Archives of: IMPAN working group in applications of set theory

Alessandro Vignati: Uniform Roe coronas

Seminar: Working group in applications of set theory, IMPAN

(joint meeting with the Geometric Group Theory Seminar, IMPAN)

Thursday, 7.02.2019, 10:15, room 403, IMPAN

NOTE ROOM CHANGE TO 403

Speaker: Alessandro Vignati (KU Leuven)

Title: “Uniform Roe coronas”

Abstact: “Given a metric space (X,d), one defines a subalgebra of the space of operators on l_2(X) called the uniform Roe algebra of (X,d), denoted C_u*(X). This is the closure of the algebra of finite-propagation operators. The study of these algebras comes from the fact that C_u*(X) catches algebraically some of the large scale geometrical properties of X. Uniform Roe algebras have therefore an intrinsic relation with coarse geometry and the coarse Baum-Connes conjecture.

In recent year, much work was dedicated to show which geometric properties are preserved by isomorphisms of Uniform Roe algebras. Namely, if C_u*(X) and C_u*(Y) are isomorphic, how much do X and Y look alike? We pose the same question for Uniform Roe corona algebras.

Since Cu*(X) contains all compact operators, we can define the natural quotient Q_u*(X)=C_u*(X)/K(l_2(X)), the Uniform Roe corona algebra of X. Which geometric properties do the spaces X and Y share, when an isomorphism between Q_u*(X) and Q_u*(Y) is given? For example, must X and Y be coarsely equivalent, or even bijectively coarsely equivalent? (Two spaces are coarsely equivalent if “they look the same when the observer is far from them”).

We answer these questions with the aid of some set theory, in particular of Forcing Axioms. Forcing Axioms are generalizations of the Baire category theorem. They are alternative to the Continuum Hypothesis, and they’re at the base of many rigidity phenomena observed in the theory of quotients (both discrete such as Boolean algebra quotient, and continuous, as the Calkin algebra or corona C*-algebras). The talk starts with introducing the objects in play. The goal is to state the main results, and at least sketch the salient points of their proofs. We conclude with a list of open questions. This is joint work with Bruno Braga and Ilijas Farah”.

This is the last meeting during the first semester of 2018/19.

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

Arturo Martínez-Celis: Porous sets on the Cantor set

Seminar: Working group in applications of set theory, IMPAN

Thursday, 31.01.2019, 10:15, room 105, IMPAN

Speaker: Arturo Martínez-Celis (IM PAN)

Title: “Porous sets on the Cantor set”

Abstact: “Given a completely metrizable space X, a subset A of X is a strongly porous set if there is a positive constant p such that for any open ball B of radius r smaller than 1, there is an open ball B’ inside of B of radius rp such that B’ evades the set A. We will study the cardinal invariants related to the σ-ideal generated by strongly porous sets on the Cantor space and its relation with other known σ-ideals of the real line. We will also uncover a deep connection between the σ-ideal of the strongly porous sets and some instances of the Martin’s Axiom. “.

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

Tomasz Weiss: Accessible points, harmonic measure and the Riemann mapping

Seminar: Working group in applications of set theory, IMPAN

Thursday, 17.01.2019, 10:15, room 105, IMPAN

Speaker: Tomasz Weiss (UKSW)

Title: “Accessible points, harmonic measure and the Riemann mapping”

Abstact: “Let D be a bounded domain in R_n, n larger than 1. We provide an elementary proof that the set of all boundary accessible points of D is an analytic set. We investigate the nature of the set of accessible points of D when n=2 using only set theoretical methods. We provide also a view of the relation between harmonic measure in D, if n=2, D simply connected, and the Riemann mapping of D. In this talk we prove new results and give easier proofs of known results”.

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

Tomasz Kochanek: Rosenthal’s lemma and its applications

Seminar: Working group in applications of set theory, IMPAN

Thursday, 10.01.2019, 10:15, room 105, IMPAN

Speaker: Tomasz Kochanek (IMPAN/UW)

Title: “Rosenthal’s lemma and its applications ”

Abstact: “In this instructional talk we will recall Rosenthal’s lemma on uniformly bounded sequences of measures and present its several classical applications in the Banach space and vector measures theory. First, we will prove the surprising Nikodym’s uniform boundedness principle and Phillips’ lemma where the application of Rosenthal’s result makes the proofs much easier than the original ones. A few further corollaries of Nikodym’s principle will be mentioned, such as the Dieudonné-Grothendieck theorem on bounded vector measures and the Seever theorem on the range of an operator into a B(Σ)-space. Next, we shall prove two beautiful consequences of Rosenthal’s lemma: the Diestel-Faires theorem and the Orlicz-Pettis theorem. If time allows, we will also briefly discuss their further deep consequences in the structural theory of Banach spaces”.

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

Damian Sobota: The Josefson–Nissenzweig theorem for Cp(X)-spaces

Seminar: Working group in applications of set theory, IMPAN

Thursday, 20.12.2018, 10:15, room 105, IMPAN

Speaker: Damian Sobota (Kurt Godel RC, Vienna)

Title: “The Josefson–Nissenzweig theorem for Cp(X)-spaces ”

Abstact: “The famous Josefson–Nissenzweig theorem asserts that for every infinite-dimensional Banach space X there exists a sequence (x_n*) in the dual space X* which is weak* convergent to 0 and each x_n* has norm 1. Despite the apparent simplicity of the theorem no constructive proof — even in the case of Banach spaces of continuous functions on compact spaces — has been known.
Recently, Banakh, Śliwa and Kąkol in their studies of separable quotients of topological vector spaces of the form Cp(X), i.e. spaces of continuous functions on Tychonoff spaces endowed with the pointwise convergence topology, have obtained several results characterizing those Cp(X)-spaces for which the Josefson–Nissenzweig theorem holds.
During my talk I will present some introductory facts concerning the theorem for Cp(X)-spaces, show that the existence of “Josefson–Nissenzweig” sequences for Cp(K)-spaces, where K is compact Hausdorff, is strongly related to a variant of the Grothendieck property of Banach spaces, as well as prove that every compact space obtained as a limit of an inverse system consisting only of minimal extensions admits such sequences (and the proof is constructive). This is a joint work with Lyubomyr Zdomskyy”.

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

Piotr Koszmider: Controlling linear operators on C(K)s through the rigidity of K

Seminar: Working group in applications of set theory, IMPAN

Thursday, 13.12.2018, 10:15, room 105, IMPAN

Speaker: Piotr Koszmider (IM PAN)

Title: “Controlling linear operators on C(K)s through the rigidity of K”

Abstact: ” In the second talk of the series devoted to classical phenomena in Banach spaces of the form C(K) we will see how linear operators on a C(K) can be represented by continuous maps from K into the space of the Radon measures on K with the weak* topology. Previously presented results concerning weakly compact subsets in M(K) will allow us to obtain a “geometric” understading of the rigidity conditions on the algebra of all linear operators (having few operators modulo weakly compact operators) as versions of topological or Boolean rigidity conditions (having few continuous maps or few Boolean endomorphisms) which can be combinatorially imposed on K.”.

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

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/