Piotr Koszmider, Mathematician

I do pure set theory and applications of set theory in diverse fields of mathematics such as Banach spaces, operator algebras, topology. This often involves elements of mathematical logic in the form of set-theoretic forcing since many results in this field are undecidable. It also often reduces to uncountable combinatorial arguments.

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Seminar: Working group in applications of set-theory

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.

Current doctoral and postdoctoral advising

Kamil Ryduchowski; Warsaw Doctoral School of Mathematics and Computer Science

Expected doctoral defence: mid 2026

Damian Głodkowski; Warsaw Doctoral School of Mathematics and Computer Science

Expected doctoral defence: mid 2024

Zdeněk Silber, postdoctoral fellow at IMPAN

2022-24
Preprints

P. Koszmider, Z. Silber, Countably tight dual ball with a nonseparable measure

Assuming the diamond principle we construct a compact K which carries a nonseparable measure but whose space of all probability measures has countable tightness. This answers questions of Plebanek and Sobota as well as of Pol concerning the property C of Corson. Classical results of Fremlin show that some additional hypothesis like the diamond is necessary.

P. Koszmider, On Ramsey-type properties of the distance in nonseparable spheres.

Assuming OCA and MA or under descriptive set-theoretic hypotheses we prove dichotomies for many classes of Banach spaces of the form: either the unit sphere admits an uncountable (r+)-separated set or else it is the union of countably many sets of diameters not exceeding r. We show applications and investigate counterexamples to these dichotomies under CH or weaker axioms. This complements classical results of Kottman, Elton and Odell in the separable case and recent results of Hájek, Kania and Russo for the densities above the continuum.

P. Koszmider, K. Ryduchowski, Equilateral and separated sets in some Hilbert generated Banach spaces

Assuming the existence of a nonmeager set of reals of the first uncountable cardinality we show that there is an equivalent renorming of a nonseparable Hilbert space with no uncountable equilateral sets. Several other consistency and independence results are shown concerning Hilbert generated spaces.

Accepted to Proceedings of the American Mathematical Society

P. Koszmider, Banach spaces in which large subsets of spheres concentrate

We construct a nonseparable Banach space X (actually, of density continuum) such that any uncountable subset Y of the unit sphere of X contains uncountably many points distant by less than 1. It relies on an almost disjoint family with special properies.

Accepted to Journal of the Institute of Mathematics of Jussieu
Recently published papers
  1. O. Guzmán, M. Hrušák, P. Koszmider, Almost disjoint families and the geometry of nonseparable spheres.

    J. Funct. Anal. 285, No. 11, Article ID 110149, 49 p. (2023).

  2. P. Koszmider, H. Wark, Large Banach spaces with no infinite equilateral sets.

    Bull. Lond. Math. Soc. 54, No. 6, 2066-2077 (2022).

  3. D. Głodkowski, P. Koszmider, On coverings of Banach spaces and their subsets by hyperplanes.

    Proc. Am. Math. Soc. 150, No. 2, 817-831 (2022).

  4. P. Koszmider, A non-diagonalizable pure state.

    Proc. Natl. Acad. Sci. USA 117, No. 52, 33085 (2021).

  5. P. Koszmider, On the existence of overcomplete sets in some classical nonseparable Banach spaces

    J. Funct. Anal. 281, No. 9, Article ID 109172, 33 p. (2021).

  6. P. Koszmider, N. J. Laustsen, A Banach space induced by an almost disjoint family, admitting only few operators and decompositions

    Adv. Math. 381, Article ID 107613, 40 p. (2021).

  7. P. Koszmider, A.Martínez-Celis, Rosenthal families, pavings and generic cardinal invariants

    Proc. Am. Math. Soc. 149, No. 3, 1289-1303 (2021).

  8. O. Guzmán, M. Hrušák, P. Koszmider, On R-embeddability of almost disjoint families and Akemann-Doner C*-algebras

    Fund. Math. 254, No. 1, 15-47 (2021).