This award will support research on the mathematics of disordered systems to be conducted by Abel Klein of the University of California at Irvine in cooperation with J. Fernando Perez and Henrique von Dreifus from the University of Sao Paulo, Brazil. They will investigate phenomena in two types of disordered systems: quantum spin systems with disorder and random Schrodinger operators. Quantum spin systems with random parameters have been introduced to study the effect of impurities in several physical systems. Examples of such systems are given by quantum X - Y, Heisenberg and Ising models in the presence of a random transverse field. These models are related to superfluidity and superconductivity. The study of disordered quantum systems that exhibit zero- temperature transitions is a new and exciting area of research in Mathematical Physics. In such models, one has to treat the combined effects of randomness and quantum fluctuations. Such combination has hardly been studied in the Physics literature or in Mathematical Physics. Such study is a long range goal of this research proposal.