# THEMATIC PROGRAMS

September 15, 2019

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

July-December 2012 Thematic Program on Forcing and its Applications

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

Organizing Committee:
Jordi Lopez Abad, Justin Tatch Moore, Stevo Todorcevic
 Confirmed Participants Back to Program index

### 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 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. Coffee break 11:00 Alan Dow, UNC Charlotte Martin's Axiom and initially W1-compact spaces (slides) 12:00-2:00 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 Frechet-Urysohn 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 still-open 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 Non-trivial 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 sigma-algebra. (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 set-theoretic assumptions or forcing arguments. 4:40 Carlos Martinez-Ranero, UNAM Invariance properties of almost disjoint families (Well quasi-ordering 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 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. 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 P-ideal 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 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.

### 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 Borodulin-Nadzieja, Piotr University of Wroclaw Brech, Christina University of São Paulo Brendle, Joerg Kobe University Brenken, Berndt University of Calgary Brodsky, Ari University of Toronto Cancino-Manríquez, Jonathan UNAM-UMSNH 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 UNAM-UMSNH 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 López-Abad, Jorge Consejo Superior de Investigaciones Cientificas (CSIC) Lupini, Martino York University Magidor, Menachem Hebrew University of Jerusalem Malliaris, Maryanthe University of Chicago Martínez-Ranero, 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 Wisconsin-Madison 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