Cycles, Plurisubharmonic Functions and Nonlinear Equations in Geometry

本课题主要研究循环及其边界、广义多次谐波函数和非线性偏微分方程。这项建议有几个相互关联的部分。第一类是关于射影变量x上的代数环和共环群，目的是了解这些群并将它们与x的整体结构联系起来。研究者在环空间的同伦群的基础上建立了代数变量的同调型理论。这一理论将用于研究代数空间的具体问题。将探讨对实际代数几何的影响。在先前的研究中出现的与拓扑学中普遍结构的惊人联系也将被调查。第二部分讨论了在射影流形中约束全纯链的环。特别是，在射影连接数和拟多次谐波函数方面的特征将被寻求。这将需要对投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，已被研究人员引入并且具有独立的兴趣。投影壳与近似理论、多能理论和巴拿赫梯度代数谱有关。第三个主题是关于黎曼流形上的完全非线性偏微分方程的Dirichlet问题。最近在这个问题上取得了有趣的进展，研究将继续着眼于进一步的应用。本研究的动机来自研究人员在校准和其他几何中的多势理论的发展，其中引入了多次谐波函数、伪凸域、容量等概念，并证明它们具有经典复杂情况中已知的许多性质。随着新的分析发展，这一领域的更深层次的问题将得到解决。该项目的这一部分将对校准几何产生重大影响，而校准几何反过来又在现代物理学中的m理论中发挥重要作用。对于辛几何和黎曼几何中的p-凸性也应该有应用。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。这些对象介于循环和平滑数据之间。在复范畴中，这涉及到Deligne上同的解析研究。它产生束和叶的不变量，并检索经典的Abel-Jacobi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互动环境的本科教育努力。几何中一个非常重要的概念是“循环”。在代数几何中，一个循环对应于一个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就是简单的例子。具有特定几何形状的循环在现代物理理论中也起着重要作用。这个提议是关于在这个广谱范围内的周期研究。在代数环境中，周期与它们周围空间的基本大尺度几何有关。这一发现揭示了代数循环空间与代数拓扑基本结构之间惊人而重要的关系，并在这两个领域带来了新的见解。这项工作将继续下去。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。在许多情况下都会出现这样的问题。提出了与近似理论和巴拿赫代数问题有关的重要循环类的猜想。成功的解决将在复杂几何中建立一系列新的结果，并将在其他几个数学领域中产生重要的新见解。第三，也是非常重要的一部分，是关于在各种几何设置下的完全非线性偏微分方程的Dirichlet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，研究将继续着眼于进一步的应用。这项研究的动机来自于研究者将经典多能理论扩展到非常一般的几何环境。这些包括校准几何，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种大背景下是成立的。随着新的分析发展，这一领域的更深层次的问题将得到解决。在某种严格意义上，这部分研究与研究这些几何图形中出现的特殊循环是双重的。它应适用于Calabi-Yau流形中的特殊拉格朗日环，以及G(2)和Spin(7)空间中的结合环和Cayley环。后者在现代物理学的m理论中起着重要的作用。第四个研究领域涉及由提议者和R. Harvey开发的一种数学仪器，用于探测循环与它们所处空间的整体结构之间的微妙关系。这个工具包含了历史上为此目的开发的一些最有效的工具，而且它更加通用。将进一步发展这一理论及其应用。这个项目也将涉及研究生的发展。学生将成为研究小组的一员。也将有一个旨在培养数学独立性和发展互动环境的本科教育努力。

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.