Previous talks this semester:
26.01.2023, Thursday 10.15, room 403
Piotr Koszmider (IMPAN)
Title: Colorings of uncountable squares, strong Luzin sets and partial orders.
Abstract: "We will present classical multidimensional transfinite inductive arguments
leading under the continnum hypothesis to a strong Luzin set (Todorcevic) and to a c.c.c. partial order whose square
is not c.c.c. (Galvin). Both of these constructions are related to some colorings of pairs
of countable ordinals which found applications in algebra, analysis and topology (e.g., see Wednesday seminar this week).
If time permits, we will discuss what happens after adding one or many Cohen reals."
19.01.2023, Thursday 10.15, room 1 (ground floor) ROOM CHANGE
Kamil Ryduchowski (UW/IMPAN)
Title: On large free sets and Lp(μ) spaces of large density
Abstract: "The aim of the talk is to present a proof of the Enflo-Rosenthal theorem stating
that Banach spaces of the form Lp(μ) (where p∈ (1,+∞), p≠2
and μ is a finite measure) of the density dens(Lp(μ))≥ ℵω
don't have unconditional bases. A combinatioral lemma similar to the
Kuratowski's free set theorem is a crucial step in the proof.
During the talk we will prove both of the aforementioned combinatorial results and proceed
to the proof of the Enflo-Rosenthal theorem.
The problem whether the assumption on the density of the space
can be weakened to dens(Lp(μ))≥ ω1
seems to be still open."
15.12.2022, Thursday 10.15, room 1 (ground floor) ROOM CHANGE
Damian Głodkowski (UW/IMPAN)
A characterization of weak compactness in spaces of Radon measures.
Abstract: "We will show a proof of the Dieudonne-Grothendieck theorem, which characterize
relatively weakly compact subsets of Banach spaces of measures. The proof will be based on the
Kadec-Pełczyński-Rosenthal subsequence splitting lemma,
which says that every bounded sequence of measures admits a subsequence that may be expressed as the sum of two sequences:
one of them consists of measures with pairwise disjoint supports and constant norm, and the second one is weakly convergent."
8.12.2022, Thursday 10.15, room 403
Piotr Koszmider (IMPAN)
Remarks on the Efimov problem, continuation.
Abstract: "We will continue the talk of 3.11.
We will present a consistent example of an Efimov space. This will be the Stone space of P(N)∩V
after adding Sacks reals. We will also see that on the other hand the Stone space of P(N)∩V
has nontrivial convergent sequences if there are Cohen reals over V.
1.12.2022, Thursday 10.15, room 403
Piotr Koszmider (IMPAN)
Topologies and subspaces of the space of measures, continuation.
Abstract: "We will continue the talk of 10.11.
We will show two theorems left from 10.11
that C(βN) has the Grothendieck property (this strengthens the nonexistence of nontrivial
convergent sequences in βN) and the conditions equivalent to the Grothendieck property
for C(K) spaces. This naturally leads to considering the weak compactness
in the space of measures and its characterizations.
24.11.2022, Thursday 10.15, room 403
Zdeněk Silber (IM PAN)
On quotients of Grothendieck C(K) spaces, continuation.
Abstract: "We continue the previous two talks and focus on C(K) spaces which satisfy the Grothendieck property.
In the beginning we recall some notions related to sequential convergence in Banach spaces, namely
weakly Cauchy sequences, and convex block subsequences (which can be understood as a generalization
of the notion of a subsequence). Finally, we will use these notions to prove that consistently every
non-reflexive Grothendieck Banach space contains l1(2ω), and thus admits l∞
as a quotient. This,
together with the result that non-Grothendieck C(K) spaces contain a complemented copy of c0, will
give us a dichotomy result about C(K) spaces."
17.11.2022, Thursday 10.15, room 403
Zdeněk Silber (IM PAN)
On quotients of Grothendieck C(K) spaces.
Abstract: "We continue the previous talk and focus on C(K) spaces which satisfy the Grothendieck property.
In the beginning we recall some notions related to sequential convergence in Banach spaces, namely
weakly Cauchy sequences, and convex block subsequences (which can be understood as a generalization
of the notion of a subsequence). Finally, we will use these notions to prove that consistently every
non-reflexive Grothendieck Banach space contains l1(2ω), and thus admits l∞
as a quotient. This,
together with the result that non-Grothendieck C(K) spaces contain a complemented copy of c0, will
give us a dichotomy result about C(K) spaces."
