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.

Ilijas Farah: A consistent failure Glimm’s theorem for nonseparable C*-algebras

Place: Fields Institute (Room 210)

Date: June 24th, 2016 (13:30-15:00)

Speaker: Ilijas Farah

Title: A consistent failure Glimm’s theorem for nonseparable C*-algebras

Abstract: A remarkable 1960 result of J. Glimm provides a sharp dichotomy for
the representation theory of separable C*-algebras. One of its
consequences is that a separable C*-algebra either has a unique
(up to the unitary equivalence) irreducible representation or continuum
many inequivalent irreducible representations. Using some ideas of
Akemann and Weaver, I’ll prove that this conclusion is consistently false
for nonseparable C*-algebras. Many open problems remain.

David Fernández Bretón: Ultrafilters on the rationals generated by perfect sets

Place: Fields Institute (Room 210)

Date: June 17th, 2016 (13:30-15:00)

Speaker: David Fernandez Bretón

Title: Ultrafilters on the rationals generated by perfect sets

Abstract:  In a 1992 paper, van Douwen defined what he calls a
“gruff ultrafilter”: an ultrafilter on the rational numbers which
is generated by perfect (this is, closed and crowded) sets; and
asked whether these ultrafilters exist, providing in the same
paper a proof that they do if cov(M)=c. The question of whether
the existence of gruff ultrafilters can be proved in ZFC alone
remains open, but further progress has been made in the
way of consistently positive answers. In this talk I will
present a proof that gruff ultrafilters exist in the Random
model, as well as in any model satisfying d=c. Joint work
with Michael Hrusak.

Dana Bartosova: Algebra in the Samuel compactification

Place: Fields Institute (Room 210)

Date: June 10, 2016 (13:30-15:00)

Speaker: Dana Bartosova

Title: Algebra in the Samuel compactification

Abstract: The Samuel compactification, or the greatest ambit, is an important compactification of a topological group for its dynamics. In the case of discrete groups, the Samuel compactification coincides with the Cech-Stone compactification and its algebra and combinatorics have been extensively studied. We remind the Samuel compactification for automorphism groups in the ultrafilter language and point out some differences and similarities with the discrete case. We will then apply algebra and combinatorics to answer a problem of Ellis for the group of permutations of the integers. This is a joint work in progress with Andy Zucker (Carnegie Mellon University).

Alessandro Vignati: CH and homeomorphisms of Stone-Cech remainders

Place: Fields Institute (Room 210)

Date: June 3rd, 2016 (13:30-15:00)

Speaker: Alessandro Vignati

Title: CH and homeomorphisms of Stone-Cech remainders

Abstract:

If X is locally compact and Polish, it makes sense to ask how many homeomorphisms does X*, the Stone Cech remainder of X, have. It is known that, if X is 0-dimensional, under the Continuum Hypothesis X* has $2^{2^{\aleph_0}}$ many homeomorphisms (Rudin+Parovicenko). The same is true if $X=[0,1)$ (Yu, Dow-KP Hart), or if X is the disjoint union of countably many compact spaces (Coskey-Farah). But the question remains open for, for example, $X=\mathbb{R}^2$. We prove that for a large class of spaces (including $\mathbb{R}^n$, for all n) CH provides $2^{2^{\aleph_0}}$ many homeomorphisms of X*.

Fulgencio Lopez: Banach spaces from Construction Schemes

Place: Fields Institute (Room 210)

Date: May 27th, 2016 (13:30-15:00)

Speaker: Fulgencio Lopez

Title: Banach spaces from Construction Schemes

Abstract: In 2011 J. Lopez-Abad and S. Todorcevic used forcing to construct a
Banach space with an $\varepsilon$-biorthogonal system that didn’t have $\delta$-biorthogonal systems for every $1\leq\delta<\varepsilon$. We show that there is a Banach space with the same property provided there is a capturing Construction Scheme.

