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.

该项目涉及循环及其边界、广义多次谐波函数和非线性偏微分方程的研究。 该提案有几个相互关联的部分。 第一个涉及射影簇 X 上的代数环和余循环群。目的是理解这些群并将它们与 X 的全局结构联系起来。研究者与其他人一起建立了基于代数簇的同源类型理论。关于循环空间的同伦群。该理论将用于研究有关代数空间的具体问题。将探讨对真实代数几何的影响。还将研究与先前研究中出现的拓扑中普遍结构的显着联系。 该提案的第二部分涉及将全纯链限制在射影流形中的循环。特别是，将寻求投影链接数和准多次谐波函数方面的表征。 这将需要对射影船体的结构进行深入分析，这是一个类似于多项式船体的概念，由研究人员引入并且具有独立的兴趣。 射影壳与逼近论、多能理论和 Banach 分级代数谱相关。第三个主题是该提案的重要组成部分，涉及黎曼流形上完全非线性偏微分方程的狄利克雷问题。最近在这个问题上取得了有趣的进展，调查将继续着眼于进一步的应用。 这项研究的动机来自于研究人员在校准几何和其他几何中对多能理论的发展，其中引入了多次谐波函数、伪凸域、容量等概念，并证明其具有经典复杂情况中已知的许多属性。随着新的分析发展，该领域更深层次的问题将得到解决。 该项目的这一部分应该会对校准几何产生重大影响，进而在现代物理学的 M 理论中发挥重要作用。 还应该应用于辛几何和黎曼几何中的 p 凸性。最后一个领域涉及由研究者和 R. Harvey 开发的差异特征和概括的分析方法。 这些对象在周期和平滑数据之间进行调解。 在复数范畴中，这涉及德利涅上同调的分析研究。 它产生束和叶状结构的不变量，并检索经典的阿贝尔-雅可比映射。 该项目还将关注学生的发展，包括旨在培养数学独立性和开发交互式环境的本科生教育工作。几何学中最重要的概念是“循环”。 在代数几何中，循环对应于多项式方程组的联立解。在微分几何中，它们以多种方式出现：作为某些微分方程的大规模解，以及作为可微映射的水平集和奇点集。 空间中的曲线和曲面是简单的例子。 具有特定几何形状的循环在现代物理理论中也发挥着基础作用。该提案涉及跨广泛范围的循环研究。在代数环境中，循环与其周围空间的基本大规模几何相关。这一发现揭示了代数环空间与代数拓扑基本构造之间令人惊讶且重要的关系，并为这两个领域带来了新的见解。 这项工作将会继续下去。另一个研究领域涉及形成具有特殊几何结构的子集边界的循环。 它们代表了分析中经典边值问题的非线性版本。此类问题在许多情况下都会出现。 提议者提出了将此类循环的重要类别与逼近论和巴拿赫代数中的问题联系起来的猜想。成功的解决方案将在复杂几何中建立一系列新结果，并应在数学的其他几个领域产生重要的新见解。该提案的第三个也是非常重要的部分涉及各种几何设置中完全非线性偏微分方程的狄利克雷问题（规定的边值问题）。 最近在这个问题上取得了有趣的进展，调查将继续着眼于进一步的应用。这项研究的动机来自于研究人员将经典多能理论扩展到非常一般的几何设置。 其中包括校准几何、辛几何和拉格朗日几何等等。大量的经典理论已经被证明在这个一般背景下是成立的。 随着新的分析发展，该领域更深层次的问题将得到解决。 从某种严格意义上来说，这部分研究与对这些几何形状中出现的特殊循环的研究是双重的。 它应该适用于 Calabi-Yau 流形中的特殊拉格朗日循环，以及 G(2) 和 Spin(7) 空间中的结合循环和凯莱循环。后面这些学科在现代物理学的 M 理论中发挥着重要作用。第四个研究领域涉及由提议者和 R. Harvey 开发的数学装置，用于检测周期与其所居住空间的全局结构之间的微妙关系。该装置包含一些历史上为此目的开发的最有效的工具，并且它更为一般。该理论及其应用的进一步发展将得到进一步发展。该项目还将关注研究生的发展。学生将成为研究团队的一部分。 还将开展本科生教育工作，旨在培养数学独立性和开发互动环境。