Set Theory and the Geometry of Banach Spaces

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The principal investigator proposes to work on three problems in the geometry of Banach spaces with a strong set theoretic and/or combinatorial flavor. Firstly, the principal investigator proposes to work on the classification problem of complemented subspaces of the classical function space of continuous functions on the unit interval, and in particular, on the fundamental question of determining the subspace structure of all complemented subspaces with separable dual.The problem is central mainly because of the importance of this space in other parts of mathematics and of mathematical physics. Secondly, the principal investigator proposes to continue his work on the existence of unconditional basic sequences in non-separable Banach spaces. Any result establishing the existence of unconditional basic sequences has immediate implications to the famous 'separable quotient problem' posed by Stefan Banach and asking whether every infinite-dimensional Banach space admits a separable infinite-dimensional quotient. Known results indicate that the 'separable quotient problem' has the deepest set-theoretic aspects among all problems in Banach space theory and is closely related to infinite combinatorics and large cardinal axioms of set theory. Thirdly, the principal investigator proposes to continue his work on applications of descriptive set theory to universality problems in Banach space theory. The project will promote the deep interaction between logic and analysis, two major disciplines of pure mathematics with far reaching applications ranking from theoretical computer science to physics.