Mathematical Sciences: Statistical Mechanics of Disordered Systems and Conformal Field Theory

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Professor Kupiainen's research project in statistical mechanics has two parts, one in the physics of disordered systems and the other in the application of conformal field theory to two-dimensional critical phenomena. In connection with disordered systems, he will endeavor to develop exact renormalization group methods for the study of random walks in random environments, of interface fluctuations in the presence of random fields or random exchange couplings, and of spin glass models. On the conformal field theory side, he will continue investigations in which the Wess-Zumino-Witten model is used to solve various conformal theories. Statistical mechanics studies the behavior of systems consisting of very large numbers of particles interacting with one another. Typically, the particles would be arrayed in a lattice and interactions would take place between nearest neighbors. The basic problem is predict the overall, long-run behavior (equilibria, phase transitions, and the like) of the system given information about the interactions. Mathematically, this is a very difficult undertaking even in the simplest models. A further complication occurs when elements of randomness are present, as in the research to be undertaken by Professor Kupiainen.