impan seminar:

working group in applications of set theory





23. 04. 2026 at 1.15 pm. Room 403.

Jarosław Swaczyna (Łódź University of Technology)
On an ideal version of Schauder bases.


Abstract: Given an ideal I on the natural numbers, one obtains a notion of I-convergence by requiring that for every ε>0, all but a set in I of terms are ϵ-close to the limit. We study this notion in the context of Schauder bases. More precisely, a sequence (en) is called an I-basis of a Banach space X if for every x∈ X there exists a unique sequence of scalars (αn) such that the series Σαnen converges to x in the sense of I-convergence of the sequence of partial sums. This concept has been considered, among others, by Kadets.

We present an overview of the current state of the art and discuss a variety of examples of such bases in concrete Banach spaces, including cases where the norms of the associated projections onto initial segments are unbounded. We also investigate general structural properties of I-bases. In particular, we show that coordinate functionals are continuous under additional assumptions on the ideal I, for instance when I is analytic. The theory of I-bases is still far from complete, and two main directions for its further development seem particularly important. The first is to determine which separable Banach spaces admit any I-basis at all — it remains conceivable that every separable space does. The second concerns understanding the structural and geometric properties of I-bases in analogy with the classical theory of Schauder bases, and identifying which classical results admit ideal-analytic counterparts.

The talk is based on joint work with Tomasz Kania, Noé de Rancourt, Adam Kwela, and ongoing work with Joanna Garbulińska-Węgrzyn, as well as on results of Kadets and collaborators.

Bibliography:

  • N. de Rancourt, T. Kania, J. Swaczyna, Continuity of coordinate functionals of filter bases in Banach spaces, Journal of Functional Analysis, Volume 284, Issue 9 (2023).
  • A. Kwela, J. Swaczyna, (2025) Zoo of ideal Schauder bases, The Journal of Symbolic Logic, pp. 1–31. doi:10.1017/jsl.2025.10136.
  • V. Kadets, M. Manskova, Norms of partial sums operators for a basis with respect to a filter, Journal of Mathematical Analysis and Applications, Volume 558, Issue 2, 2026.




Previous talks this semester:




16. 04. 2026 at 1.15 pm. Room 403.

Kacper Kucharski (UW)
Killing homeomorphisms of function spaces on separable compact lines.


Abstract: A compact line is any linearly ordered compact topological space. During the talk we will provide a ZFC construction of an example of a separable compact line K of weight 2ω, whose spaces of continuous functions with the pointwise convergence topology Cp(K) and the weak topology Cw(K) are not homeomorphic to their square. The talk will be related to the talk at Wednesday seminar on 15.04 at MIMUW but can be attended independently.



9. 04. 2026 at 1.15 pm. Room 403.

Michał Wojciechowski (IM PAN)
Compact spaces with Nevanlina-Pick property.


Abstract: Recall that the Gelfand space Δ(A) of a (complex) unital commutative Banach algebra A is the set of all multiplicative functionals on A with the weak* topology. The Gelfand transform sends a in A to a continuous function â in C(Δ(A)) given by â(φ)=φ(a). A is a semisimple Banach algebra whenever the Gelfand tranform is injective.

We say that A has the Nevanlina-Pick property if for every finite subset F of Δ(A) the quotient norm in the finite dimensional quotient {â|F: a∈A} is the supremum norm. This condition called the Nevanlina-Pick property of A is necessary for A to be of the form C(K) for K compact Hausdorff. One may ask when it is sufficient.

A compact space K has the Nevanlina-Pick property whenever every commutative semisimple algebra with the Nevanlina-Pick property whose Gelfand space is K is of the form C(K). We investigate the class of compact Hausdorff spaces with the Nevanlina-Pick property. The talk is based on a joint paper with P. Ohrysko (in preparation).



26. 03. 2026 at 1.15 pm. Room 403.

