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.

Read More

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 advising

Małgorzata Rojek; Warsaw Doctoral School of Mathematics and Computer Science

Expected doctoral defence: mid 2028

Kamil Ryduchowski; Warsaw Doctoral School of Mathematics and Computer Science

Expected doctoral defence: mid 2026
Preprints

D. Głodkowski, P. Koszmider, Products of C*-algebras that do not embed into the Calkin algebra

We show that in the Cohen model of set theory ZFC there is no embedding of the product (c0(2ω))N of infinitely many copies of the abelian C*-algebra c0(2ω) into the Calkin algebra. This enlarges the collection of the known examples due to Vaccaro and to McKenney and Vignati of abelian algebras, asymptotic sequence algebras, reduced products and coronas of stabilizations which consistently do not embed into the Calkin algebra.

P. Koszmider, Z. Silber, On Subspaces of Indecomposable Banach Spaces

We show that every Banach space of density at most continuum not admitting l∞ as its quotient is isometric to a subspace of an indecomposable Banach space (of density at most continuum). This includes all WLD or Asplund Banach spaces of density at most continuum.

Accepted to Advances in Mathematics

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.

Accepted to Transactions of the American Mathematical Society
Recently published papers
  1. P. Koszmider, Z. Silber, Countably tight dual ball with a nonseparable measure.

    J. Lond. Math. Soc., II. Ser. 110, No. 4, Article ID e12988, 25 p. (2024).

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

    J. Inst. Math. Jussieu 23, No. 2, 737-752 (2024)

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

    Proc. Am. Math. Soc. 152, No. 3, 1003-1017 (2024)

  4. 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).

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

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

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

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

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

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

  8. 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).

  9. 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).