Mathematical Sciences: Application of Measure Theory to Probability and Banach Spaces

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This is a project for research at the interface of two areas of mathematics, Banach space theory and probability theory. The former has to do with the convergence of series. Classically, the terms of the series are functions, but there is a conceptual advantage in taking them simply to be vectors in an abstract Banach space. Respecializing, the terms of the series can be random variables, whose study is at the heart of probability theory. This point of view has lately turned out to be very fruitful. More specifically, Professor Talagrand will continue the introduction of new combinatorial and isoperimetric methods in probability theory. A particular objective is better understanding of necessary and sufficient conditions for sample boundedness of p-stable processes. He will also tackle a famous open problem in Banach space theory having to do with the classical space of absolutely integrable functions on the unit interval.