Our mathematical research centers around the following problems in partial differential equations of both theoretical and practical interest. The first part is concerned with the regularity of solutions for the sum of squares operator, or sublaplacian. We will investigate necessary and sufficient conditions for the local and global analytic regularity of the sublaplacian when the bracket condition is satisfied. Moreover, in the absence of the bracket condition we will study the weaker properties of global analytic and smooth regularity of the sublaplacian on the torus. Also we will consider regularity problems for CR structures, and principal type operators. The second part is concerned with the study of the Cauchy problem for nonlinear partial differential equations of shallow water type under low regularity initial data using harmonic analysis techniques. In particular we plan to investigate the local and global well-posedness of the initial value problem for the completely integrable Camassa-Holm equation. And finally, in the third part we plan to show that the exponential map of an appropriate metric on the group of volume--preserving diffeomorphisms of a compact riemannian manifold is a nonlinear Fredholm map of index zero. This would be significant because, as is well known, geodesics of this metric correspond to solutions of the Euler equations of hydrodynamics.
Partial differential equations is a many-faceted subject. Our understanding of the fundamental processes of the natural world is based to a large extent on partial differential equations. For example, the equations considered in the first part of our proposal arise in the diffusion of chemicals, in the spread of heat, and in many other physical processes influenced by diffusion. Moreover diffusion processes are closely connected to random (Brownian) motions. The synthesis of the theory of these diffusion partial differential equations and probability provides the theoretical framework for studying problems where diffusion and randomness are present. An example outside mathematics and physics is the field of Finance, where powerful techniques from stochastic analysis have been brought to bear on almost all aspects of mathematical finance: pricing of financial derivative products such as options and bonds, hedging, interest rates and so on. The equations in the second and third parts arise from problems in hydrodynamics. They are mathematical models describing fluid flow. The study of these equations will contribute in our understanding of wave formation, the structure of their singularities, and their long time behavior. It may also contribute in the very big problem of understanding turbulence. At the same time the theory for studying these partial differential equations problems forms a vast subject that interacts with many other branches of mathematics, such as complex analysis, differential geometry, harmonic analysis, probability, and mathematical physics.
