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.

## Yinhe Peng: Basis of Countryman lines

Place: Fields Institute (Room 210)
Date: July 13 , 2018 (13:30-15:00)
Speaker:
Title: Basis of Countryman lines
Abstract: U. Abraham and S. Shelah proved that it is consistent to have a 2 element basis for Countryman lines. J. T.  Moore proved that, under PFA, these 2 Countryman lines serve as a basis for Aronszajn lines. We will show that for any positive integer $n$, it is consistent that the basis of Countryman lines has size $2^n$. We will also list some conditions that imply the basis to have maximal size – $2^{\omega_1}$.

## Hossein Lamei Ramandi: A minimal Kurepa tree with respect to club embeddings

Place: Fields Institute (Room 210)
Date: July 6, 2018 (13:30-15:00)
Speaker: Hossein Lamei Ramandi
Title:  A minimal Kurepa tree with respect to club embeddings
Abstract:  We will prove that it is consistent with CH that there is a Kurepa tree which club embeds into all of its Kurepa sub trees. Moreover, the Kurepa tree we introduce has no Aronszajn sub tree.

## Ilya Shapirovsky: Locally finite varieties of modal algebras

Place: Fields Institute (Room 210)
Date: June 15, 2018 (13:30-15:00)
Speaker: Ilya Shapirovsky
Title: Locally finite varieties of modal algebras
Abstract: A modal algebra is a Boolean algebra enriched with an additive operation. Equational theories of modal algebras are called modal logics. A logic L is said to be n-tabular if, up to the equivalence in L, there exist only finitely many n-variable formulas.  L is locally tabular if it is n-tabular for all finite n. Algebraically, n-tabularity of a logic means that its n-generated free algebra is finite (thus,  local tabularity of a logic is equivalent to local finiteness of its variety).

It is known that a variety of closure algebras  is locally finite iff its one-generated free algebra is finite (Larisa Maksimova, 1975). The following question has been open since 1970s: does this equivalence hold for every variety of modal algebras? (The analogous problem is open for varieties of Heyting algebras: does 2-tabularity of an intermediate logic imply local tabularity?)

Recently, in our joint work with Valentin Shehtman, it was shown how local tabularity of modal logics can be characterized in terms of partitions of relational structures. I will discuss this criterion and then use it to construct the first example  of a 1-tabular but not locally tabular modal logic.

## Andrea Vaccaro: Embedding C*-algebras into the Calkin algebra

Place: Bahen Centre Information (Room BA 2165)
Date: June 8, 2018 (13:30-15:00)
Speaker: Andrea Vaccaro
Title: Embedding C*-algebras into the Calkin algebra
Abstract:

Given a separable Hilbert space H, the Calkin algebra is the quotient C(H) = B(H)/K(H), B(H) being the algebra of all linear continuous operators from H into itself, and K(H) the closed ideal of compact operators. The Calkin algebra can be considered a noncommutative analogue of P(omega)/Fin, and it is known that these two objects share many structural properties. We show that yet another property of P(omega)/Fin has a noncommutative analogue for C(H). In particular, it is known that for every poset P there is a ccc poset H_P which forces the existence of an embedding of P into P(omega)/Fin. We prove that for any C*-algebra A there exists a ccc poset which forces the existence of an embedding of A into C(H).

## Mikołaj Krupski: The functional tightness of infinite products

Place: Bahen Centre (BA 2165)
Date: June 1, 2018 (13:30-15:00)
Speaker: Mikołaj Krupski
Title: The functional tightness of infinite products
Abstract: The functional tightness $t_0(X)$ of a space $X$ is a cardinal invariant related to both the tightness $t(X)$ and the density character $d(X)$ of $X$. While the tightness $t(X)$ measures the minimal cardinality of sets required to determine the topology of $X$, the functional tightness
measures the minimal size of sets required to guarantee the continuity of real-valued functions on $X$.
A classical theorem of Malykhin says that if $\{X_\alpha:\alpha\leq\kappa\}$ is a family of compact spaces such that $t(X_\alpha)\leq \kappa$, for every $\alpha\leq\kappa$, then $t\left( \prod_{\alpha\leq \kappa} X_\alpha \right)\leq \kappa$, where $t(X)$ is the tightness of a space $X$.
In my talk I will prove the following counterpart of Malykhin’s theorem for functional tightness:
Let $\{X_\alpha:\alpha<\lambda\}$ be a family of compact spaces such that $t_0(X_\alpha)\leq \kappa$. If $\lambda \leq 2^\kappa$ or $\lambda$ is less than the first measurable cardinal, then $t_0\left( \prod_{\alpha<\lambda} X_\alpha \right)\leq \kappa$, where $t_0(X)$ is the functional tightness of a space $X$. In particular, if there are no measurable cardinals the functional tightness is preserved by arbitrarily large products of compacta.

## Otmar Spinas: Why Silver is special

Place:   Bahen Center BA6183

Date: May 25, 2018 (13:30-15:00)
Speaker: Otmar Spinas
Title:  Why Silver is special
Abstract: I will try to give some insight into the challenging combinatorics of two amoeba forcings, one for Sacks forcing, the other one for Silver forcing. They can be used two obtain some new consistencies of inequalities between the additivity and the cofinality coefficients of the associated forcing ideals which are the Marcewski and the Mycielski ideal, respectively, and of the ideals associated with Laver forcing and Miller forcing.

## Alessandro Vignati: Homeomorphisms of Cech-Stone remainders

Place: Bahen Centre Information T (Room BA 2165)
Date: May 18, 2018 (13:30-15:00)
Speaker: Alessandro Vignati
Title: Homeomorphisms of Cech-Stone remainders
Abstract: From a locally compact space X one construct its Cech-Stone remainder X*=beta X minus X. We analyze the problem on whether X* and Y* can be homeomorphic for different spaces X and Y. In the 0-dimensional case, a solution to this problem has been proved to be independent of ZFC, by the work of Parovicenko, Farah, Dow-Hart and Farah-McKenney among others.
We prove, under PFA, the strongest possible rigidity result: for metrizable X and Y, we prove that X* is homeomorphic to Y* only if X and Y are homeomorphic modulo compact subsets. We also show that every homeomorphism X* –> Y* lifts to an homeomorphism between cocompact subsets of X and Y.

## Osvaldo Guzman: hm and the ultrafilter number

Place: Bahen Centre Information T .   BA 2165
Date: May 11, 2018 (13:30-15:00)
Speaker: Osvaldo Guzman
Title: hm and the ultrafilter number
Abstract: The cardinal invariant $\mathfrak{hm}$ is defined as the minimum size of a family of $\mathsf{c}_{\mathsf{min}}$-monochromatic sets that cover $2^{\omega}$ (where $\mathsf{c}_{\mathsf{min}}\left( x,y\right)$ is the parity of the biggest initial segment both $x$ and $y$ have in common). We prove that $\mathfrak{hm}=\omega_{1}$ holds in the Shelah’s model of $\mathfrak{i<u}$ so the inequality $\mathfrak{hm<u}$ is consistent with the axioms of $\mathsf{ZFC.}$ This answers a question of Thilo Weinert.

## 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.