# Category Archives: Seminars

## 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.

## David Fernández-Bretón: Variations and analogs of Hindman’s theorem

Mathematical logic seminar – May 22 2018
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         David Fernández-Bretón
Department of Mathematics
University of Michigan

Title:     Variations and analogs of Hindman’s theorem

Abstract:

Hindman’s theorem is a Ramsey-theoretic result asserting that, whenever one colours the set of natural numbers with finitely many colours, there will be an infinite set such that all numbers that can be obtained by adding finitely many elements from the set (no repetitions allowed) have the same colour. I will explore generalizations and extensions of this theorem: replacing “natural numbers” with “abelian group” and varying the number of colours, as well as the size of the desired monochromatic set, yields a plethora of very interesting results.

## Harry Altman: Lower sets in products of well-ordered sets and related WPOs

Thursday, May 24, 2018, from 4 to 5:30pm
East Hall, room 4096

Speaker: Harry Altman (University of Michigan)

Title: Lower sets in products of well-ordered sets and related WPOs

Abstract:

Following last week’s talk on maximum order types of well partial orders, we’ll compute the maximum order type of the set of bounded lower sets in N^m, as well as generalizations to finite products of other well-ordered sets, and discuss the maximum order types of some other related well partial orders also.

## Damian Sobota: Rosenthal families and ultrafilters

Dear all,

The seminar meets on Wednesday May 23rd at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Damian Sobota — Rosenthal families and ultrafilters

Rosenthal’s lemma is a simple technical result with numerous
applications to measure theory and Banach space theory. The lemma in its
simplest form reads as follows: “For every infinite real-entried matrix
(m(n,k): n,k in N) such that every entry is non-negative and the sum of
every row is <=1, and every epsilon>0, there exists an infinite subset A
of N such that for every k in A we have sum_{n in A, n\neq
k}m_n^k<epsilon.” A natural question arises whether we can choose the
set A from a previously fixed family F of infinite subsets of N. If F
has such a property, then we call it Rosenthal. Thus, Rosenthal’s lemma
states that [N]^omega is Rosenthal. During my talk I’ll present some
necessary or sufficient conditions for a family to be Rosenthal and
prove that under MA(sigma-centered) there exists a P-point which is a
Rosenthal family but not a Q-point. (No Banach space will appear during
the talk.)

Best,
David

## 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.