impan seminar:

working group in applications of set theory



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

Abstract: "By the theorem of J. J. Parovicenko from 1963, every Boolean algebra of size at most ω1 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 zero-dimensional space of weight at most ω1 is a continuous image of N* (i.e. the remainder of the Cech-Stone compactification of the natural numbers) and it implies that there exists a Banach space isometrically universal for Banach spaces of density ω1. 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 product-forcing construction of a model of ZFC in which there is no universal Boolean algebra of size continuum. "

Previous talks this semester

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 ω1 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 zero-dimensional space of weight at most ω1 is a continuous image of N* (i.e. the remainder of the Cech-Stone compactification of the natural numbers) and it implies that there exists a Banach space isometrically universal for Banach spaces of density ω1. 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 product-forcing 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 "Q-set" 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 corona-algebras which are the noncommutative analogues of the Cech-Stone 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 2-dimensional cardinals"

Continuation from 02.11

Thursday, 02.11. 2017, 10:00, room 105, Piotr Koszmider (IMPAN) "Applications of 2-dimensional 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 2-dimensional von Neumann ordinals. These are quite powerful combinatorial tools allowing to reach the uncountable with the finite (omitting the countable), or to reach ω2 only with countable initial fragments (omitting ω1). 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 set-theoretic 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 well-founded relations".

Thursday, 26.10. 2017, 10:00, room 105, Tomasz Kochanek (IMPAN/MIMUW) "Three-space 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 three-space 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- 9-13.10, 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 self-adjoint 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 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.

Time and place: Thursdays 10.00-11.45 am, room 105, Sniadeckich 8


The scope of the seminar: Set-theoretic combinatorial and topological methods in diverse fields of mathematics, with a special emphasis on abstract analysis like Banach spaces, Banach algebras, C*-algebras, Here we include both the developing of such methods as forcing, descriptive set theory, Ramsey theory as well as their concrete applications in the fields mentioned above.

Working group style: We will make efforts so that this seminar has more a working character rather than the presentation style. This means that we encourage long digressions, discussions, background preparations and participation of everyone. We would like to immerse ourselves into the details of the mathematical arguments studied. Also the talks are usualy devoted to research in progress or fascinating results leading to some project not yet resolved. While ready final results could be presented at other seminars at IM PAN or UW.

Participants this semester so far:

  • Tristan Bice (IM PAN/WCNM)
  • Michał Tomasz Godziszewski (Ph. D. student IMPAN)
  • Yurii Khomskii (University of Hamburg)
  • Tomasz Kochanek (IM PAN/UW)
  • Piotr Koszmider (IM PAN)
  • Eric Reckwerdt (IM PAN)
Forthcoming talks :

  • No more seminars in 2017, the meetings will resume in January 2018
  • 11.01.2018 Tomasz Kochanek on rich families
  • 18.01.2018 Tristan Bice on the Stone duality and grupoids
  • 25.01.2018 Scientific Council of IMPAN - no seminar
  • 08.02.2018 Alessandro Vignati