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 
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 isometry groups and maximal 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 in abstract elementary classes
We will present some applications of iterated generic ultrapowers to the study of models of cardinality in various abstract elementary classes. The impetus for this work was the stillopen question of absoluteness of categoricity for the class of models of a fixed sentence of . 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
TBA 

Coffee break 
11:00 
Matteo Viale, University of Torino
Forcing with forcings
Let be a class of partial orders and be a class of complete embeddings between elements of closed under composition.Then is a category whose objects are elements of and whose arrows are elements of . Moreover is a partial order. Depending on the nature of and this can be an interesting or trivial partial order. If is the class of all posets and is the class of all complete embeddings and is limit is a trivial partial order since all elements of this partial order are compatible.
We shall study the case in which is the class of stationary set preserving (semiproper, proper) posets, and is the class of complete embeddings between elements of with a stationary set preserving (semiproper, proper) quotient.
We show that if has some degree of largeness which depends on the choice of the category , is a STATIONARY SET PRESERVING partial order which collapses to become but it should NEVER be a proper one and can be a semiproper one only if 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 . However this will be the subject of a future talk.


Lunch Break 
2:00 
Hiroshi Sakai, Kobe University 

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 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
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
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 maximal antichains, in specific classes of posets that preserve both and .


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 , every Banach space of density with weak* sequentially compact dual ball has a quotient of density with a Schauder basis. Together with an example of Todorcevic, it follows that under the PID, b= 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.


