简介

My research has concerned mathematical analysis of issues of physics, in particular: • Disorder effects on the spectra and dynamics of operators of quantum mechanics, and related random matrix phenomena. • Critical behavior in systems with many degrees of freedom. • Scaling limits, and field theory. • Spin glass issues, and the dynamics of competing particle systems; the topics are related through the 'cavity perspective' on SG. Typically, many of the interesting phenomena encountered in statistical mechanics are beyond the reach of exact solutions and perturbative methods. Our goal has been to develop rigorous methods which can shed light on the critical behavior even in such situations. A recurring theme has been the appearance of stochastic geometric effects which play important roles in the behavior of critical system. The work has contributed to the nascence of the current field of random fractal structures which capture the scaling limits of many critical two dimensional systems, and which bear close relations with conformal field theories. Our previous works have rigorously established the existence of model-dependent upper critical dimensions, above which the critical behavior simplifies. Works on quantum spectra and dynamics have provided new methods for dealing with localization caused by disorder, and conversely for establishing the persistence of extended states in the presence of disorder. On the latter topic the success has been rather limited; the issues continues to provide a mathematical challenge. Other goals of current work include shedding light on spectral gaps for systems with disorder, and conjectured relations with the distributions known from random matrix models.