Three topics in mathematical quantum theory of systems of many degrees of freedom will be investigated in this project. Of main interest are problems related to the structure and stability of matter. First, rigorous estimates will be established on the effects of correlations between many electron on the binding and ionization in atomic and molecular systems. Second, the structure of matter on the surface of a neutron star under large magnetic fields will be mathematically analyzed. This study involves the fairly new area of semiclassical analysis in the presence of a magnetic field. Third, are problems related to the statistical mechanics of quantum spin systems, especially the systems with continuous symmetry. Quantum mechanics is probably one of the most successful attempts to give a mathematical description of the world around us. However, while many complex quantum mechanically systems are fairly well understood at a somewhat heuristic level, a complete mathematical analysis of these systems is very difficult and very often lacking. The lack of a detailed mathematical analysis of these systems is not separate from a lack of complete understanding of the underlying physics. The mathematical analysis of a series of these complex physical systems will be attempted and it is the hoped that new mathematical as well as physical insight will emerge.
