Cycles, Nonlinear Differential Equations, and Geometric Pluripotential Theory

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AbstractAward: DMS 1301804, Principal Investigator: H. Blaine LawsonThese projects are concerned with the study of cycles, nonlinear partial differential equations, and geometric generalizations of pluripotential theory. The first project concerns fully nonlinear differential equations in riemannian geometry. Recent work of the principal investigator and R. Harvey on the Dirichlet problem will be continued, and questions concerning singularities and tangents to solutions will be investigated. Motivation for this study came from the investigators' development of pluripotential theory in calibrated and other geometries, where notions of plurisubharmonic functions, pseudo-convex domains, capacity, etc. were introduced and many basic properties established. This should have an important impact in calibrated geometry, which in turn plays an important role in M-theory in modern physics. There should also be applications to symplectic geometry and to p-convexity in Riemannian geometry. The second part of the proposal concerns the groups of algebraic cycles and cocycles on a projective algebraic variety. Here the aim is to understand these groups and relate them to the global structure of the variety itself. The investigator has, with others, established a theory of homology type for algebraic varieties based on the homotopy groups of cycles spaces. This theory will be applied to concrete questions about algebraic spaces, and implications for real algebraic geometry will be explored. Striking connections to universal constructions in topology which emerged in prior research will also be probed. A strongly related part of the proposal concerns cycles which bound holomorphic chains in projective manifolds. Characterizations in terms of projective linking numbers and quasi-plurisubharmonic functions will be studied. This will involve analyzing the structure of projective hulls, a concept analogous to polynomial hulls, which has been introduced by the investigator and is of independent interest. Projective hulls are related to approximation theory, pluripotential theory, and the spectrum of Banach graded algebras. The final area concerns analytic approaches to differential characters, and their generalizations, developed by the PI. The central objects mediate between cycles and smooth data. In the complex category this involves an analytic study of Deligne cohomology. It yields invariants for bundles and foliations, and retrieves the classical Abel-Jacobi mappings. Variational methods will be brought to bear on the study and connect it to the 1-Laplacian and the theory of minimal hypersurfaces. A concept of central importance in geometry is that of a "cycle." In algebraic geometry a cycle corresponds to the simultaneous solution of a system of polynomial equations. In differential geometry cycles arise in many ways: as the large scale solutions of certain differential equations, and as the level sets and singularity sets of differentiable mappings. Curves and surfaces in space are simple examples. Cycles with a particular geometry (discovered by the investigator and his collaborators) play a fundamental role in modern physical theories This project will also be concerned with graduate student development. Students will be part of the research team. There will also be an undergraduate educational effort aimed at fostering mathematical independence.