The PI will explore and apply logic, set-theoretic and combinatorial methods to the geometry of Banach spaces. The isomorphic classification of the complemented subspaces of the spaces of p-power integrable functions, especially those contained in some nice subspaces, will be considered. Necessary and sufficient conditions for bounded linear operators from these spaces to factor through spaces with simpler and better structures will be studied. The investigator will also study operators from spaces with certain asymptotic structures. Characterizations of subspaces and quotients of separable Banach spaces with shrinking unconditional basis and subsequences of unconditional sequences will be investigated.
Banach space theory is about vector spaces in which there is a very natural notion of distance. These spaces are of fundamental importance in many areas, including mathematical models in quantum mechanics. The understanding of the geometry of Banach spaces has been and will continue to be useful in many areas of mathematics and engineering. In particular, the use of logic, set-theoretic and combinatorial methods will provide rich information about the geometric structure of Banach spaces.
