The objective of this research program is to develop a broad conceptual understanding of geometric analysis on manifolds which takes as its paradigm the role of sharp inequalities in Fourier analysis on the manifold as encoding geometric information about intrinsic invariants, symmetries and the underlying geometric structure of the manifold. Our direct focus is to obtain new and sharper results on isoperimetric inequalities, critical Sobolev embedding, log Sobolev and Levy-Gromov functionals, and multilinear fractional integrals that incorporate the framework of the Hardy-Littlewood-Sobolev inequality, Stein-Weiss integrals, Birman-Schwinger kernels and Riesz potentials, especially in the setting of Lie groups, symmetric spaces, complex manifolds and affine geometry including hyperbolic space, SL(2,R) and the Heisenberg group. Sharp embedding estimates are a critical tool to establish existence and regularity for solutions to pde's, and to control oscillatory behavior on a manifold. This program reflects a natural interplay and stimulus with problems in conformal deformation, fluid dynamics, geometric probability, statistical mechanics, string theory and turbulence. Computational methods are essential tools to obtain a clearer understanding of the complexity of the geometric symmetry.
Mathematical models that characterize physical phenomena and the analysis of differential equations that describe dynamical processes require a rich and diverse analytic framework. Scientists say that "mathematical laws underpin the fabric of our universe." The aim of this research program is to develop a supporting mathematical structure that helps us to understand relations that connect uncertainty, entropy and disorder, geometric symmetry, renormalization and scaling arguments, non-homogeneity of space-time models, isoperimetric comparison between surface area and volume, and the fine detail of hyperbolic geometry. We want to obtain broad and rigorous understanding of mathematical tools that support a quantitative description of the fundamental laws of nature, and to develop and explore mathematical structures that model physical phenomena ranging from fluid dynamics to string theory, especially by using patterns of symmetry and intrinsic constants that encode geometric information about the underlying theoretical framework.