10.11.2022, Thursday 10.15, room 403
Piotr Koszmider (IMPAN)
Topologies and subspaces of the space of measures.
Abstract: "We will discuss the usefullness of considering the weak topology
in the dual of a Banach space, in particular for a space of the form C(KA), where KA
is the Stone space of a Boolean algebra A and of comparing it with the weak* topology. This naturally leads to considering the weak compactness
and to introducing the Grothendieck property. We will also discuss some aspects of the interplay
between compact spaces (or Boolean algebras) and the Banach spaces of continuous function defined on them.
The talk should motivate some notions which will appear during the following talk by Zdeněk Silber
on some dichotomy under Martin's axiom."
3.11.2022, Thursday 10.15, room 403
Piotr Koszmider (IMPAN)
Remarks on the Efimov problem.
Abstract: "The following is
known as the Efimov problem: is there an infinite compact Hausdorff space which does not contain
a nontrivial convergent sequence and does not contain a copy of βN? Such a space is called an Efimov space. Up till now there are
many consistent examples of Efimov spaces but no ZFC example has been constructed and no one proved the consistency of the nonexistence
of them. In this talk we give a simple (but not elementary)
construction of an Efimov space using Sacks forcing and show some interesting sufficient conditions for a compact space to contain a copy of βN
(based on the paper P. Koszmider, S. Shelah;
Independent families in Boolean algebras with some separation properties. Algebra Universalis 2013)
The problem has a translation to Boolean algebras: "Is there an infinite Boolean algebra without an independent family of cardinality continuum
and without an infinite countable Boolean homomorphic image?". The Banach spaces version: "Is there an infinite dimensional
nonreflexive Banach space
whose quotients do not include spaces isomorphic to c0 nor l∞"
has been established as an undecidable problem (Talagrand, Haydon-Levy-Odell).
We plan to discuss these results."
20.10.2022, Thursday 10.15, room 403
Damian Głodkowski (UW/IMPAN)
Sums of Fredholm operators and strictly singular operators, continuation.
Abstract: "Let X and Y be Banach spaces. We say that a bounded linear operator T: X ---> Y is Fredholm, if ker T and Y/TX are both finite-dimensional as vector spaces, and we define the index of T as i(T)=dim(ker T)- dim(Y/TX). I will show a classical result that if T: X ---> Y is Fredholm and S: X ---> Y is strictly singular (i.e. S is not an isomorphism when restricted to any infinite-dimensional subspace), then T+S is also Fredholm, and i(T+S)=i(T). We will illustrate the talk with natural examples concerning Banach spaces of the form C(K), where K is compact and Hausdorff. Some applications will be discussed as well."
13.10.2022, Thursday 10.15, room 403
Damian Głodkowski (UW/IMPAN)
Sums of Fredholm operators and strictly singular operators.
Abstract: "Let X and Y be Banach spaces. We say that a bounded linear operator T: X ---> Y is Fredholm, if ker T and Y/TX are both finite-dimensional as vector spaces, and we define the index of T as i(T)=dim(ker T)- dim(Y/TX). I will show a classical result that if T: X ---> Y is Fredholm and S: X ---> Y is strictly singular (i.e. S is not an isomorphism when restricted to any infinite-dimensional subspace), then T+S is also Fredholm, and i(T+S)=i(T). We will illustrate the talk with natural examples concerning Banach spaces of the form C(K), where K is compact and Hausdorff. Some applications will be discussed as well."
Talks in the second semester of 2021-22.
Talks in the first semester of 2021-22.
Talks in the second semester of 2020-21.
Talks in the first semester of 2020-21.
Talks in the second semester of 2019-20.
Talks in the first semester of 2019-20.
Talks in the second semester of 2018-19.
Talks in the first semester of 2018-19.
Talks in the second semester of 2017-18.
Talks in the first semester of 2017-18.
Talks in the second semester of 2016-17.
Talks in the first semester of 2016-17.
Talks in the second semester of 2015-16.
Talks in the first semester of 2015-16.
Talks in the second semester of 2014-15.
Talks in the first semester of 2014-15.
Talks in the second semester of 2013-14.
Talks in the first semester of 2013-14.
Talks in the second semester of 2012-13.
Talks in the first semester of 2012-13.
Talks in the second semester of 2011-12.
Talks in the first semester of 2011-12.
|