Małgorzata Rojek (UW/IMPAN)
Around decompositions of C(N*). Continuation.



19. 03. 2026 at 1.15 pm. Room 403.

Małgorzata Rojek (UW/IMPAN)
Around decompositions of C(N*)


Abstract: A closed subspace Y of a Banach space is called complemented if there exists a closed subspace Z of X such that any element x from X can be uniquely written as a sum of a vector from Y and a vector form Z. In such case we say that X has a direct sum decomposition X = Y⊕Z. A Banach space X is primary if for any direct sum decomposition X = Y⊕Z at least one of the summands Y or Z is isomorphic to X.

During the talk, we will discuss certain properties of complemented subspaces and decompositions of the Banach space C(N*), where N*=βN-N is the Čech-Stone reminder of N. We will show that in any direct sum decomposition of C(N*) at least one of the summands contains a closed subspace isomorphic to C(N*). Under CH, we will show that C(N*) is isomorphic to l∞(C(N*)) and that C(N*) is primary. The talk is mostly based on the article "On the primariness of the Banach space l∞/c0" by L. Drewnowski and J. W. Roberts.



12. 03. 2026 at 1.15 pm. Room 403.

Piotr Koszmider (IMPAN)
Minimal extensions of Boolean algebras and the Efimov Problem


Abstract: We will discuss the notion of a minimal extension of a Boolean algebra and minimally generated Boolean algebras (obtained by transfinite sequence of consecutive minimal extensions). The Stone spaces of such algebras do not contain the Cech-Stone compactification βN of the integers. We will show how to construct such space which aditionally does not have nontrivial convergent sequences in the Cohen model. This is related to the Efimov problem: does every compact space either contain a copy of βN or a nontrivial convergent sequence. It is still not known if the positive solution of this problem is consistent.



5. 03. 2026 at 1.15 pm. Room 403.

Piotr Koszmider (IMPAN)
On covers and tilings of infinite dimensional Banach spaces


Abstract: A body in a Banach space is a set which is the closure of its interior. A tiling is a cover by bodies which may intersect only at their boundaries. We will discuss classical and recent results concerning covers and tilings of infinite dimensional Banach spaces. Some of the results depend on the properties of the cardinals which are the densities of the spaces, other results depend on the minimal cardinality of a pairwise disjoint family of closed sets which cover the interval [0,1] (known to have its value dependent on additional set theoretic hypotheses).







Talks in the first semester of 2025-26.

Talks in the second semester of 2024-25.

Talks in the first semester of 2024-25.

Talks in the second semester of 2023-24.

Talks in the first semester of 2023-24.

Talks in the second semester of 2022-23.

Talks in the first semester of 2022-23.

Talks in the second semester of 2021-22.

Talks in the first semester of 2021-22.

Talks in the second semester of 2020-21.

Talks in the first semester of 2020-21.

Talks in the second semester of 2019-20.

Talks in the first semester of 2019-20.

Talks in the second semester of 2018-19.

Talks in the first semester of 2018-19.

Talks in the second semester of 2017-18.

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 13.15-15.00 pm, room 403, Śniadeckich 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:

  • Piotr Koszmider (IM PAN)
  • Kacper Kucharski (PH. D. student UW)
  • Małgorzata Rojek (PH. D. student UW/IMPAN)
  • Jan Rossa (M. A. student UW)
  • Kamil Ryduchowski (PH. D. student UW/IMPAN)
  • Piotr Szewczak (MIM UW)
  • Michał Wojciechowski (IM PAN)
  • Krzysztof Zakrzewski (SGH)
Forthcoming talks:

  • 30. 04 J. Swaczyna will continue.

  • 7.05 No talk.

  • 14.05 Antonio Acuaviva (Ph.D. student of Lancaster University)

  • 21. 05 Mikołaj Marsy (Ph. D student UWr).

  • 28.05 No talk.

  • 4. 06 ?

  • 11.06 Tanmay Inamdar.