Matemáticas 
Hathaway, Daniel 
University of Michigan 
Haydon, Richard 
Brasenose College, Brasenose College 
Hrusak, Michael 
Universidad Nacional Autónoma de México 
Juhász, István 
Hungarian Academy of Sciences 
Kibedi, Francisco 
York University 
Koszmider, Piotr 
Polish Academy of Sciences 
Kubis, Wieslaw 
Academy of Sciences of the Czech Republic 
Laflamme, Claude 
University of Calgary 
Larson, Paul 
Miami University 
Li, Zhiqiang 
Fields Institute 
LópezAbad, Jorge 
Consejo Superior de Investigaciones Cientificas (CSIC) 
Lupini, Martino 
York University 
Magidor, Menachem 
Hebrew University of Jerusalem 
Malliaris, Maryanthe 
University of Chicago 
MartínezRanero, Carlos 
UNAM 
Mathias, A. R. D. 
University of La Réunion 
May, Natasha 
York University 
McKenney, Paul 
Carnegie Mellon University 
Mejia, Diego 
Kobe University 
Miller, Arnold W. 
University of WisconsinMadison 
Mitchell, William 
University of Florida 
Moore, Justin Tatch 
Cornell University 
Mota, Miguel Ángel 
The Fields Institute 
Nashaat, Sherif 
McGill University 
Neeman, Itay 
University of California, Los Angeles 
Ojeda Aristizabal, Diana Cristina 
Cornell University 
Pachl, Jan 
Fields Institute 
Pawlikowski, Janusz 
University of Wroclaw 
Peng, Yinhe 
National University of Singapore 
Raghavan, Dilip 
National University of Singapore 
Rinot, Assaf 
The Fields Institute 
Rosendal, Christian 
University of Illinois at Chicago 
Sabok, Marcin 
Polish Academy of Sciences 
Sakai, Hiroshi 
Kobe University 
Shani, Assaf 
Hebrew University 
Shi, Xianghui 
Beijing Normal University 
Soukup, Dániel Tamás 
University of Toronto 
Steprans, Juris 
York University 
Tall, Franklin 
University of Toronto 
Todorcevic, Stevo 
University of Toronto and C.N.R.S., Paris 
Trujillo, Timothy 
University of Denver 
Unger, Spencer 
Carnegie Mellon University 
Venturi, Giorgio 
Université Paris Diderot 
Viale, Matteo 
University of Torino 
Vignati, Alessandro 
York University 
Weiss, William 
University of Toronto 
Wilson, Trevor 
The Fields Institute 
Yorioka, Teruyuki 
Shizuoka University 
Zapletal, Jindrich 
University of Florida 
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