This is the second talk of  Kingman’s subadditive ergodic theorem.  I will finish the proof during the talk.

We will show that if a discrete d-dimensional Schr\"odinger operator has only discrete spectrum outside the interval [-2d, 2d], then its essential spectrum must be [-2d, 2d].

This talk will be based on a recent work by Killip,  Molchanov and  Safronov.

We will discuss recent results of M. Hochman and B. Solomyak.

The goal of this Seminar is to give an introduction of the dispersive and Strichartz estimates. I will present a new way to the proof of dispersive estimate which is easy to be extended to manifold. Using $TT*$ arguments and dispersive estimate, I will set up the Strichartz estimate. In order to simplify the Seminar, I will focus on the free laplacian.

In this talk, we will discuss Furstenberg correspondence principle, which connects ergodic theory and combinatorial number theory.