It was observed in the 1950s that the eigenvalues of a (non-Hermittian) random matrix seem to distribute uniformly in a circle (circular law). A rigorous explanation was given by Ginibre/Mehta (60s) for the Gaussian case, and by Bai (1997, following a work of Girko from 1984) for general continuous models. Bai's proof, however, does not extend to the discrete models.

Another observation in a completely different area: linear programming. It is a very well known phenomenon in the programming community that the simplex method, while proven to be exponential in the worst case, usually runs very fast, beating all other algorithms, even those are proven to be polynomial in all cases. Few years ago, Spielman and Teng came up with a nice explanation (smooth analysis) which relies on the existence of noise in the calculation process. Their analysis assumed continuous noise. The proof for the discrete (and more natural) case has been missing.

I will discuss recent developments which fill these missing parts, using tools from additive combinatorics, in particular the so-called Inverse Littlewood-Offord theorem.