# Category Archives: Seminars

## 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.

## Mirna Džamonja: Higher order versions of the logic of chains close

Dear all,

The seminar meets on Wednesday June 20th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Mirna Džamonja — Higher order versions of the logic of chains
close

First order logic of chains was discovered by Carol Karp and revisited
in recent work of Dz. with Jouko Vaananen. The results have shown that
the logic, defined through a singular cardinal of countable cofinality,
behaves very much like the first order logic. In our new joint work, we
study higher order versions of the logic of chains and their fragments
to defend the thesis that in this context we can also recover
similarities with the ordinary logic. We also discuss the idea of
infinite computation.

Best,
David

## Joshua Brot, Mengyang Cao, David J. Fernández-Bretón: Finiteness classes arising from Ramsey-theoretic statements in set theory without choice

Thursday, June 14, 2018, from 4 to 5:30pm
East Hall, room 4096

Speaker: Joshua Brot, Mengyang Cao, David J. Fernández-Bretón (University of Michigan)

Title: Finiteness classes arising from Ramsey-theoretic statements in set theory without choice

Abstract:

We investigate infinite sets that witness the failure of certain Ramsey-theoretic statements, such as Ramsey’s or (appropriately phrased) Hindman’s theorem; such sets may exist if one does not assume the Axiom of Choice. We will show very precise information as to where such sets are located within the hierarchy of infinite Dedekind-finite sets. The proofs involve both very pleasant combinatorial arguments (to establish certain implications) and the Fränkel-Mostowski technique to obtain permutation models of ZFA (to prove that certain other implications are not provable).

## Zoltán Vidnyánszky: Borel chromatic numbers: basis and antibasis results

Talk held by Zoltán Vidnyánszky (KGRC) at the KGRC seminar on 2018-06-14.

Abstract: We give a full description of the existence of a homomorphism basis for Borel graphs of given Borel chromatic number. In particular, we show that there is a Borel graph with Borel chromatic number 3 that admits a homomorphism to any Borel graph of Borel chromatic number at least 3. We also discuss the relation of these results to Hedetniemi’s conjecture.

## 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).

## Viera Šottová: Ideal version of selection principle S1(P,R)

The seminar meets on Wednesday June 6t at 11:00 in the Institute of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Speaker: Viera Šottová

Title: Ideal version of selection principle $S_1(\mathcal P,\mathcal R)$.

Abstract: attached.

## 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.

## Martin Goldstern: Higher Random Reals

HUJI Logic Seminar
Tuesday 29/5 13:30-15:00 Ross 63
Title: Higher Random Reals
Speaker: Martin Goldstern
Abstract:
The set of real numbers is often identified with
Cantor Space 2^omega, with which it shares many important
properties: not only the cardinality, but also other
“cardinal characteristics” such as cov(null), the smallest
number of measure zero sets needed to cover the whole space,
and similarly cov(meager), where meager=”first category”;
or their “dual” versions non(meager) (the smallest
cardinality of a nonmeager set) and non(null).

Many ZFC results and consistency results (such as
“cov(meager) lessequal non(null), but no inequality between
cov(null) and cov(meager) is provable in ZFC”)
are known.

Recent years have seen a renewed interest in “higher reals”,
i.e., elements of 2^kappa, where kappa is usually an inaccessible
cardinal. Meager sets have a natural generalisation to this
context, namely “kappa-meager” sets (using the <kappa-box product
topology), but what is the natural generalisation of the
ideal of null sets?

In my talk I will present an ideal null_kappa recently introduced
by Saharon Shelah, and some ZFC and consistency results from a
forthcoming joint paper with Thomas Baumhauer and Saharon Shelah,
such as “cov(null_kappa) lessequal non(null_kappa)”, and
“consistently, cov(meager_kappa) > cov(null_kappa)”.

## 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.

## Borisha Kuzeljevic: P-ideal dichotomy and some versions of the Souslin Hypothesis

Talk held by Borisha Kuzeljevic (Czech Academy of Sciences, Prague) at the KGRC seminar on 2018-05-24.

Abstract: The talk will be about the relationship of PID with the statement that all Aronszajn trees are special. This is joint work with Stevo Todorcevic.