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Abstract
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The concentration of measure phenomenon, which developed during the last century in various parts of mathematics, is major tool in the study of quantitative bounds in functional and geometric analysis and probability theory, typically for models involving an infinite number of variables. Its numerous illustrations and applications range from geometric analysis and probability to statistical mechanics, mathematical statistics and learning theory, randomized algorithms, complexity, random matrix theory etc.
After a review of the basic methods and tools towards measure concentration, including spectral and functional inequalities, optimal transport and isoperimetric inequalities, we will address some mostly open recent issues in the form of superconcentration inequalities beyond the standard concentration rates, related to quantitative bounds on Gaussian fields, random matrices, spin glasses and related models.
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