This project is concerned with the study of cycles and their boundaries, generalized plurisubharmonic functions, and nonlinear partial differential equations. The proposal has several interrelated parts. The first concerns the groups of algebraic cycles and cocycles on a projective variety X. The aim is to understand these groups and relate them to the global structure of X. 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 used to study concrete questions about algebraic spaces. Implications for real algebraic geometry will be explored. Striking connections to universal constructions in topology which emerged in prior research will also be investigated. The second part of the proposal concerns cycles which bound holomorphic chains in projective manifolds. In particular, characterizations in terms of projective linking numbers and quasi plurisubharmonic functions will be sought. This will entail a deep analysis of the structure of projective hulls, a concept analogous to polynomial hulls, which has been introduced by the investigators and is of independent interest. Projective hulls are related to approximation theory, pluripotential theory, and the spectrum of Banach graded algebras. The third topic, an important part of the proposal, concerns the Dirichlet problem for fully nonlinear partial differential equations on riemannian manifolds. Interesting progress was recently made on this question and the investigation will continue with an eye to further applications. 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 shown to have many of the properties known in the classical complex case. With the new analytic developments, deeper questions in this field will be addressed. This part of the project should have a major 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 final area concerns analytic approaches to differential characters and generalizations, developed by the investigator and R. Harvey. These objects mediate betweeen 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. This project will also be concerned with student development, including an undergraduate educational effort aimed at fostering mathematical independence and developing interactive environments.
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 they 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 also play a fundamental role in modern physical theories. This proposal is concerned with the study of cycles across this broad spectrum. In the algebraic setting, cycles have been related to fundamental large-scale geometry of their surrounding space. This discovery has revealed surprizing and important relationships between spaces of algebraic cycles and fundamental constructions in algebraic topology and has led to new insights in both fields. This work will be continued. Another area of investigation concerns cycles which form the boundary of subsets with special geometric structure. They represent non-linear versions of classical boundary value problems in analysis. Such questions arise in many contexts. The proposer has formulated conjectures relating important classes of such cycles to questions in approximation theory and Banach algebras. Successful resolution will establish a series of new results in complex geometry and should lead to significant new insights in several other fields of mathematics. A third, and very important, part of the proposal concerns the Dirichlet problem (the prescribed boundary-value problem) for fully nonlinear partial differential equations in various geometric settings. Interesting progress was recently made on this question and the investigation will continue with an eye to further applications. Motivation for this study came from the investigators' extension of classical pluripotential theory to very general geometric settings. These include calibrated geometries, symplectic and Lagrangian geometries, and much more. An uncanny amount of the classical theory has already been shown to hold in this general context. With the new analytic developments, deeper questions in this field will be addressed. This part of the study is, in a certain strict sense, dual to the study of the special cycles appearing in these geometries. It should apply to Special Lagrangian cycles in Calabi-Yau manifolds, and associative and Cayley cycles in G(2) and Spin(7) spaces. These latter subjects play an important role in M-theory in modern Physics. A forth domain of investigation concerns a mathematical apparatus developed by the proposer and R. Harvey to detect subtle relationships between cycles and the global structure of the space they live in. This apparatus encompasses some of the most effective tools historically developed for this purpose, and it is much more general. Further development of this theory and its applications will be pursued. 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 and developing interactive environments.
Calibrated geometry, a subject introduced by the PI and Reese Harvey, has played a central role in mirror symmetry conjectures and in modern M-theory. It has also been of importance in riemannian geometry. Historically, research has been focused on the ``calibrated submanifolds''. These are certain special multidimensional spaces which exhibit many of the properties of solutions to complex polynomial equations. One of the principal outcomes of this project has been the development of the ``dual side'' of the theory -- a pluripotential theory on calibrated spaces.This means the study of functions whose restriction to every calibrated submanifold is classically subharmonic.These developments have made many new analytical tools available in the field. They have also led to a number of results in other areas. The main project entailed understanding certain nonlinear partial differential equations, and this part of the research turned out to be highly successful. Results concerning the existence and uniqueness of solutions to the Dirichlet Problem on geometric spaces (riemannian manifolds) were established in much greater generality than expected. These results turned out to have applications to many other areas. Long-term existence of solutions to the associated parabolic equations was also established. Furthermore, certain new techniques were introduced -- such as the notion of ``jet equivalence'' of equations -- which have been of great use in the study of problems that arise in geometric analysis. One area for which the results have been useful is that of p-convexity. This is a weakening of the standard notion of convexity for any real number p ≥ 1. It occurs naturally in geometry and analysis. A number of new results were proved in this area, including a solution of the Levi Problem. Another area in which substantial progress was made concerns almost complex manifolds. These are spaces where one has a weak (non-integrable) form of the Cauchy-Riemann equations. They play a central role in the highly active field of symplectic topology. Two major results were established. The first was a proof of the Pali Conjecture. This completed the foundations for doing analysis on these spaces. The second was the proof of existence and uniqueness of solutions to the Dirichlet Problem for the homogeneous and inhomogeneous Monge-Ampere equation on almost complex manifolds. A series of ``removable singularity'' theorems for nonlinear partial differential equations were proved. These are results which say when solutions (or subsolutions) can be extended as solutions (or subsolutions) acrosscertain kinds of subsets. Often the equations of interest in geometry can be expressed as homogeneous polynomials in the second derivatives of the function. Many times this polynomial P(A), where A is a symmetric matrix, has the property of being Garding hyperbolic with respect to the identity. For example P(A) = det(A) has this property. For this reason the investigators looked into the theory of these polynomials, and they were able to prove a new result concerning the real analyticity of the Garding eigenvalues. Some work was done on the subject of paracomplex geometry. It is an interesting fact that if one changes the complex numbers by requiring the square of ``i'' to be 1 instead of -1 (this gives the other Clifford algebra of dimension 2), an astonishing amount of complex analysis and complex geometry can be established in this setting. This includes the existence of a special Lagrangian calibration and special Lagrangian submanifolds. A survey article was written on this subject, and a new regularity theorem was proved. A long survey article was written on the subject of nonlinear partial differential equations on riemannian manifolds. Educational activities were largely concerned with the training of graduate students to do mathematical research.Three students completed their Ph.D theses under the direction of the PI. One of these was in the field of calibrations. end
