The Toronto Set Theory Seminar is normally held on Fridays, from 1:30 to 3pm, in room 210 of the Fields Institute. For listing of talks from years earlier than 2006, see this page.

Archives of: Toronto Set Theory Seminar

David Fernandez Breton: Partition theorems on uncountable abelian groups

Place: Fields Institute (Room 210)

Date: April 27, 2018 (13:30-15:00)

Speaker: David Fernandez Breton

Title: Partition theorems on uncountable abelian groups

Abstract:

In the past two years, a number of Ramsey-theoretic results concerning the additive structure of uncountable abelian groups have been investigated by diverse subsets of the set {Komjáth, Rinot, D. Soukup, W. Weiss, myself} (among others). From Ramsey results generalizing Hindman’s theorem for certain groups and colourings, to anti-Ramsey statements asserting the existence of ”rainbow colourings” for sets of finite sums, I plan to provide a panoramic overview of this exciting line of research, and point towards possible future lines of enquiry.

Anthony Bonato: The new world of infinite random geometric graphs

Place: Fields Institute (Room 210)

Date: April 20, 2018 (13:30-15:00)

Speaker: Anthony Bonato

Title: The new world of infinite random geometric graphs

Abstract:

The \emph{infinite random} or \emph{Rado graph} $R$ has been of interest to graph theorists, probabilists, and logicians for the last half-century. The graph $R$ has many peculiar properties, such as its \emph{categoricity}: $R$ is the unique countable graph satisfying certain adjacency properties. Erd\H{o}s and R\'{e}nyi proved in 1963 that a countably infinite binomial random graph is isomorphic to $R$.

Random graph processes giving unique limits are, however, rare. Recent joint work with Jeannette Janssen proved the existence of a family of random geometric graphs with unique limits. These graphs arise in the normed space $\ell _{\infty }^{n}$, which consists of $\mathbb{R}^{n}$ equipped with the $L_{\infty }$-norm. Balister, Bollob\'{a}s, Gunderson, Leader, and Walters used tools from functional analysis to show that these unique limit graphs are deeply tied to the $L_{\infty }$-norm. Precisely, a random geometric graph on any normed, finite-dimensional space not isometric $\ell _{\infty}^{n}$ gives non-isomorphic limits with probability $1$.

With Janssen and Anthony Quas, we have discovered unique limits in infinite dimensional settings including sequences spaces and spaces of continuous functions. We survey these newly discovered infinite random geometric graphs and their properties.

Menachem Magidor: Compactness and Incompactness for being Corson and Eberline compacta

Place: Fields Institute (Room 210)

Date: April 6, 2018 (13:30-15:00)

Speaker: Menachem Magidor

Title: Compactness and Incompactness for being Corson and Eberline compacta

Abstract:

The problems we shall discuss are examples of compactness (or dually a
reflection) problems. A typical compactness property is a statement that a
given structure has a certain property, provided smaller some structures have
this property. Reflection property is a dual statement, namely if a structure
has the property then there is a smaller substructure having the property. The
notion of “smaller substructure ” may depend on the domain we talk about. Thus
for an algebraic structure “smaller substructure” typically means a subalgebra
having a smaller cardinality. For topological spaces “smaller substructure ”
may mean a continuous image of the space of smaller weight.

Compactness problems tend to form clusters, which share the same pattern. For
instance sharing the same cardinals which are compact for the given property.
In this talk we shall survey some of these patterns. But we shall concentrate
on the problem of compactness for a compact space being Corson.

A compact space is a Corson compact if it can be embedded into $\Sigma\left(
R^{k}\right) $ where $\Sigma\left( R^{k}\right) $ is the the subspace of $R^{k}$ (with the product topology) of those sequences which are non zero only
on countably many coordinates. The compactness problem for Corson compacta is whether a space is Corson compact if compact given that all its continuous images of small weight are Corson. We shall report on some ongoing work about this problem.

Carlos Di Prisco: Chromatic numbers of Borel graphs

Place: Fields Institute (Room 332)

Date: March 28, 2018 (13:30-15:00)

Speaker: Carlos Di Prisco

Title: Chromatic numbers of Borel graphs

Abstract: A graph G=(X, R) defined on a Polish space X is Borel if the binary relation R is a Borel subset of the cartesian product of X with itself.
The Borel chromatic number of such a graph is the least cardinal k such that there is a Borel measurable function c from X to k coloring of the graph, that is, connected elements of X get different images under c.
The study of Borel chromatic numbers was initiated by Kechris, Solecki and Todorcevic (Advances in Mathematics 141 (1999) 1-44) and has received considerable attention since then.
We will survey some of the basic results regarding this concept and mention some open questions.

