In probabilistic combinatorics, one often needs to have some rough estimate for the distribution of a complicated random variable. There are two main kinds of estimates:
(1) Concentration: If one take an interval I far from the mean, then the probability that X in I is very small.
(2) Anti-concentration: If one take a "short" interval I, then the probability that X in I is also very small.
I am going to discuss a few recent results in these areas, together with applications. Most of the talk will be based on my papers "Small Ball Probability" with H. Nguyen, and "Random Weighted projections" with K. Wang, available on the arxiv.