# Set Theory and its Applications in Topology (16w5053)

Arriving in Oaxaca, Mexico Sunday, September 11 and departing Friday September 16, 2016

## Organizers

Michael Hrusak (Universidad Nacional Autónoma de México)

Stevo Todorcevic (University of Toronto and Institut de Mathématiques de Jussieu)

Justin Tatch Moore (Cornell University, Ithaca, NY, USA)

## Objectives

The symbiotic and mutually beneficial relationship between set theory and topology is undeniable. The workshop aspires to bring together leading experts from both areas in order to keep the exchange of ideas between the fields flowing. The inclusion of functional analysis is the result of recent developments, which showed that not only topological, but also Ramsey-theoretic and forcing techniques can be successfully used to solve outstanding problems in this field.

The centerpiece of the proposed workshop %, and of the connections between the areas of mathematics involved, is the the

Algebraic aspects of $\beta\mathbb N$ are also of utmost interest here. Addition on $\mathbb N$ can be naturally extended to a binary operation on $\beta\mathbb N$, turning $\beta\mathbb N$ into a compact left-topological semigroup. The structure of the semigroup can be used to prove strong Ramsey-theoretic statements such as Hindman's

There is a strong link between Ramsey-theoretic statements and amenability properties of infinite groups [kechris-pestov-todorcevic, moore]. For instance, work of Kechris-Pestov-Todorcevic [kechris-pestov-todorcevic] showed that the question of whether the automorphism group of a countable ultrahomogeneous structure such as $(\mathbb{Q},<)$ is

In [moore], Moore translated the problem of the amenability of a given countable discrete group into a Ramsey-theoretic problem. He showed that the well studied question of whether Thompson's group $F$ is amenable is equivalent to a Ramsey-theoretic statement and would follow from a natural generalization of Hindman's theorem to the setting of nonassociative binary systems. This analysis suggests new ways that the dynamics of the space of ultrafilters on a countable set can be used to study the question of whether Thompson's group is amenable.

The methods of set theory, topology, and functional analysis are linked though the study of the structure and geometry of function spaces. In order to construct examples of Banach spaces with complex structural or geometric properties, for instance, one often first constructs a Boolean algebra $\mathbb B$ --- either using forcing, or some other suitable set-theoretic tool or axiom. By Stone duality, this algebra can be regarded as a compact topological space $K=St(\mathbb B)$. Finally, one considers the Banach space $C(K)$ of continuous real-valued functions on $K$, see e.g. [koszmider].

While this is a rather straightforward connection, it leads to much more important and sophisticated ones. For example topological properties of the dual ball of a Banach space $X$ are closely related to purely analytical properties of the norm of $X$. One of the deepest connection of this sort where set-theoretic methods are relevant are the Tsirelson-type constructions of Banach spaces $X$ by first constructing their dual balls. While this method has seen great success in the realm of separable Banach spaces, such as the solution of the unconditional basic sequence problem by Gowers and Maurey [gowers] or the scalar-times-identity plus compact-operator problem by Argyros and Haydon [argyros-haydon], the nonseparable theory has seen recently great advances exactly because of its deep connection to set theory. Of these advances we could mention the construction of a nonseparable reflexive Banach space with no infinite unconditional basic sequence by Argyros--Lopez-Abad--Todorcevic [argyros-lopez-todorcevic] or the isolation of the threshold $\aleph_\omega$ as the minimal cardinal which can have the property that every weakly null sequence of that length must contain an infinite unconditional basic subsequence by Dodos--Lopez-Abad--Todorcevic [dodos-lopez-todorcevic].

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The centerpiece of the proposed workshop %, and of the connections between the areas of mathematics involved, is the the

*\v Cech-Stone compactification*of a countable discrete space $\beta\mathbb N$, also known as the space of ultrafilters on $\mathbb N$. Astonishingly, many fundamental problems about $\beta\mathbb N$ are still open. A short sample:- (Efimov) Does every compact space contain either a convergent sequence or a copy of $\beta\mathbb N$?
- (Szyma\'nski) Can $\beta\mathbb N\setminus \mathbb N$ and $\beta\omega_1\setminus \omega_1$ ever be homeomorphic?
- (Leonard and Whitfield) If the Banach space $C(\beta\mathbb N)\simeq \ell_\infty/c_0$ is written as a direct sum of two closed spaces $X$ and $Y$, must either $X$ or $Y$ be isomorphic to $C(\beta\mathbb N)$? (I.e. is $\ell_\infty/c_0$

*primary*?)

