Previous talks this semester:
Thursday, 08.02. 2018, 10:00, room 105, Alessandro Vignati (Paris 7)
"Triviality and nontriviality of homeomorphisms of CechStone remainders "
Abstract: "Given a locally compact space X, we define a homeomorphism of its
CechStone remainder βXX to be trivial if it is induced by an homeomorphism between
cocompact subsets of X. For example, if X=ω, trivial homeomorphisms correspond to almost permutations on ω.
Are all homeomorphisms of βXX of this form? Rudin, Shelah, and Velickovic showed that in case X=ω the answer to this question depends on set theory.
We report on recent successful attempts to extend this intuition to more general locally compact spaces. If time permits, we will discuss the noncommutative situation".
Thursday, 11.01. 2018, 10:00, room 105, Tomasz Kochanek (IMPAN/UW) "On noncommutative variants of richness"
Abstract: "In 1953, A. Grothendieck proved that for every compact Hausdorff space K,
the Banach space C(K) has Pelczynski's property (V) which implies that its dual is weakly sequentially complete.
His result was considerably generalized in 1983 by J. Bourgain who introduced the
socalled rich subspaces of C(K,E)spaces (here, E can be any finite dimensional Banach space)
and proved that any such subspace satisfies property (V) as well.
This notion of richness turns out to be rich enough to cover many important cases,
like the polydisc algebras or the spaces of ntimes continuously differentiable functions
on finite dimensional tori.
The goal of my talk is to discuss possible noncommutative versions of richness
which would hopefully lead to some new results concerning property (V) for operator algebras.
It should be mentioned that the core of this issue is the classical result of H. Pfitzner (1994)
which says that weak compactness in duals of C*algebras is determined commutatively and hence every C*algebra satisfies (V)."
Thursday, 07.12. 2017, 10:00, room 105, Michał Tomasz Godziszewski (Ph. D. student IMPAN)
"On the existence of universal Boolean algebras of cardinality continuum"
Continuation from 30.11
Thursday, 30.11. 2017, 10:00, room 105, Michał Tomasz Godziszewski (Ph. D. student IMPAN) "On the existence of universal Boolean algebras of cardinality continuum"
Abstract: "By the theorem of J. J. Parovicenko from 1963, every Boolean algebra of size at most
ω_{1} embeds into the algebra P(N)/fin.
The result has some nice applications in topology and analysis,
since by Stone's duality it is equivalent to the statement that each compact and zerodimensional
space of weight at most ω_{1} is a continuous image of N* (i.e. the remainder of the CechStone compactification of the natural numbers)
and it implies that there exists a Banach space isometrically universal for Banach spaces of density ω_{1}.
It thus follows that under CH the algebra P(N)/fin is universal for Boolean algebras of cardinality at most continuum
(and that each compact space of weight at most continuum is a continuous
image of N* and that C(K), where K is the Stone space of P(N)/fin, is isometrically universal for Banach spaces of density continuum).
In 2000, A. Dow and K. P. Hart demonstrated that not only the universality of P(N)/fin,
but even the existence of universal Boolean algebra of cardinality continuum is actually independent from ZFC.
During the talk we will give a proof of Parovicenko's theorem and
show the consistency result of Dow and Hart via a productforcing construction
of a model of ZFC in which there is no universal Boolean algebra of size continuum. "
Thursday, 23.11. 2017, 10:00, room 105, Piotr Koszmider (IMPAN) "Constructions of various almost disjoint families in P(N)"
Abstract: "We will present classical constructions of diverse almost disjoint families in P(N)
as the Luzin family, the Mrowka family, the "Qset" family. These famlies form combinatorial foundations
of many interesting constructions in topology (e.g., the Stone spaces of Boolean algebras
generated by almost disjoint families), Banach spaces (e.g., spaces of continuous functions
on such Stone spaces) or C*algebras (e.g., subalgebras of continuous functions on such Stone spaces into matrices)."
Thursday, 16.11. 2017, 10:00, room 105, Tatiana Shulman (IMPAN) "On some lifting problems in C*algebras"
Abstract: "We will start with discussing general lifting problems in C*algebras
and a notion of projective C*algebras introduced by Blackadar. Then I will talk
about question of Loring and Pedersen on lifting of nilpotent contractions. This
is my old work but I chose
to speak about it because it involves multipliers and coronaalgebras
which are the noncommutative analogues of the CechStone compactifications and
their remainders and thus
fits to the topic of the seminar".
Thursday, 09.11. 2017, 10:00, room 105, Piotr Koszmider (IMPAN) "Applications of 2dimensional cardinals"
Continuation from 02.11
Thursday, 02.11. 2017, 10:00, room 105, Piotr Koszmider (IMPAN) "Applications of 2dimensional cardinals"
Abstract: "
The goal of the talk is to present the main ideas of simplified morasses in the language
of Velleman in which they can be considered as 2dimensional von Neumann ordinals. These are
quite powerful combinatorial tools allowing to reach the uncountable with the finite (omitting the countable),
or to reach ω_{2} only with countable initial fragments (omitting ω_{1}).
After developing basic machinery we plan to show
combinatorial applications (squares, trees, Hausdorff gaps, colorings) and topological application (nonreflecting nonmetrizable spaces),
we may also mention how to construct interesting Banach spaces or other structures
using these settheoretic tools.
The talk is
related to a recent survey paper in Arch. Math. Logic
but will deal with simpler applications. In particular we assume from the audience only the knowledge of
von Neumann ordinals with the order topology on them, transfinite induction and wellfounded relations".
Thursday, 26.10. 2017, 10:00, room 105, Tomasz Kochanek (IMPAN/MIMUW) "Threespace properties of asymptotic ideal seminorms"
Abstract: "We introduce two "flavors" of seminorms of Banach spaces
(or, more generally, operators acting on them), each having both a type and a cotype version,
and each of these having both a weak and a weak* version, resulting in eight families of seminorms.
The intuition behind these new quantities is to define asymptotic versions of
the classical Rademacher and Pisier (martingale) types and cotypes.
We will describe our seminorms in terms of asymptotic structures,
discuss some ideal and duality properties. As an application of this theory,
we will prove that having Szlenk power type at most p is a threespace property.
This strengthens and, in fact, gives a sharp version of an earlier
result by Brooker and Lancien (J. Math. Anal. Appl. 2013) who proved
that any twisted sum of Banach spaces with Szlenk power types p and q has power type at most pq.
The talk is based on a joint work with R.M. Causey and S. Draga.".
Thursday, 19.10. 2017, no seminar due to Scientific Council of IM PAN
Thursday, 12.10. 2017, no seminar due to 14th International Workshop in Set Theory 913.10, Luminy
Thursday, 05.10. 2017, 10:00, room 105, Tristan Bice (IMPAN) "Monotone Complete C*algebras"
Abstract: "Based on a recent book by Saito and Wright, we give an
introduction to the theory of C*algebras whose positive unit ball is directed complete,
i.e. every bounded increasing net of selfadjoint elements has a least upper bound.
Von Neumann algebras are the most famous examples but there are many more,
large classes of which can be distinguished by a simple classification semilattice defined by certain linear maps.
The general theory is also more algebraic/order theoretic,
as there is generally no close analog of the weak or strong topology".
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.
