Georgia Institute of Technology College of Sciences

Consider a sample of a centered random vector with unit covariance matrix. We show that under certain regularity assumptions, and up to a natural scaling, the smallest and the largest eigenvalues of the empirical covariance matrix converge, when the dimension and the sample size both tend to infinity, to the left and right edges of the Marchenko-Pastur distribution. The assumptions are related to tails of norms of orthogonal projections. They cover isotropic log-concave random vectors as well as random vectors with i.i.d. coordinates with almost optimal moment conditions. The method is a refinement of the rank one update approach used by Srivastava and Vershynin to produce non-asymptotic quantitative estimates. In other words we provide a new proof of the Bai and Yin theorem using basic tools from probability theory and linear algebra, together with a new extension of this theorem to random matrices with dependent entries. Based on joint work with Djalil Chafai.

We will summarize what we did so far in this sequence of seminars, among other things, the convergence of eigenvalues of Wigner random matrices and also GUE in expectation. This time we will explore concentration inequalities and use these to go from the convergence in expectation to convergence almost surely.

We will discuss the problem of restricting the Fourier transform to manifolds for which the curvature vanishes on some nonempty set. We will give background and discuss the problem in general terms, and then outline a proof of an essentially optimal (albeit conditional) result for a special class of hypersurfaces.

In previous talks, we discussed an algorithm (Nash-Moser iteration) to compute invariant whiskered tori for fibered holomorphic maps. Several geometric and number-theoretic conditions are necessary to carry out each step of the iteration. Recently, there has been interest in studying what happens if some of the conditions are removed. In particular, the second Melnikov condition we found can be hard to verify in higher dimensional problems. In this talk, we will use a method due to Eliasson, Moser and Poschel to obtain quasi-periodic solutions which, however, lose an important geometric property relative to the solutions previously constructed.