IDEAS IN THE THEORY OF RANDOM-MEDIA

These lectures discuss the ideas of localization, intermittency, and random fluctuations in the theory of random media. These ideas are compared and contrasted with the older approach based on averaging. Within this framework, the topics discussed include: Anderson localization, turbulent diffusion and flows, periodic Schrodinger operators and averaging theory, longwave oscillations of elastic random media, stochastic differential equations, the spectral theory of Hamiltonians with (an infinite sequence of) wells, random Schrodinger operators, electrons in a random homogeneous field, influence of localization effects on the propagation of elastic waves, the Lyapunov spectrum (Lyapunov exponents), the Furstenberg and Oseledec theorems for an n-tuple of identically distributed unimodular matrices and their relation with the spectral theory of random Schrodinger or string operators, Rossby waves, averaging on random Schrodinger operators, percolation mechanisms, the moments method in the theory of sequences of random variables, the evolution of a magnetic field in the turbulent flow of a conducting fluid or plasma (the so-called kinematical dynamo problem), heat transmission in a randomly flowing fluid.