Asger Tornquist: An invariant set-theoretic approach to Mathias’ theorem

Place: Fields Institute (Room 210)

Date: May 20, 2016 (13:30-15:00)

Speaker: Asger Tornquist

Title: An invariant set-theoretic approach to Mathias’ theorem

Abstract: Mathias proved in 1969 that there are no infinite analytic
maximal almost disjoint families of subsets of $\omega$. His proof is
essentially Ramsey theoretical. A few years ago I found a “classical” proof
of this theorem which uses a tree/derivative argument. But in this talk, I
will give yet another proof, this time one that is closer in spirit to
Mathias’ original proof, but which avoids the Ramsey theoretical machinery
by instead using a bit of garden-variety invariant descriptive set theory.

Peter Koellner: Large Cardinals Beyond Choice

Place: Fields Institute (Room 210)

Date: May 13, 2016 (13:30-15:00)

Speaker: Peter Koellner

Title: Large Cardinals Beyond Choice

Abstract:

The hierarchy of large cardinals provides us with a canonical means to
climb the hierarchy of consistency strength. There have been many
purported inconsistency proofs of various large cardinal axioms. For
example, there have been many proofs purporting to show that
measurable cardinals are inconsistent. But to date the only proofs
that have stood the test of time are those which are rather
transparent and simple, the most notable example being Kunen’s proof
showing that Reinhardt cardinals are inconsistent. The Kunen result,
however, makes use of AC, and long standing open problem is whether
Reinhardt cardinals are consistent in the context of ZF.

In this talk I will survey the simple inconsistency proofs and then
raise the question of whether perhaps the large cardinal hierarchy
outstrips AC, passing through Reinhardt cardinals and reaching far
beyond. There are two main motivations for this investigation. First,
it is of interest in its own right to determine whether the hierarchy
of consistency strength outstrips AC. Perhaps there is an entire
“choiceless” large cardinal hierarchy, one which reaches new
consistency strengths and has fruitful applications. Second, since the
task of proving an inconsistency result becomes easier as one
strengthens the hypothesis, in the search for a deep inconsistency it
assumptions and then work ones way down. This will lead to the
formulation of large cardinal axioms (in the context of ZF) that start
at the level of a Reinhardt cardinal and pass upward through Berkeley
cardinals (due to Woodin) and far beyond. Bagaria, Woodin, and I
have been charting out this new hierarchy. I will discuss what we have
found so far.

Place: Fields Institute (Room 210)

Date: May 2nd , 2016 (13:30-15:00)

Speaker: Ari Brodsky

Abstract:  We propose a parametrized proxy principle from which $\kappa$-Souslin trees with various additional features can be constructed, regardless of the identity of $\kappa$. We then introduce the microscopic approach, which is a simple method for deriving trees from instances of the proxy principle. As a demonstration, we give a construction of a coherent $\kappa$-Souslin tree that applies also for $\kappa$ inaccessible.

Slides: AriBrodskyMay2016

Juris Steprans: Graph embedding and the P-ideal dichotomy

Place: Fields Institute (Room 210)

Date: April 22nd , 2016 (13:30-15:00)

Speaker: Juris Steprans

Title: Graph embedding and the P-ideal dichotomy

Abstract: I will discuss a family of proofs of the consistency of a universal graph on $\omega_1$ with the failure of CH that rely on iterating reals and using the P-ideal dichotomy.

Francisco Guevara: Analytic group topologies

Place: Fields Institute (Room 210)

Date: April 15th , 2016 (13:30-15:00)

Speaker: Francisco Guevera

Title: Analytic group topologies

Abstract: We study the effective version Malykhin’s question about the metrizability of (countable) Frechet groups and its natural generalization to metrizability of (countable) sequential groups of higher sequential order. A countable topological space $(X,\tau)$ is analytic if $\tau$ is analytic as a subset of the Cantor set $2^X$. By effective we mean the group topology is analytic. A space is sequential if all sequentially closed sets are closed. In sequential spaces, the sequential order is defined as the minimal ordinal $\alpha$ so that the closure of every set is obtained by applying the operation of adding limit points $\alpha$-many times. A sequential space has order $1$ iff it is Frechet. The results presented in the talk come from some works of A. Shibakov,  S. Todorcevic, and C. Uzcategui.