impan seminar:

working group in applications of set theory

19.11.2015, 1115 room 105
Speaker: Tomasz Kochanek (IMPAN)
Title: A non-separable reflexive Banach space without the Elton-Odell property

Abstract: "We shall discuss a construction of a non-separable, reflexive Banach space X for which there is no e > 0 and an uncountable subset A of the unit ball of X so that ||x - y|| > 1 + e for all distinct x,y from A. In other words, X fails the "uncountable" version of the Elton-Odell property; recall that a non-reflexive example, namely c01), was given by Elton and Odell themselves. The space in question can be described as a "quasi-symmetric version of a long Tsirelon space with weights". A crucial role in the construction is played by certain Todorcevic's pseudo-metric on ω1 which we shall also discuss. The construction moreover shows that the two theorems obtained jointly with T. Kania (discussed during this seminar on October 1) are, in a sense, sharp. "

Previous talks this semester:

05.11.2015, 1115 room 105
Speaker: Eva Pernecka (IMPAN)
Title: Uniformly differentiable mappings from l

Abstract: "We will discuss the rigidity of l and ln with respect to uniformly differentiable mappings. Our main result is a non-linear analogy of the result on rigidity of l with respect to non-weakly compact linear operators by Rosenthal, and it generalises the theorem on non-complementability of c0 in l due to Phillips."

29.10.2015, 1115 room 105
Speaker: Saeed Ghasemi (IMPAN)
Title: The rigidity of isomorphisms between some corona algebras - continuation

15.10.2015, 1115 room 105
Speaker: Saeed Ghasemi (IMPAN)
Title: The rigidity of isomorphisms between some corona algebras - continuation

08.10.2015, 1115 room 105
Speaker: Saeed Ghasemi (IMPAN)
Title: The rigidity of isomorphisms between some corona algebras

Abstract: "The rigidity question was first studied for the automorphisms of Boolean algebras. A groundbreaking result of Shelah shows that it is consistent with the usual axioms of mathematics (ZFC) that all automorphisms of the Boolean algebra P(N)/Fin are trivial, in the sense that they are implemented by almost permutations of the natural numbers N. While assuming the Continuum Hypothesis, a transfinite induction due to Rudin shows that there are many nontrivial automorphisms of P(N)/Fin. Using the Stone duality, the results of this kind can be translated in the category of (zero-dimensional locally compact and Hausdorff) topological spaces. Motivated by a question of Brown-Douglas-Fillmore, the rigidity question was studied in the non-commutative settings for the corona of C*-algebras. It is proved by Philips-Weaver that assuming the Continuum Hypothesis the Calkin algebra has outer automorphisms. On the other hand Farah showed that the Open Coloring Axiom implies that all the automorphisms of the Calkin algebras are inner (implemented by almost unitary elements of B(H)). In my talks we will take a closer look at Farah's result and study the rigidity question for the isomophisms between the corona of Finite-Dimensional Decomposition algebras (reduced products of matrices) and its relevance to Farah's result. We will also look at the later question in more general setting, motivated by the more recent results of Farah-Shelah about the Boolean algebra counterparts."

01.10.2015, 1115 room 105
Speaker: Tomasz Kochanek (IMPAN/UW)
Title: Uncountable sets of unit vectors that are separated by more than 1

Abstract: "Kottman proved that in the unit ball of any infinite-dimensional Banach space one can find an infinite subset such that the distance between any two distinct elements is larger than 1. Elton and Odell, employing Ramsey theory, strengthened this result by showing that it is possible to have all distances at least equal to 1+c for some c>0 depending only on the given space. Both these results are far-reaching generalizations of the Riesz lemma. We will investigate their non-separable versions focusing on two results: (1) the analogue of Kottman's theorem is valid for every non-separable reflexive Banach space (the set from the assertion is then uncountable); (2) the analogue of the Elton-Odell theorem is valid for every non-separable superreflexive Banach space."

24.09.2015, 1115 room 105
Speaker: Antonio Aviles (Universidad de Murcia)
Title: Compact subsets of the first Baire class.

Abstract: "Compact spaces consisting of Baire first class functions in the pointwise topology have been extensively studied with a number of remarkable results by Rosenthal, Bourgain, Fremlin, Talagrand, Todorcevic and others. They play a role in the category of compact spaces that is analogous to that of Borel sets in the class of all subsets of the real line. We shall discuss some history and some recent results about them in collaboration with Stevo Todorcevic."

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 11.15-13.00 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:

  • Saeed Ghasemi (IM PAN)
  • Tomasz Kochanek (IM PAN/UW)
  • Piotr Koszmider (IM PAN)
  • Eva Pernecká (IM PAN)
  • Roman Pol (UW)
  • Adam Skalski (IM PAN)
  • Damian Sobota (Ph. D. student IM PAN)
  • Karen Strung (IM PAN)
  • Michał Świętek (Ph. D. student UJ)
  • Michał Wojciechowski (IM PAN)
  • Przemysław Wojtaszczyk (ICM/IMPAN)
  • Piotr Zakrzewski (UW)
Forthcoming talks (approximately):

  • 14.01 Maciej Malicki, on amenable groups;
  • 21.01 Damian Sobota, Nikodym and Grothendieck properties of von Neumann algebras