Cycles, Characters and Pluripotential Theory in Calibrated Geometry

本课题的第二部分是关于循环及其边界、形式和广义多次调和函数的研究。该提案有几个相互关联的部分。第一个是关于射影簇X上的代数圈群和上循环群。目的是将这些群与X的整体结构联系起来。研究者与其他人一起，建立了基于圈空间的同伦群的代数簇的同调型理论。这一理论将被用来研究有关代数空间的具体问题。本课程将探讨实代数几何的含义。拓扑学中出现的与普遍结构的惊人联系也将被调查。该建议的第二部分涉及限制射影流形中全纯链的圈。特别地，将寻求关于射影连接数和准多重亚调和函数的刻画。这将需要对射影壳的结构进行深入的分析，这是一个类似于多项式壳的概念，由研究人员引入，是一种独立的兴趣。射影壳与逼近理论、多位势理论以及Banach分次代数的谱有关。第三个主题是该提案的主要部分，涉及校准几何和其他几何中多势理论的广泛发展。我们将在非常一般的背景下研究广义Monge-Amp方程的Dirichlet问题的复次调和函数、伪凸域、容量和解的概念。这个已经在进行中的项目应该会对校准几何学产生影响，而校准几何学反过来又在现代物理学的M理论中扮演着重要的角色。也应该应用于辛几何和黎曼几何中的p-凸性。提案的第四部分涉及火花和火花复合体。这些对象在周期和平滑数据之间起到中介作用，并给出了差异特征及其推广的具体表示。在复范畴中，这涉及到Deligne上同调的分析研究，并涉及算术Chow群。它产生丛和叶层的不变量，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发展互动环境的本科教育努力。第二部分几何中最重要的一个概念是“循环”。在代数几何中，循环对应于多项式方程组的同时解。在微分几何中，它们以多种方式出现：作为某些微分方程组的大规模解，以及作为可微映射的水平集和奇异集。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扮演着基本的角色。这个建议涉及到对这种广泛谱上的循环的研究。在代数环境中，循环与其周围空间的基本大尺度几何有关。这一发现揭示了代数圈空间和代数拓扑学中基本结构之间令人惊讶的重要关系，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圈。它们代表了分析中经典边值问题的非线性版本。这样的问题在许多情况下都会出现。提出者提出了一些猜想，这些猜想把重要的这类圈与逼近理论和Banach代数中的问题联系起来。成功的解决方案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产生重大的新见解。提案的第三部分旨在将经典的多势理论扩展到非常一般的几何环境。这些几何包括校准几何、辛几何和拉格朗日几何，以及更多。大量的经典理论已经被证明在这个一般的背景下是成立的。在这种情况下，将寻求关联的Monge-Ampere型方程的Dirichlet问题的解。在某种严格意义上，这项研究与这些几何中出现的特殊循环的研究是双重的。它适用于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Cayley循环。后两个主题与物理学中的规范场理论和引力有关。第四个研究领域涉及由提出者和R.哈维开发的一种数学仪器，用于探测周期和它们所处空间的全球结构之间的微妙关系。这种仪器包含了历史上为此目的开发的一些最有效的工具，而且它的通用性要强得多。这一理论及其应用的进一步发展将继续进行。这个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。还将有一个旨在培养数学独立性和发展互动环境的本科生教育努力。

Part IThis project is concerned with the study of cycles and their boundaries, forms, and generalized plurisubharmonic functions. 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 relate these groups 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 investigator and is ofindependent interest. Projective hulls are related to approximation theory, pluripotential theory, and the spectrum of Banach graded algebras. The third topic, a major part of the proposal, concerns the broad development of a pluripotential theory in calibrated and other geometries. The notions of plurisubharmonic function, pseudo-convex domain, capacity, and solutions to the Dirichlet problem for generalized Monge-Amp\`ere-type equations will be studied in a very general setting. This project, already underway, should have an 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 forth part of the proposal concerns sparks and spark complexes. These objects mediate betweeen cycles and smooth data, and give a concrete presentation of differential characters and their generalizations. In the complex category this involves an analytic study of Deligne cohomology and relates to arithmetic Chow groups. 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. Part II 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 theoriesThis 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 part of the proposal aims at extending 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. Solutions to the Dirichlet problem for associated Monge-Ampere type equations will be sought in this setting. 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 relate to gauge field theory and gravity in PhysicsA 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.