Cycles, Differential Characters and Global Problems in Geometry

几何中的循环、微分特征和全局问题

• 批准号: 0102525
• 负责人: H. Blaine Lawson
• 金额: $ 32.97万
• 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: 2001
• 资助国家: 美国
• 起止时间: 2001-06-15 至 2004-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0102525&HistoricalAwards=false
• 关键词: Cycles; Differential; Characters; Global; Problems

Abstract for DMS - 0102525 (Blaine Lawson)This project is concerned with global problems in geometry and inparticular with the study of cycles residues and differential characters.It focuses on the relationship between certain important families of cyclesin a space and the geometry of the space itself. Of particular interestare algebraic cycles and the cycles associated to singularities of mappingsor the higher order contact of geometric structures. These objects -- ofimportance in themselves -- have been shown to have ties to other areas ofmathematics. A major aim here is the discovery and development of suchties. The proposal has several interrelated parts. The first concerns groupsof algebraic cycles and cocycles on a projective variety. A theory ofhomology-type based on cycles has been developed by the proposer andothers. It will be used to study concrete questions about algebraicspaces. In a variant of the theory involving real algebraic cycles,surprizing connections to equivariant homotopy theory have been found. The implications for real algebraic geometry will be explored, and thequaternionic analogues will be studied. A second part of the proposal concerns differential characters, objects which mediate between cycles and smooth data, and lead to importantgeometric invariants. Recent discoveries have been made concerning them --for example, the existence of a fundamental duality theorem. Furtherdevelopment of the theory is proposed. Geometric results will be sought bybringing the calculus of variations to bear in this domain. A third area of the proposal concerns the study of singularities and characteristic forms. The subject includes a generalization of Chern-Weil theory which gives canonical homologies between singularities of bundle maps and characteristic forms. Many applications concerning the globalgeometry of singularities, and its relation to characteristic classes anddifferential characters, will be investigated. A forth area is concerned with special cycles in geometry: Special Lagrangian cycles in Calabi-Yau manifolds, and associative and Cayley cycles in G(2) and Spin(7) spaces. These latter subjects relate to gaugefield theory and gravity in Physics as well as many areas of geometry andalgebra. A concept of central importance in geometry is that of a ``cycle''.In algebraic geometry a cycle corresponds to the simultaneous solution of asystem of polynomial equations. In differential geometry cycles arise inmany 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. This proposal isconcerned with the study of certain important classes of cycles which arisein geometry. Part of the study aims at relating them to fundamentallarge-scale geometry of the surrounding space. In the algebraic case thishas led to the establishment of surprizing and important relationshipsbetween spaces of algebraic cycles and fundamental constructions inalgebraic topology that have led to new insights in both fields. This workwill be continued with the intent of obtaining further concreteapplications. A second part of the proposal concerns differential characters,objects which mediate between cycles and smooth data. They lead toimportant geometric invariants and have appeared in discussions of the``Mirror Symmetry Conjecture'' from modern physics. The proposer has madesome recent discoveries about characters, including a basic DualityTheorem. Further development of the theory and its applications isproposed. Another area of investigation is concerned with relationsbetween cycles and geometry which arise from connections. Connections arefundamental in mathematics, where they constitute differentiation laws, andin physics, where they represent the fundamental forces of nature at theclassical level. The investigator has developed a theory of singular connections whichencompasses much previously unrelated phenomena and has applications tomany areas of geometry. The proposal will continue this work withemphasis on applications. Yet another area of the proposal is concernedwith very special cycles in geometry which relate to gauge field theory andgravity in Physics as well as many areas of geometry and algebra. This project will also be concerned with graduate student development.Students will be part of the research team. There will also be anundergraduate educational effort aimed at fostering mathematicalindependence and developing interactive environments.

DMS 摘要 - 0102525（Blaine Lawson）该项目关注几何中的全局问题，特别是循环留数和微分特征的研究。它重点关注空间中某些重要的循环族与空间本身的几何形状之间的关系。 特别令人感兴趣的是代数循环和与映射奇点或几何结构的高阶接触相关的循环。 这些对象本身就很重要，已被证明与其他数学领域有联系。 这里的一个主要目标是发现和开发此类物质。该提案有几个相互关联的部分。第一个涉及射影簇上的代数环和余环群。 提出者和其他人已经发展了基于循环的同调型理论。 它将用于研究有关代数空间的具体问题。 在涉及实代数循环的理论变体中，发现了与等变同伦理论的令人惊讶的联系。 将探讨对实代数几何的影响，并研究四元数类似物。该提案的第二部分涉及微分特征，即在循环和平滑数据之间进行调解的对象，并导致重要的几何不变量。最近对它们有了一些发现——例如，基本对偶定理的存在。 提出了该理论的进一步发展。 通过将变分计算应用于该领域，可以寻求几何结果。 该提案的第三个领域涉及奇点和特征形式的研究。该主题包括 Chern-Weil 理论的概括，该理论给出了束映射的奇点和特征形式之间的规范同源性。 将研究有关奇点的全局几何及其与特征类和微分特征的关系的许多应用。 第四个领域涉及几何中的特殊循环：卡拉比-丘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。后面这些科目涉及物理学中的规范场理论和重力以及几何和代数的许多领域。 几何中最重要的概念是“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，循环以多种方式出现：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。 该提案涉及几何中出现的某些重要循环类别的研究。 该研究的一部分旨在将它们与周围空间的基本大尺度几何联系起来。在代数案例中，这导致了代数环空间与代数拓扑中的基本构造之间令人惊讶且重要的关系的建立，从而在这两个领域带来了新的见解。 这项工作将继续进行，以获得进一步的具体应用。该提案的第二部分涉及差异特征，即在循环和平滑数据之间进行调解的对象。它们导致了重要的几何不变量，并出现在现代物理学“镜像对称猜想”的讨论中。提议者最近对字符有了一些发现，包括基本的对偶定理。提出了该理论及其应用的进一步发展。 另一个研究领域涉及由联系产生的循环和几何之间的关系。 连接在数学中是基础，它们构成了微分定律；在物理学中，它们代表了经典层面上的基本自然力。 研究人员发展了一种奇异联系理论，它涵盖了许多以前不相关的现象，并应用于几何学的许多领域。 该提案将继续这项工作，重点是应用程序。该提案的另一个领域涉及几何中非常特殊的循环，这些循环与物理学中的规范场理论和引力以及几何和代数的许多领域相关。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。 还将开展本科教育工作，旨在培养数学独立性和开发互动环境。