## Piotr Koszmider: The Grothendieck property for Banach spaces of continuous functions

Seminar: Working group in applications of set theory, IMPAN

Thursday, 29.11.2018, 10:15, room 105, IMPAN

Speaker: Piotr Koszmider (IM PAN)

Title: “The Grothendieck property for Banach spaces of continuous functions”

Abstact: “In the first talk of the series devoted to classical phenomena in Banach spaces of the form C(K) we will see how weakly compact sets in the dual space to C(K) generalize finite subsets of K. The concrete goal will be to motivate the Grothendieck property (weak and weak* convergence of sequences coincide in the dual) for C(K)s as a generalization of K having no nontrivial convergent sequence and to prove that l∞≡C(βN) has the Grothendieck property. This will require the proof of the Grothendieck-Dieudonne characterization of weakly compact sets in the spaces of measures. All the results and proofs presented during the talk are in classical texbooks, but we will try to represent combinatorial and topological bias, leading in the following talks to more set-theoreic issues. The purpose of this series of talks is to introduce particpants with the set-theoretic topological background to some topics related to Banach spaces of the form C(K). The area is quite sensitive to infinitary combinatorics, e.g., Talagrand: CH implies that there is infinite K such that C(K) is Grothendieck but does not have l∞ as its quotient; Haydon, Levy, Odell: p=2^ω>ω_1 implies that every Grothendieck C(K) for K infinite has l∞ as its quotient “.

Visit our seminar page which may include some future talks at https://www.impan.pl/~set_theory/Seminar/

## Tomasz Kochanek: Bases of Banach spaces with respect to filters

Seminar: Working group in applications of set theory, IMPAN

Thursday, 22.11. 2018, 10:15, room 105, IMPAN

Speaker: Tomasz Kochanek (IM PAN / MIM UW)

Title: “Bases of Banach spaces with respect to filters”

Abstact: “In 2011, Vladimir Kadets proposed the following problem: Given a filter F of subsets of natural numbers and a Banach space X, we say that a sequence (e_n) in X forms an F-basis, provided that every x in X has a unique representation as a series of linear combinations of e_n’s, where the convergence is understood in the norm topology and with respect to F. Thus, for F being the filter of cofinite sets we obtain the classical notion of Schauder basis for which it is well-known that all the coordinate functionals are automatically continuous. The question is whether, they must be continuous for a general filter F. I shall present a positive answer to this questions in the case where the character of F is smaller than the pseudointersection number (published in Studia Math. 2012). Unfortunately, the answer is still not known in the important case where F is the statistical filter consisting of all sets of asymptotic density 1. We will also discuss some other related open problems concerning bases with brackets and with individual brackets”.

Visit our seminar page which may include some future talks at https://www.impan.pl/~set_theory/Seminar/

## Fulgencio Lopez: A capturing construction scheme from the diamond

Seminar: Working group in applications of set theory, IMPAN

Thursday, 15.11. 2018, 10:15, room 105, IMPAN

Speaker: Fulgencio Lopez (IM PAN)

Title: “A capturing construction scheme from the diamond principle” Continuation from 8.11.2018.

Abstact: “S. Todorcevic introduced the concept of a capturing construction scheme and showed it is consistent with the diamond principle. A construction scheme is a well-founded family of finite subsets of ω1. We give a quick presentation of the history and motivation for this tool and show that it follows from the diamond principle. “.

Visit our seminar page which may include some future talks at https://www.impan.pl/~set_theory/Seminar/

## Fulgencio Lopez: A capturing construction scheme from the diamond

Seminar: Working group in applications of set theory, IMPAN

Thursday, 08.11. 2018, 10:15, room 105, IMPAN

Speaker: Fulgencio Lopez (IM PAN)

Title: “A capturing construction scheme from the diamond principle”

Abstact: “S. Todorcevic introduced the concept of a capturing construction scheme and showed it is consistent with the diamond principle. A construction scheme is a well-founded family of finite subsets of ω1. We give a quick presentation of the history and motivation for this tool and show that it follows from the diamond principle. “.

Visit our seminar page which may include some future talks at https://www.impan.pl/~set_theory/Seminar/

## Arturo Martínez-Celis: On the Michael Space Problem

Seminar: Working group in applications of set theory, IMPAN

Thursday, 25.10. 2018, 10:15, room 105, IMPAN

Speaker: Arturo Martínez-Celis (IM PAN)

Title: “On the Michael Space Problem”

Abstact: “A Lindelöf Topological space is Michael if it has non-Lindelöf product with the space of the irrational numbers. These kind of spaces were introduced by Ernest Michael in 1963 and it is still unknown if one can be constructed in ZFC. We will introduce the notion of Michael ultrafilter, which implies the existence of a Michael space. We will also discuss the relation between this kind of ultrafilters and some classical cardinal invariants and we will use this to study the behaviour of this notion in some models of set theory”.

Visit our seminar page which may include some future talks at https://www.impan.pl/~set_theory/Seminar/

## Piotr Koszmider: Uncountable constructions from CH using generic filters

Seminar: Working group in applications of set theory, IMPAN

Thursday, 11.10. 2018, 10:15, room 105, IMPAN

Speaker: Piotr Koszmider (IMPAN)

Title: “Uncountable constructions from CH using generic filters”

Abstact. We will present some old CH constructions due to S. Shelah. As usual they use transfinie induction, diagonalization and enumeration of all relevant objects in the first uncountable type. However, the use of the Martin’s axiom type arguments makes them additionally powerful.

Visit our seminar page which may include some future talks at https://www.impan.pl/~set_theory/Seminar/