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 MoreSet-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.
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.
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.
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.
J. Inst. Math. Jussieu 23, No. 2, 737-752 (2024)
Proc. Am. Math. Soc. 152, No. 3, 1003-1017 (2024)
J. Funct. Anal. 285, No. 11, Article ID 110149, 49 p. (2023).
Bull. Lond. Math. Soc. 54, No. 6, 2066-2077 (2022).
Proc. Am. Math. Soc. 150, No. 2, 817-831 (2022).
Proc. Natl. Acad. Sci. USA 117, No. 52, 33085 (2021).
J. Funct. Anal. 281, No. 9, Article ID 109172, 33 p. (2021).
Adv. Math. 381, Article ID 107613, 40 p. (2021).
Proc. Am. Math. Soc. 149, No. 3, 1289-1303 (2021).