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

Stevo Todorcevic: A proof of Galvin’s Conjecture

Place: Fields Institute (Room 210)
Date: October 26, 2018 (13:30-15:00)
Speaker: Stevo Todorcevic
Title: A proof of Galvin’s Conjecture
Abstract: We prove that for every finite colouring of the set of unordered
pairs of real numbers there is a set of reals homeomorphic to the rationals whose pairs use no more than two colours. This solves a problem of F. Galvin from the 1970’s. The proof uses large cardinals. This is a joint work with Dilip Raghavan.

Ilya Shapirovsky: Modal logics of model-theoretic relations

Place: Fields Institute (Room 210)
Date: October 19 , 2018 (13:30-15:00)
Speaker: Ilya Shapirovsky
Title: Modal logics of model-theoretic relations
Abstract: Consider a unary operation f on the set of sentences of a model-theoretic language L, and a set T of sentences of L.  Properties of f in T can be studied using  propositional modal language: variables are evaluated as sentences of L, and f interprets the modal operator. The modal theory of f in T is defined as the set of those modal formulas which are in T under every valuation.

An example of this approach is Solovay’s theorem providing a complete modal axiomatization of formal provability in Peano arithmetic. Another example is the theorem of Hamkins and Loewe axiomatizing the modal logic of forcing, where the modal operator expresses satisfiability in forcing extensions. Both these logics  have good semantic and algorithmic properties: in particular, they have the finite model property, are finitely axiomatizable, and hence decidable.

This raises the question of modal theories of other model-theoretic relations R (e.g., the submodel relation or the homomorphic image relation). These theories can be defined in the case when satisfiability in R-images is expressible in L. In this talk we will discuss general properties of such modal systems, and then provide a complete axiomatization for the case of the submodel relation.  This talk is based on a joint work with D.I. Saveliev.

Vera Fischer: More ZFC inequalities between cardinal invariants

Place: Fields Institute (Room 210)
Date: September 21, 2018 (13:30-15:00)
Speaker: Vera Fischer
Title: More ZFC inequalities between cardinal invariants
Abstract: We will discuss some recent ZFC results concerning the
generalized Baire spaces, and more specifically the generalized
bounding number, relatives of the generalized almost disjointness
number, as well as generalized reaping and domination.

Damjan Kalajdzievski: Bounding, splitting, almost disjointness and covering of the meager ideal

Place: Fields Institute (Room 210)
Date: , 2018 (13:30-15:00)
Speaker: Damjan Kalajdzievski
Title: Bounding, splitting, almost disjointness and covering of the meager ideal
Abstract:This talk is on joint work with Osvaldo Guzman. This talk will go over forcing $\omega_1=\mathfrak{b}=cov(\mathcal{M})<\mathfrak{s}=\omega_2$ with a countable support iteration of proper forcings. In doing so we will introduce the forcings $\mathbb{PT}(\mathcal{F})$, which are Miller trees that satisfy a restriction on splitting nodes relative to the filter $\mathcal(F)$, and discuss their properties when $\mathcal{F}$ is Canjar. The result is achieved by iterating the forcing $\mathbb{F}_\sigma*\mathbb{PT}(\mathcal{F})$, where $\mathbb{F}_\sigma$ is the forcing of $F_\sigma$ filters on $\omega$ ordered by reverse inclusion.

Osvaldo Guzman: On restricted MADness

Place: Fields Institute (Library)
Date: September 7, 2018 (13:30-15:00)
Speaker: Osvaldo Guzman
Title: On Restricted MADness
Abstract: Let $\mathcal{I}$ be an ideal on $\omega.$ We define \textsf{cov}$^{\ast}\left( \mathcal{I}\right) $ as the least size of a family
$\mathcal{B\subseteq I}$ such that for every infinite $X\in\mathcal{I}$ there is $B\in\mathcal{B}$ for which $B\cap X$ is infinite. We say an \textsf{AD} family $\mathcal{A\subseteq I}$ is a \emph{\textsf{MAD} family restricted to }$\mathcal{I}$ if for every infinite $X\in\mathcal{I}$ there is $A\in \mathcal{A}$ such that $\left\vert X\cap A\right\vert =\omega.$ The cardinal invariant $\mathfrak{a}\left( \mathcal{I}\right) $ is defined as the least size of an infinite \textsf{MAD} family restricted to $\mathcal{I}.$ The cardinal invariants $\mathfrak{o}$ and $\mathfrak{a}_{s}$ may be seen as
particular cases of this class of invariants. In this talk, we will prove that
if the maximum of $\mathfrak{a}$ and  textsf{cov}$^{\ast}\left(\mathcal{I}\right) $ is $\omega_{1}$ then $\mathfrak{a}\left( \mathcal{I}%
\right) =\omega_{1}.$ We will obtain some corollaries of this result. This is part of a joint work with Michael Hru\v{s}\'{a}k and Osvaldo Tellez.

Yinhe Peng: Basis of Countryman lines

Place: Fields Institute (Room 210)
Date: July 13 , 2018 (13:30-15:00)
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

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.