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.

对DMS-0102525(Blaine Lawson)来说，这个项目是关于几何中的整体问题，特别是关于圈、剩余和微分特征的研究。它集中在空间中某些重要的圈族与空间本身的几何之间的关系。特别感兴趣的是代数圈和与映射的奇点或几何结构的高阶接触有关的圈。这些物体本身很重要，已经被证明与数学的其他领域有联系。这里的一个主要目标是发现和发展这样的东西。该提案有几个相互关联的部分。第一个是关于射影簇上的代数圈和上圈群。作者和其他人发展了基于圈的同调类型理论。它将被用来研究关于代数空间的具体问题。在涉及实代数圈的理论的一个变体中，发现了与等变同伦理论的惊人联系。我们将探索实代数几何的含义，并研究四元数的类似物。该建议的第二部分涉及差异字符，即在周期和平滑数据之间起中介作用的对象，并导致重要的几何不变量。最近已有关于它们的发现--例如，存在一个基本的对偶定理。并对该理论的进一步发展进行了展望。几何结果将通过将变分演算应用于该领域来寻求。提案的第三个领域涉及奇点和特征形式的研究。这个主题包括了Chern-Weil理论的推广，该理论给出了丛映射的奇点和特征形式之间的典范同调。本课程将研究有关奇点的整体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：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。后者涉及物理中的规范场理论和重力，以及几何和代数的许多领域。在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，循环以多种方式出现：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。这一建议涉及对几何中某些重要的循环类的研究。这项研究的一部分旨在将它们与周围空间的基本大尺度几何图形联系起来。在代数的情况下，这导致了代数圈空间和代数拓扑的基本结构之间惊人而重要的关系的建立，从而在这两个领域都有了新的见解。这项工作将继续进行，以期获得更多的具体应用。该提案的第二部分涉及差异字符，即在周期和平滑数据之间起调节作用的对象。它们导致了重要的几何不变量，并出现在现代物理学的“镜像对称猜想”的讨论中。这位提出者最近在人物方面有了一些疯狂的发现，包括一个基本的对偶定理。提出了该理论及其应用的进一步发展方向。另一个研究领域是关于圈和由连通产生的几何之间的关系。联系在数学中和物理学中都是基本的，在数学中，它们构成了微分定律，在物理学中，它们在经典水平上代表了自然的基本力量。这位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，并在几何学的许多领域都有应用。提案将继续这项工作，重点放在应用上。该提案的另一个领域是关于几何学中非常特殊的循环，这些循环与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有关。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。还将有一项旨在培养数学独立性和发展互动环境的本科生教育努力。