Combinatorics, Banach Spaces, Probability, Spin Glasses

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9703879 Talagrand This research centers on two main tools of abstract probability, concentration of measure (a versatile tool to bound the fluctuations of complicated functions) and the theory of majorizing measures (a versatile tool to find upper and lower bounds for the expected values of complicated functions). The investigator will try to pursue the development of the underlying theories, and will also try to find new applications, in particular to the geometry of Banach spaces and Hilbert spaces. He will also try to use abstract methods to achieve progress on the understanding of mean field models for spin glasses, a subject where rigorous results are very hard to obtain. A separate project consists in the study of the subsets of the discrete cube of large dimension, and of their extremal properties. Real life phenomenon are usually affected by random events outside our control. Random variables that depend on many independent influences are therefore a good model for the outcome of complicated real procedures. Surprisingly general tools have recently been developed to rigorously control the fluctuations of these random variables in a wide variety of situations, and they demonstrate the power and the relevance of abstract methods to concrete situations. The development of these methods and the research of new applications will be continued. A particular effort will be made to use these methods to obtain rigorous results for spin glass models. These are mathematical models for some very intriguing states of condensed matter, that are of considerable interest in the physics community.