Mengentheoretische Kombinatorik in Banach- und Maßräumen

1) Wider research context. Applying set-theoretic techniques to functional analysis and measure theory has always delivered many interesting and surprising results, which could not be obtained using other methods, especially purely analytic ones. The reason behind this is that many problems naturally occurring in analysis, although simply stated and seemingly being of analytic character, rely on the existence of special set-theoretic and combinatorial structures and thus are sensitive to the assumed system of axioms. As examples one can mention here questions asking about the existence of such objects as outer automorphisms of the Calkin algebra, universal Banach spaces of a fixed density, or uncountable biorthogonal systems in Banach spaces. In this project we plan to follow this line of research and to investigate selected problems originating in analysis and measure theory, which may have hidden settheoretic nature, and to solve them by applying techniques of infinitary combinatorics and forcing.

2) Research questions. We plan to investigate problems related to the existence of such objects as Grothendieck C(K)-spaces of small density in the Laver model, large quotients of non-reflexive Grothendieck Banach spaces in models where Martin's axiom does not hold, or universal Banach spaces of density continuum. Besides, we plan to study additivity properties of probability measures on countable sets and their connections with the existence of P-points as well as to establish which subsets of the unit interval and which products of the real line are Prokhorov spaces.

3) Methods. We plan to apply methods from set theory (e.g. Laver and random forcing, combinatorics involving cardinal characteristics of the continuum and such structures as dominating families or filters), topological measure theory (e.g. constructions of non-Polish Prokhorov spaces based on special subsets of the Cantor set, e.g. Hurewicz subspaces), and Banach space theory (e.g. 'sliding hump' techniques of constructing sequences in Banach spaces, methods related to weak topologies, constructions of embeddings, etc.).

4) Level of originality. The main innovative aspect of this interdisciplinary project is to investigate and fill evident gaps in the current state of knowledge by answering questions related to the existence of special Banach spaces and measures spaces in various models of set theory, as well as by applying miscellaneous methods related to such set-theoretic structures as, e.g., dominating families, filters, almost disjoint families, etc. Such investigations will lead to constructions of new Banach and measure spaces as well as to better understanding of the hidden nature of already standard and apparently well-known objects.

5) Primary researchers involved. The Principal Investigator is Damian Sobota and the Mentor of the project is Vera Fischer. Both of the researchers currently work at the Kurt Gödel Research Center at the University of Vienna, Austria.