9705095 Koch Professor Koch is investigating continuous-spin models near criticality, quasiperiodic orbits for classical Hamiltonian flows, and lattice gases related to aperiodic tilings and quasicrystals. The first two of these projects involves the use, and further development, of renormalization group techniques. In a previous renormalization group analysis of ferromagnetic spin models, some fundamental questions were answered in a simplified hierarchical setting. Ongoing work now deals with the problem of restoring some of the ingredients that have been missing. The investigation of quasiperiodic motion for Hamiltonian flows is based on a new class of renormalization group transformations that was introduced in a previous project. One of the goals is to develop these transformations into tools for studying non-perturbative phenomena, such as the breakup of smooth invariant tori. Other interesting questions concern the accumulation of closed orbits at invariant tori and the interplay between renormalization group transformations that correspond to different frequency vectors. The third project deals with a promising but largely unknown class of statistical mechanics models: Lattice gases with non-periodic Gibbs states. The current investigation focuses on a few examples, based on aperiodic tilings, for which the minimum energy configurations are well known. The goal is to identify and understand the low temperature properties of these models and to develop the appropriate methods for analyzing them. Professor Koch's study of ferromagnetic spin models is part of a long-term effort toward a mathematical foundation of the modern theory of critical phenomena in condensed matter physics. One of the striking phenomena is that there are observable quantities (critical indices) which seem to be independent of the system considered, within large classes of different systems. Starting from some basic assumptions, the current theory allows an approximate computation of these universal quantities. But it is still an open problem to show that these assumptions hold, within a class of reasonably realistic models. Ongoing investigations in this area use computer-assisted proofs and involve further development of these techniques. This includes validated numerics -- a technique which is of increasing interest also in engineering and modern industrial design. Another critical phenomenon investigated in this project is the loss of stability of quasi- periodic orbits in classical Hamiltonian systems. In certain cases, this loss of stability is believed to be associated with a significant increase in "chaotic" motion. The process appears to be universal, in the sense described above, but it is not yet understood. Interest in this problem stems from celestial mechanics and plasma physics, where questions of stability play an important role. And of course, the mathematics involved is interesting in itself. The third project deals with statistical mechanics models whose minimum energy configurations are non-periodic. Some of these models are believed to describe quasicrystals, and others may have features similar to spin glasses. They are unusual by current standards, but not exceptional. In fact, non-periodicity is a generic property for ground states in a standard class of statistical mechanics models. So far, almost all investigations of such models have been limited to studying the zero temperature state. The goal now is to obtain some useful information about the behavior at positive temperatures, using a combination of (large scale) numerical simulations and analytical techniques.