Antonio Aviles: Free Banach lattices

Place: Fields Institute (Room 210)

Date: March 23, 2018 (14:05-15:05)

Speaker: Antonio Aviles

Title: Free Banach lattices

Abstract: A Banach lattice has compatible structures of both Banach space and lattice. In this talk we present free constructions of Banach lattices based on a given Banach space or based on a given lattice, and we discuss some of their properties, like chain conditions ccc and others.

Slawomir Solecki: Polishable equivalence relations

Place: Fields Institute (Room 210)

Date: March 23, 2018 (13:00-14:00)

Speaker: Slawomir Solecki

Title: Polishable equivalence relations

Abstract: We introduce the notion of Polishable equivalence relations. This class of equivalence relations contains all orbit equivalence relations induced by Polish group actions and is contained in the class of idealistic equivalence relations of Kechris and Louveau. We show that each orbit equivalence relation induced by a Polish group action admits a canonical transfinite sequence of Polishable equivalence relations approximating it. The proof involves establishing a lemma, which may be of independent interest, on stabilization of increasing ω1-sequences of completely metrizable topologies.

Yasser Fermán Ortiz Castillo: Crowded pseudocompact spaces of cellularity at most the continuum are resolvable

Place: Fields Institute (Room 210)

Date: March 9 , 2018 (13:30-15:00)

Speaker: Yasser Fermán Ortiz Castillo

Title: Crowded pseudocompact spaces of cellularity at most the continuum are resolvable

Abstract: It is an open question from W. Comfort and S. Garcia-Ferreira if it is true that every crowded pseudocompact space is resolvable. In this talk will be present a partial positive answer for spaces of cellularity at most the continuum.

Bruno Braga: On the rigidity of uniform Roe algebras of coarse spaces

Place: Fields Institute (Room 210)

Date: March 2 , 2018 (13:30-15:00)

Speaker: Bruno Braga

Title: On the rigidity of uniform Roe algebras of coarse spaces.

Abstract: (joint with Ilijas Farah) Given a coarse space (X,E), one can define a $C^*$-algebra $C^∗_u(X)$ called the uniform Roe algebra of (X,E). It has been proved by J. \v{S}pakula and R. Willet that if the uniform Roe algebras of two uniformly locally finite metric spaces with property A are isomorphic, then the metric spaces are coarsely equivalent to each other. In this talk, we look at the problem of generalizing this result for general coarse spaces and on weakening the hypothesis of the spaces having property A.

Frank Tall: Co-analytic spaces, K-analytic spaces, and definable versions of Menger’s conjecture

Place: Fields Institute (Room 210)

Date: February 23 , 2018 (13:30-15:00)

Speaker: Frank Tall

Title: Co-analytic spaces, K-analytic spaces, and definable versions of Menger’s conjecture

Abstract: We continue the study of K-analytic and related spaces started last time, especially the connections between descriptive set theory as presented by Rogers and Jayne, and generalized metric spaces. We shall mention a number of unsolved problems and also give applications to productively Lindelof spaces and to topological groups.

Will Brian: Autohomeomorphisms of ω∗ : the quotient relation

Place: Fields Institute (Room 210)

Date: February 16, 2018 (13:30-15:00)

Speaker: Will Brian

Title: Autohomeomorphisms of ω∗ : the quotient relation

Abstract: Given two autohomeomorphisms f and g of N*, we say that f is a quotient of g when there is a continuous surjection Q from N* to N* such that Qg = fQ. In other words, f is a quotient of g if it is the “continuous image” of g, in the appropriate sense.

I have been investigating this relation, and will present some of the results of that investigation in my talk. For example, under CH: there are many universal autohomeomorphisms (an autohomeomorphism is universal if everything else is a quotient of it); the quotient relation has uncountable chains and antichains; there is an exact description of the quotients of a given trivial map. Under OCA+MA the picture is still murky: for example, there is a jointly universal pair of autohomeomorphisms (meaning everything else is a quotient of one or the other), but I do not know if there is a single universal automorphism. I will sketch some of these results and include several open questions.