Abstract:  In 1958 P.Anderson dicovered absense of difusion of waves in disordered medium. This phenomenon named after Anderson, who suggested a mechanism for electron localization is in a lattice potential, provided that the degree of randomness (disorder) in the lattice is suciently large.  Anderson considered so-called one-electron approximation which runs in three dimensional space or lattice.
It was understood later that the dimension of the problem is critically important. Mathematicians noted that the Anderson localization phenomenon is a very hard eld which has a large numbers of natural conjectures and possibly applications. That applies even to the simplest case of one dimensional lattice.

In this talk I will discuss the main principles of Anderson localization for one dimensional setting. Most of the material I will discuss was developed in last 20 years in the works of W.Schlag and myself. The main part of the discussion will be around the spliting between the eigenvalues and spectral resolution for the second order linear dierence equation on Z. The only non-constant coecient of the equation comes in the form V(Tⁿx) where T : T^d→ T^d a smooth map and V is a smooth function on the torus. I will speculate on applications of the spectral resolution problem to Lyapunov exponent of Standard map, which is a project in progress.