Cycles, Characters and Pluripotential Theory in Calibrated Geometry

本项目主要研究循环及其边界、形式和广义多次谐波函数。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，目的是将这些群与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，该概念已被研究者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。第三个主题是提案的主要部分，涉及校准几何和其他几何中多能理论的广泛发展。广义Monge-Amp\ ' e型方程的多重次谐波函数、伪凸域、容量和Dirichlet问题的解的概念将在一个非常一般的环境中进行研究。这个项目已经在进行中，应该会对校准几何产生影响，而校准几何反过来又在现代物理学中的m理论中起着重要作用。对于辛几何和黎曼几何中的p-凸性也应该有应用。提案的第四部分涉及火花和火花复合物。这些对象介于循环和平滑数据之间，并给出微分特征及其概括的具体表示。在复范畴中，这涉及到Deligne上同调的解析研究，并涉及到算术周群。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。几何中的一个重要概念是“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。本提案涉及对这种广谱循环的研究。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。提案的第三部分旨在将经典多能理论扩展到非常一般的几何设置。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。在这种情况下，将寻求相关蒙日-安培型方程的狄利克雷问题的解。在某种严格意义上，这项研究是对这些几何图形中出现的特殊周期的研究的双重研究。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。后面的这些主题涉及到规范场论和引力在物理学的第四个研究领域，涉及到由提议者和R. Harvey开发的一种数学装置，用于探测循环和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。

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.