Mathematics Colloquia and Seminars

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Description

There are two complementary directions in probability theory. One is the theory of large deviations, which seeks to control the probability of deviations of a random variable X from its mean M. The other (more recent) direction is the theory of small deviations, or "the small ball probability", which seeks to control the probability of X being very small, i.e. it looks for upper bounds on Prob (|X| < t M). A general impression is that the latter direction is harder. For example, it is harder to estimate the least eigenvalue of a random matrix than its largest eigenvalue (and there are fascinating conjectures on that problem). I had a very general conjecture on the small ball probability for a norm of a gaussian random variable. This conjecture was proved this summer by Latala and Oleszkiewicz. They reduced it to the "B-coinjecture" on Gaussian measures of convex symmetric sets K, described in Latala in his Beijing Congress talk: meas[tK] meas[(1/t)K] > meas(K)2 for all real t. The B-conjecture was recently solved by Fradelizi, Cordero and Maurey using transportation of measure, deep results due to Brenier. McCann and Caffarelli. I will describe the rich topic of the transportation of measure, the conjectures and known results on Gaussian measures of convex sets, and the small ball probability theory. There are (obvious) connections to Gaussian processes and less obvious connections to high dimensional geometry.