Cycles, Plurisubharmonic Functions and Nonlinear Equations in Geometry

几何中的循环、多次谐波函数和非线性方程

• 批准号: 1004171
• 负责人: H. Blaine Lawson
• 金额: $ 34.6万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1004171&HistoricalAwards=false
• 关键词: Cycles; Plurisubharmonic; Functions; Nonlinear; Equations

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.