impan seminar:

working group in applications of set theory

No more seminars this academic year. The talks will start in October 2018

Previous talks this semester:

Thursday, 07.06. 2018, 10:15, room 105, Michał Tomasz Godziszewski (Ph. D. student IMPAN) "The rearrangement number"

Abstract: "The Riemann Series Theorem says that for any conditionally convergnet series Σn ∈ Nan of real numbers and for any number S in the extended real line there exists a permutation σ of N such that the series Σn ∈ N aσ(n) converges to S or diverges by oscillation. A natural set-theoretic question is: what is the minimal cardinality of a family C of permuations of natural numbers such that for any conditionally convergent series Σn ∈ Nan of real numbers there is a permutation σ in C such that the series Σn ∈ N aσ(n) converges to a different real number, diverges to infinity or diverges by oscillation? Such a minimal cardinality is denoted by the rearrangement number rr. The question has been investigated in a recent paper by A. Blass, J. Brendle, W. Brian, J. Hamkins, M. Hardy, P. Larson on "The Rearrangement Number". During the talk, we will demonstrate some results from the paper, in particular comparing (in ZFC) the number rr and its variants to some well-known cardinal characteristics of the continuum, as e.g. the bounding number b, the dominating number d, as well as the covering and uniformity for measure and category, i.e. cov(N), cov(M), non(N), non(M). If the time permits, we will also give some consistency results concerning the rearrangement numbers and show open questions in the subject. "

Thursday, 17.05. 2018, 10:15, room 105, Piotr Koszmider (IMPAN) "Uncountable irredundant sets in large algebras"

Abstract: "A subset X of a structure A is irredundant if no x from X belongs to the substructure generated by the remaining elements of X. For example irredundant subsets of Banach spaces correspond to biorthogonal systems, and it is known that the existence of uncountable biorthogonal systems in nonseparable Banach spaces is undecidable. Similar results hold for Boolean algebras. In my talk I will focus on irredundant sets in C*-algebras. Even in the commutative case of C(K) it is not known if there is an absolute example of a nonmetrizable compact K such that C(K) has no uncountable irredundant set. But the totally disconnected case behaves like the Banach space case. I will start with the review of the commutative case where most of the results are due to Todorcevic and will touch the noncommutative case where partial results were obtained together with my Ph.d. student from São Paulo, Clayton Suguio Hida"

Thursday, 10.05. 2018, 10:15, room 105, Tomasz Kochanek (IMPAN/UW) "On the property (X) of Banach spaces"

Abstract: "The property (X) was introduced by Godefroy and Talagrand in 1981 in order to study the problem of uniqueness of preduals. It is a relatively simply formulated condition for veryfing whether a given element of the bidual space actually belongs to the original space, and which implies that the underlying Banach space is the strongly unique predual of its dual. We will discuss some reformulations of this property (for example in terms of a certain Edgar's ordering between Banach spaces) and its connections with the better known Pelczynski's properties (V) and (V*). We shall focus, in particular, on the still open problem posed by Godefroy and Lerner whether the Lipschitz-free spaces over finite dimensional cubes have property (X). Some possible approaches will be discussed, based on a joint paper with E. Pernecka, where it was shown that these Lipschitz-free spaces have a somewhat weaker property (V*)."

Thursday, 12.04. 2018, 10:15, room 105, Piotr Koszmider (IMPAN) "Applications of Todorcevic's construction schemes"

Abstract: "Recently Todorcevic (S. Todorcevic, A construction scheme for non-separable structures. Adv. Math. 313 (2017), 564-589.) proposed combinatorial structures whose existence follows from the diamond principle. These ideas can be traced back to the previous attempts of Magidor and Malitz (1977), Shelah (1985) and Velleman (1984)), but construction schemes seems to a more elegant framework providing quick constructions of complicated objects ranging from combinatorics to functional analysis. I will show at least a construction of a Souslin tree from such a scheme."

Thursday, 05.04. 2018, 10:15, room 105, Damian Sobota (TU Wien) "Grothendieck C(K)-spaces of small density"

Abstract: "A Banach space X is Grothendieck if every weak* convergent sequence of bounded functionals on X is weakly convergent. A typical example of a Grothendieck space is the space l or more generally every Banach space C(K) of real-valued continuous functions on the Stone space K of a σ-complete Boolean algebra. During my talk I will investigate relations between densities of the Grothendieck spaces of the form C(K) for K being the Stone space of a non-σ-complete Boolean algebra and classical cardinal characteristics of the continuum."

Thursday, 22.03. 2018, 10:15, room 105, Tomasz Kochanek (IMPAN/UW) "A short proof of a theorem of Pfitzner"

Abstract: "In 1953 Grothendieck proved a classical result on characterization of weakly compact subsets of duals of C(K)-spaces. His criterion of (non-)weak compactness defines the so-called Pelczynski property (V). A noncommutative analogue was provided by Pfitzner (Math. Ann. 1994) who showed that all C*-algebras have that property (V), which can be also expressed by saying that weak compactness in duals of C*-algebras is determined commutatively. During the talk, I shall present a more elementary proof of this result due to Fernández-Polo and Peralta (Quart. J. Math. 2009).

Thursday, 01.03. 2018, 10:15, room 105, Wieslaw Kubis (Czech Academy and UKSW) "On the weak amalgamation property "

Abstract: "The somewhat technical weak amalgamation property (weak AP) of models with embeddings was identified first by Ivanov in 1999 during his study of generic automorphisms, later re-discovered by Kechris and Rosendal (2007), again studying generic automorphisms. Recently, we have discovered, jointly with Adam Krawczyk, that the weak amalgamation property plays essential role in characterizing the existence of a winning strategy in the abstract Banach-Mazur game, played with models and embeddings. We shall describe the main results relating this game with the weak AP. Next, we shall present some new examples of hereditary classes of finite structures with the weak AP, failing its stronger version called the cofinal AP. The material comes from joint works with A. Krawczyk, A. Kruckman, and A. Panagiotopoulos."

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.

Time and place: Thursdays 10.15-12.00 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:

  • Tristan Bice (IM PAN)
  • Michał Tomasz Godziszewski (Ph. D. student IMPAN)
  • Tomasz Kochanek (IM PAN/UW)
  • Piotr Koszmider (IM PAN)
  • Marek Miarka (UKSW)
  • Witold Marciszewski (MIMUW)
  • Jarosław Swaczyna (Lodz UT)
  • Piotr Zakrzewski (MIMUW)
Forthcoming talks :

  • No more seminars this academic year. The talks will start in October 2018