This project in mathematical physics has to do with the rigorous analysis of renormalization group techniques used in a heuristic fashion by physicists. ( The basic physical problem is to determine macroscopic properties of large systems of particles with known short range interactions. ) Some of the work involves a novel use of the computer to assist in finding proofs of theorems that require intricate schemes for estimating quantities. Renormalization group techniques have proved to be a powerful tool in statistical mechanics, especially in the theory of critical phenomena. The same techniques also play a dominant role in the modern approach to constructive quantum field theory. Much of the current research in these two areas is concerned with the extension of present methods from relatively simple to more realistic systems. Professor Koch will continue his investigation of models relevant to the theory of critical phenomena. His second main emphasis is an analysis of disordered systems such as the Hopfield model and the Sherrington - Kirkpatrick model, which exhibit spin glass behavior for a certain range of parameters.
