Anderson localization for dynamically generated and random potentials and applications

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Quantum mechanics explains the laws of nature of electrons and similar objects and such phenomena as for instance electrical conductance. Schrodinger equation plays in this mechanics the same role as the Newton equations of classical mechanics. In 1958 Anderson had discovered some very special characteristics of the dynamics of an electron propagating in a metal with considerable number of impurities. This fundamental discovery in the field of condensed matter physics suggested a number of very deep and hard mathematical problems. These problems were studied extensively in last fifty years. However the hardest and most important problems in this field remain unsolved. In the effort to solve these problems mathematicians have introduced new technologies which proved to be effective in various domains of modern mathematics. In this proposal we plan to bring in some new ideas in this field and to build new methods upon these ideas.