Previous talks this semester:
10.01.2019. 10.15, room 105;
Tomasz Kochanek
(IMPAN/UW)
Rosenthal's lemma and its applications
Abstract:
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 DieudonnéGrothendieck 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 DiestelFaires theorem and the OrliczPettis theorem. If time allows, we will also briefly discuss their further deep consequences in the structural theory of Banach spaces.
20.12.2018. 10.15, room 105;
Damian Sobota
(Kurt Godel RC, Vienna)
The JosefsonNissenzweig theorem for C_{p}(X)spaces
Abstract:
The famous JosefsonNissenzweig theorem asserts that for every
infinitedimensional 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 C_{p}(X), i.e. spaces
of continuous functions on Tychonoff spaces endowed with the pointwise
convergence topology, have obtained several results characterizing
those C_{p}(X)spaces for which the JosefsonNissenzweig theorem holds.
During my talk I will present some introductory facts concerning the
theorem for C_{p}(X)spaces, show that the existence of
"JosefsonNissenzweig" sequences for C_{p}(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.
13.12.2018. 10.15, room 105;
Piotr Koszmider
(IM PAN)
Controlling linear operators on C(K)s through the rigidity of K.
Abstract:
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 imposed on K.
29.11.2018. 10.15, room 105;
Piotr Koszmider
(IM PAN)
The Grothendieck property for Banach spaces of continuous functions
Abstract:
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 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 GrothendieckDieudonne 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 settheoreic issues. The purpose of this series of talks is to introduce
particpants with the settheoretic 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;
22.11.2018. 10.15, room 105;
Tomasz Kochanek
(IM PAN / MIM UW)
Bases of Banach spaces with respect to filters.
Abstract:
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 Fbasis, 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 wellknown 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.
15.11.2018. 10.15, room 105;
Fulgencio Lopez
(IM PAN)
A capturing construction scheme from the diamond principle. Continuation from 8.11.18.
08.11.2018. 10.15, room 105;
Fulgencio Lopez
(IM PAN)
A capturing construction scheme from the diamond principle.
Abstract:
S. Todorcevic introduced the concept of a capturing construction scheme
and showed it is consistent with the diamond principle. A construction scheme is a wellfounded
family of finite subsets of ω_{1}. We give a quick presentation of
the history and motivation for this tool and show that it follows from the diamond principle.
25.10.2018. 10.15, room 105;
Arturo MartínezCelis (IM PAN)
On the Michael Space Problem
Abstract:
A Lindelöf Topological space is Michael if it has nonLindelöf 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.
11.10.2018. 10.15, room 105;
Piotr Koszmider (IMPAN)
Uncountable constructions from CH using generic filters
Abstract:
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 type ω_{1}. However, the use of the Martin's axiom type arguments
makes them additionally powerful.
Talks in the second semester of 201718.
Talks in the first semester of 201718.
Talks in the second semester of 201617.
Talks in the first semester of 201617.
Talks in the second semester of 201516.
Talks in the first semester of 201516.
Talks in the second semester of 201415.
Talks in the first semester of 201415.
Talks in the second semester of 201314.
Talks in the first semester of 201314.
Talks in the second semester of 201213.
Talks in the first semester of 201213.
Talks in the second semester of 201112.
Talks in the first semester of 201112.
