Professor de la Llave will study several maximal operators that control many problems. In dynamical systems, he will investigate cocycle equations and their applications to rigidity problems, and computational Kolmogorov . Arnold . Moser theory. He will also develop methods to obtain numerically accurate lower bounds for Schrodinger operators. This research project concetrates on several questions in mathematical analysis that are relevant to dynamical systems and to the N.body problem in quantum mechanics. A wide variety of methods will be used. On the one hand, regularity problems will be approached by the traditional methods of hard analysis, defining appropriate spaces and proving sharp estimates for operators acting on them. On the other hand, the research will systematically use computers, not only as a heuristic aid but also in a framework being developed by de la Llave and others for automating the routine parts of proofs of analytic estimates. This will make it possible to employ proof strategies inaccessible to a human mathematician just because of the large amount of elementary work involved.
