The focus of this research project is a family of nonlinear partial differential equations (PDE) that are the sub-Riemannian analogues of the mean curvature flow PDE, the minimal surface PDE, and the conformal n-Laplacian. Together with his students and his collaborators, the principal investigator will build on his previous work with Mario Bonk and Giovanna Citti in order to investigate questions of existence, regularity, and uniqueness of solutions to the equations and to explore the qualitative features of these solutions (e.g., convexity, self-similarity, asymptotic behavior). In collaboration with Bonk, Scott Pauls, and Jeremy Tyson, the principal investigator will study the differential geometry of submanifolds in a sub-Riemannian space, thereby providing the geometric background for the PDE theory. The principal investigator will also continue his work with Michael Cowling and Loredana Lanzani and study a geometric function theory/nonlinear PDE approach to the boundary regularity of biholomorphic mappings between strictly pseudoconvex sets.
The study of sub-Riemannian geometry has been driven since its origins by "real world" problems and applications. Roughly speaking, sub-Riemannian spaces are those whose structure can be viewed as a "constrained geometry": motion is possible only along a given set of directions, which changes from point to point yet nevertheless guarantees global accessibility. The scope of sub-Riemannian geometry is startlingly broad, including applications to areas as diverse and varied as the following: complex analysis, motion of robot arms and control theory, hyperbolic geometry, quantum computing, satellites, quantum mechanics, the structural function of the mammalian visual cortex. The study of partial differential equations in this setting allows one to create mathematical models of real-life situations. One then uses the models to achieve a better understanding of the systems under study and, as a result, to make accurate forecasts of their future behavior.
