Projektdetails
Abstract
Primary researchers involved: Piotr Borodulin-Nadzieja, Barnabas Farkas, Grzegorz Plebanek, Damian Sobota, Lyubomyr Zdomskyy.
Wider research context: The general aim of the project is to study classical and recently discovered interactions between certain combinatorial structures of set theory and functional analysis, focusing on analytic P-ideals, Banach spaces, Stone spaces, and the random forcing.
Research questions:
(1) concerns the interactions between combinatorics of families of finite subsets of the natural numbers, geometry of the associated Banach spaces, and complexity of the analytic P-ideals generated by the bases of these spaces. We want to characterize properties of Banach spaces (such as l1-saturation, the Schur property, etc) and properties of these ideals (such as compactly supported, totally bounded, etc) in terms of the other two structures. This research direction is motivated by a recent characterisation of precompact families by Borodulin-Nadzieja and Farkas. Furthermore, this approach leads to potential new candidates of (consistently) universal Banach spaces of density continuum.
(2) focuses on problems related to the classical Borel measure algebra B: (2a) Investigations of specific subalgebras of B, and looking for a ZFC example of a Boolean algebra which is Grothendieck and not Nikodym (based on constructions of Sobota and Zdomskyy). (2b) Possible (consistent) embeddings of large subalgebras of B in the powerset of the natural numbers modulo finite sets. (2c) The interplay between properties of ultrafilters (e.g. P-points) in the random model and the homomorphisms generating their names (based on works of Borodulin-Nadzieja and Sobota).
Methods: In (1) we are going to present how techniques developed in the theory of Banach spaces can be directly applied to analytic P-ideals and vice versa. The research on universal Banach spaces requires developing new combinatorial forcing methods. In (2a) we are planing to work with algebras of Borel sets which can be ``fastly'' approximated by clopen sets. In (2b) we will discuss how certain combinatorial axioms effect embeddability of subalgebras of B. In (2c) we are conducting a systematic study of translating properties of homomorphisms to properties of ultrafilters in the random model.
Innovation: This line of research in (1) opens up a new multidisciplinary approach both to analytic P-ideals and to Banach spaces, and shows strong potential to provide us with new structure theorems and peculiar examples of these objects. The use of the specific algebras mentioned in (2a) recently appeared to be very useful in similar constructions. In (2b) our approach, through understanding how additional axioms turn certain embeddings impossible, gives us a deeper insight into the structure of the quotient algebra. In (2c) our approach through homomorphism is a new novel method and will help us deepen our understanding of the structure of ultrafilters in the random model.
Wider research context: The general aim of the project is to study classical and recently discovered interactions between certain combinatorial structures of set theory and functional analysis, focusing on analytic P-ideals, Banach spaces, Stone spaces, and the random forcing.
Research questions:
(1) concerns the interactions between combinatorics of families of finite subsets of the natural numbers, geometry of the associated Banach spaces, and complexity of the analytic P-ideals generated by the bases of these spaces. We want to characterize properties of Banach spaces (such as l1-saturation, the Schur property, etc) and properties of these ideals (such as compactly supported, totally bounded, etc) in terms of the other two structures. This research direction is motivated by a recent characterisation of precompact families by Borodulin-Nadzieja and Farkas. Furthermore, this approach leads to potential new candidates of (consistently) universal Banach spaces of density continuum.
(2) focuses on problems related to the classical Borel measure algebra B: (2a) Investigations of specific subalgebras of B, and looking for a ZFC example of a Boolean algebra which is Grothendieck and not Nikodym (based on constructions of Sobota and Zdomskyy). (2b) Possible (consistent) embeddings of large subalgebras of B in the powerset of the natural numbers modulo finite sets. (2c) The interplay between properties of ultrafilters (e.g. P-points) in the random model and the homomorphisms generating their names (based on works of Borodulin-Nadzieja and Sobota).
Methods: In (1) we are going to present how techniques developed in the theory of Banach spaces can be directly applied to analytic P-ideals and vice versa. The research on universal Banach spaces requires developing new combinatorial forcing methods. In (2a) we are planing to work with algebras of Borel sets which can be ``fastly'' approximated by clopen sets. In (2b) we will discuss how certain combinatorial axioms effect embeddability of subalgebras of B. In (2c) we are conducting a systematic study of translating properties of homomorphisms to properties of ultrafilters in the random model.
Innovation: This line of research in (1) opens up a new multidisciplinary approach both to analytic P-ideals and to Banach spaces, and shows strong potential to provide us with new structure theorems and peculiar examples of these objects. The use of the specific algebras mentioned in (2a) recently appeared to be very useful in similar constructions. In (2b) our approach, through understanding how additional axioms turn certain embeddings impossible, gives us a deeper insight into the structure of the quotient algebra. In (2c) our approach through homomorphism is a new novel method and will help us deepen our understanding of the structure of ultrafilters in the random model.
| Status | Nicht begonnen |
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Projektbeteiligte
- Universität Wien
- Technische Universität Wien (Leitung)
- Uniwersytet Wrocławski