- ([dow}) (In {\sf ZFC]) $\beta\mathbb N$ contains a non-trivial copy of $\beta\mathbb N$.

Algebraic aspects of $\beta\mathbb N$ are also of utmost interest here. Addition on $\mathbb N$ can be naturally extended to a binary operation on $\beta\mathbb N$, turning $\beta\mathbb N$ into a compact left-topological semigroup. The structure of the semigroup can be used to prove strong Ramsey-theoretic statements such as Hindman's

*finite sum theorem*. Research in this area is very active, see [hindmand-strauss, solecki, kechris-pestov-todorcevic, todorcevic-book].There is a strong link between Ramsey-theoretic statements and amenability properties of infinite groups [kechris-pestov-todorcevic, moore]. For instance, work of Kechris-Pestov-Todorcevic [kechris-pestov-todorcevic] showed that the question of whether the automorphism group of a countable ultrahomogeneous structure such as $(\mathbb{Q},<)$ is

*extremely amenable*is equivalent to the question of whether the class of finite substructures form a so-called*Ramsey class*. Here a topological group is extremely amenable if all of its continuous actions on compact spaces have a fixed point. The notion of a Ramsey class was the central object of study in a part of combinatorics known as*structural Ramsey theory*which was developed by Graham, Ne\v{s}et\v{r}il, R\"odel, Rothschild and others in the 1970s and 1980s. The techniques of [kechris-pestov-todorcevic] have since been adapted to other settings and used to study the homeomorphism groups of the pseudo-arc and the Lelek fan of continuua theory [bartosova-kwitkowska, solecki].In [moore], Moore translated the problem of the amenability of a given countable discrete group into a Ramsey-theoretic problem. He showed that the well studied question of whether Thompson's group $F$ is amenable is equivalent to a Ramsey-theoretic statement and would follow from a natural generalization of Hindman's theorem to the setting of nonassociative binary systems. This analysis suggests new ways that the dynamics of the space of ultrafilters on a countable set can be used to study the question of whether Thompson's group is amenable.

The methods of set theory, topology, and functional analysis are linked though the study of the structure and geometry of function spaces. In order to construct examples of Banach spaces with complex structural or geometric properties, for instance, one often first constructs a Boolean algebra $\mathbb B$ --- either using forcing, or some other suitable set-theoretic tool or axiom. By Stone duality, this algebra can be regarded as a compact topological space $K=St(\mathbb B)$. Finally, one considers the Banach space $C(K)$ of continuous real-valued functions on $K$, see e.g. [koszmider].

While this is a rather straightforward connection, it leads to much more important and sophisticated ones. For example topological properties of the dual ball of a Banach space $X$ are closely related to purely analytical properties of the norm of $X$. One of the deepest connection of this sort where set-theoretic methods are relevant are the Tsirelson-type constructions of Banach spaces $X$ by first constructing their dual balls. While this method has seen great success in the realm of separable Banach spaces, such as the solution of the unconditional basic sequence problem by Gowers and Maurey [gowers] or the scalar-times-identity plus compact-operator problem by Argyros and Haydon [argyros-haydon], the nonseparable theory has seen recently great advances exactly because of its deep connection to set theory. Of these advances we could mention the construction of a nonseparable reflexive Banach space with no infinite unconditional basic sequence by Argyros--Lopez-Abad--Todorcevic [argyros-lopez-todorcevic] or the isolation of the threshold $\aleph_\omega$ as the minimal cardinal which can have the property that every weakly null sequence of that length must contain an infinite unconditional basic subsequence by Dodos--Lopez-Abad--Todorcevic [dodos-lopez-todorcevic].

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