I will describe our recent proof of localization at the bottom of the spectrum for Schrodinger operators with Poisson random potentials. Poisson random potentials are the most natural model for describing a material with impurities. This has been a longstanding open problem. I will give a very informal talk on work in progress.

In my talk I will discuss recent results obtained in collaboration with Th. Bodineau, D. Ioffe and R. Schonmann concerning the fine details of the geometry of the random macroscopic droplet of minus-phase, floating in the plus-phase of the 3D Ising model.

Einstein's kinetic theory of the Brownian motion, based upon light
water molecules continuously bombarding the heavy pollen, provided an explanation of diffusion from the Newtonian mechanics. Since the discovery of quantum mechanics it has been a challenge to verify the emergence of diffusion from the Schrodinger equation. In this talk I will report on a mathematically rigorous derivation of a diffusion equation
as a long time scaling limit of a random Schr\"odinger equation in a weak, uncorrelated disorder potential. This is a joint work with M. Salmhofer and H.T. Yau.