Workshop on Forcing Axioms and their Applications, October 22-26, 2012

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 P-ideal 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.

Answering a 1978 problem of Malykhin we show that it is relatively consistent with ZFC that every separable Frechet-Urysohn topological group is metrizable. (Joint work with Ulises Ariet Ramos Garcia)

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 still-open 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.

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.

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 sigma-algebra. (Joint work with Grzegorz Plebanek and Jose Rodriguez)

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.

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.

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 set-theoretic assumptions or forcing arguments.

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.

A strong Gurarii space is a Banach space containing isometric copies of all finite-dimensional spaces which is additionally homogeneous with respect to finite-dimensional subspaces. The latter means that every linear isometry between its finite-dimensional subspaces extends to a bijective isometry of the entire space. It is well-known (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 set-theoretic 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.

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 .

We show that under the assumptions of the P-ideal 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.

Recent work of Malliaris and Shelah on model-theoretic 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.