Mathematical Sciences: Banach Spaces and Operator Theory

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Arias will investigate several problems in Banach space theory with applications to operator theory and probability. The main areas of research will be the local theory of Banach spaces, nest algebras for a Banach space perspective, and finite- dimensional operator theory. Banach space theory is that part of mathematics that attempts to generalize to infinitely many dimensions the structure of 3-dimensional Euclidean (i.e.ordinary) space. The axioms for the distance function in a Banach space are more relaxed than those for Euclidean distance (For example, the "parallelogram law" is not required to hold.), and as a result, the "geometry" of a Banach space can be quite exotic. Much of the research in this area concerns studying the structure theory of Banach spaces.