9401028 Brydges The proposal is to study problems in the following related areas: (1)Models of polymers with attractive self interactions. (2) Infrared behavior of weakly perturbed Gaussian measures arising in statistical mechanics of systems that are near a second order phase transition. (3) Epsilon expansions for models with nearly Gaussian fixed points for the renormalization group. (4) a perturbative study of the renormalization group evolution of the extrinsic geometry of target manifolds in the nonlinear sigma model. (5) Non-perturbative renormalization group for strong coupling problems such as the non-linear sigma model. The theory of elementary particles, the prediction of the size and shape of long chain molecules in solution, and the study of phase transitions are related problems from the view of mathematics; they require a calculus for the approximate evaluation of integrals over spaces of infinite dimension. In the 1970's the Nobel Laureate K. Wilson, made fundamental progress in this calculus, which is called the renormalization group. He left mathematicians with the challenge of filling in his many leaps of faith and creating a systematic theory. The first three parts ofthis proposal hope to achieve this for a class of integrals that are almost close to Gaussian integrals - a special class that can be evaluated exactly. The last two parts are more speculative attempts to create the same theory for integrals that are much further from being equal to Gaussian integrals. ***
