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Description

I will survey the recent progress in understanding small ball probabilities and their connections to additive combinatorics and random matrix theory. Small ball probabilities help us to bound random variables *away* from the mean. This direction of probability theory is opposite to the classical theory of concentration of measure. Unlike concentration bounds, small ball probabilities are sensitive to arithmetic structure of random variables. The two major challenges are: (1) locate the arithmetic structure which is the obstruction to small ball probability, and (2) how to remove that structure. The work on the first challegne was initiated by Littlewood, Offord and Erdos. I will discuss the recent progress by Tao, Vu and a joint work with Mark Rudelson on this program. One application of the newly developed theory settles the distance problem: how close a random vector can be to a given subspace in R^N? The distance problem in turn leads to advances in random matrix theory in arbitrary bounded dimensions, in particular to computing the least singular value of random matrices.