impan seminar:

working group in applications of set theory

29.01.2015, Omar Selim (USP) - 1115 room 105:

Title: An abstract version of Choquet's theorem
Abstract: I will discuss an abstract version of Choquet’s theorem concerning the existence of Borel measures that are realised by capacities of infinite order. The version stated here concerns only Boolean algebras and the classical result concerning particular locally compact topological spaces follows. I will also discuss some possible applications.

Previous talks this semester:

November 27th, Grzegorz Plebanek (Wrocław) - 1115 room 105:

Title: Weak* separability in C(K)*
Abstract: The plan is to recall some older results on weak* separability in the spaces of the form C(K)* as well as a recent result due to Aviles, Rodriguez and myself: There is a compact space K such that the space C(K)* is weak* separable but its unit ball is not. (Talagrand's construction of such a space K required CH, ours is in ZFC). Finally, we shall discuss a related open problem on the Baire measurability of the ball in C(K).

November 20th, Leandro Candido (IMPAN/USP) - 1115 room 105:

Some Banach spaces satisfying a pointwise chain condition in the weak topology
Abstract: The talk is about a ZFC example by Dow, H. Junilla and J. Pelant of a Banach space C(K), K scattered of countable height, such that every linear bounded operator from C(K) into c01) has separable range. These are examples 2.15 and 2.16 of Dow, A.; Junnila, H.; Pelant, J. Chain conditions and weak topologies. Topology Appl. 156 (2009), no. 7, 1327-1344.
This should be compared with a result of Todorcevic which says that under Martin's Maximum every Banach space of density ω1 admits a bounded linear operator into c01) with nonseparable range. This is subsequently used by Todorcevic to prove that MM implies that every nonseparable Banach space has an uncountable biorthogonal system and that every Banach space of density ω1 has a separable quotient.

November 13th, Michal Doucha (IMPAN/IMPACT) - 1115 room 105:

Introduction to Lipschitz-free Banach spaces
Abstract: Lipschitz-free spaces are Banach spaces constructed from metric spaces which are characterized by the universal property that Lipschitz maps from the starting metric space into some Banach space uniquely extend into linear operators with the same Lipschitz constant. Their norm is a variant of the Kantorovich transportation distance which is a concept that penetrated many areas of mathematics. Alternatively, one can view them as certain preduals of Banach spaces of real-valued Lipschitz functions on metric spaces. I will present both of these views and show their equivalence. Then I will proceed to basic properties of these spaces and show some examples. Finally, I will focus on problems that people have been recently studying on these spaces, namely approximation properties of these spaces. That is related to the problem of linear extension of Lipschitz maps on metric spaces.

November 6th, Michał Świętek (Ph. D. student UJ/WCMCS-IM PAN) - 1115 room 105:

Continuation of the talk of 23.10.2014.

October 30th - Marek Cuth (WCMCS-IM PAN) - 1115 room 105:

Elementary submodels in Banach space theory
Abstract: The method of elementary submodels can be viewed as a tool of set theory which enables us to handle very complicated inductive constructions. I will present how elementary submodels can be used in Banach space theory. More precisely: in 1993 Argyros and Mercourakis introduced a class of Banach spaces called weakly Lindelof determined (WLD). It appeared that this class of WLD Banach spaces has many nice properties and it is possible to find nice characterizations of it. I will present how elementary submodels can be used in order to prove some of those characterizations and prove a characterization of WLD spaces in terms of elementary submodels.

October 23rd, Michał Świętek (Ph. D. student UJ/WCMCS-IM PAN) - 1115 room 105:

Exotic Banach spaces via Boolean algebras and Stone spaces
Abstract: One of the oldest questions in the theory of Banach spaces was whether every Banach space is isomorphic to its hyperplanes. This was answered negatively by Gowers and Mauray. During my talk I will present a construction of a classical C(K) space, based on the paper Piotr Koszmider, Banach spaces of continuous functions with few operators. Math. Ann. 330 (2004), no. 1, 151–183, which also answers the above question negatively.

October 16th, Gabriel Salazar (IMPAN) - 1115 room 105:

Some Applications of Shelah's Black Box
Abstract:Shelah’s Black Box is a combinatorial principle that allows us to partially predict a given map under specific cardinal conditions. Very roughly speaking, if you can get a result using Jensen’s Diamond Principle then you can get a weak version of it in ZFC using the Black Box. In this talk I will present some algebraic constructions realized by means of this principle, focusing in the combinatorial aspect behind them.

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:

  • Leandro Candido (IM PAN/USP)
  • Marek Cuth (WCMCS-IM PAN)
  • Michal Doucha (IMPACT/IM PAN)
  • Tomasz Kochanek (IM PAN/UW)
  • Piotr Koszmider (IM PAN)
  • Maciej Malicki (SGH)
  • Samuel Roth (Purdue/WCMCS-IM PAN)
  • Gabriel Salazar (IM PAN)
  • Damian Sobota (Ph. D. student IM PAN)
  • Michał Świętek (Ph. D. student UJ/WCMCS-IM PAN)
  • Michał Wojciechowski (IM PAN)
Forthcoming talks:

  • No more talks this semester