This research will focus on a geometric approach to mathematical problems of harmonic analysis. The geometry and symmetry structure of a manifold determine a priori analytic inequalities for the natural operators that characterize analysis on function spaces over the manifold. These include the Fourier transform, Green's functions and the Laplace-Beltrami operator. The work seeks to develop and understand analysis on Lie groups and Riemannian manifolds through the study of differential operators, singular integrals and variational problems intrinsic to the manifold. Principal directions are (1) the role of renormalization arguments in controlling the behavior of oscillatory phenomena and (2) geometric analysis of higher-order multilinear differential operators with tensor structure. Central to this research is the interplay between geometric structure of manifolds and harmonic analysis of the operators using conformal invariance and geometric symmetrization. Sharp constants for variational problems provide a rich source of geometric and probabilistic information. Insight into the mathematical framework is gained from exact model calculations in conformal string theory and statistical mechanics using the conformal symmetry of critical phenomena. Computation analysis, especially computation and computer graphics, is used to provide intuition and strategy for problems where the geometric structure is too complex for elementary analysis.
