The purpose of this research project is to develop Wilson's renormalization group as a technique for proving mathematical results in statistical mechanics and quantum field theory. In particular, these methods offer the possibility of proving results (1) on the asymptotic behavior for large times of the end-to-end distance of self-avoiding walks, (2) on the low temperature phase and its correlations of the non-linear sigma model in more than two dimensions, (3) on the construction of the non-linear sigma models in two dimensions, and (4) on the absence or presence of a mass gap for these models. This research project is in the general area of mathematical physics and, in particular, deals with research in the subareas statistical mechanics and quantum field theory.
