Mathematical Sciences: Probability Theory with Minimum Structures

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The main theme of this work is "probabilities with minimal structures", and in particular the research of upper and lower bounds for stochastic processes under general conditions. Connected sidelines are applications to Banach spaces, to combinatorics, to probabilistic analysis of hard problems, to random Fourier series and of course to more classical probabilistic questions such as trajectories of stochastic processes and gaussian processes. The central philosophy is to give priority to tools that require very little structure over more specific arguments. The ideas of isoperimetry in product spaces, that have given a new understanding of the notion of probabilistic independence will continue to be a central tool. The true scope of these ideas has been realized only recently, and the potential for future progress and applications should not be underestimated. The main thrust of this research is to pursue the development of abstract tools of probability in particular in relationship with the classical idea of isoperimetry. It was realised by the investigator that the classical idea of isoperimetry can be made meaningful in a much more general setting than was previously believed. This gave rise to new families of inequalities that have solved numerous problems in abstract probability, and had impact in areas where probability is a tool rather than an object of study in itself, and in particular operation research and theoretical computer science. The potential of these idea, both at the theoretical level and the level of applications starts only to be understood, and appears very promising.