Cycles, Differential Characters and Global Problems in Geometry

DMS - 0102525（Blaine Lawson）的摘要本项目关注几何学中的整体问题，特别是圈、剩余和微分特征的研究，重点是空间中某些重要的圈族与空间本身几何学之间的关系。 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。 这些物体本身就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种生物。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群和上圈群。 提出者等人发展了一种基于循环的同系型理论。 它将被用来研究有关代数空间的具体问题。 在涉及真实的代数圈的理论的一个变体中，已经发现了与等变同伦理论的联系。将探索对真实的代数几何的影响，并研究四元数类似物。第二部分的建议涉及差分字符，对象之间的调解周期和光滑的数据，并导致importantgeometric不变量。最近的发现已经取得了关于他们-例如，存在一个基本的对偶定理。 提出了该理论的进一步发展。 几何结果将寻求通过使变分法承担在这一领域。 该建议的第三个领域涉及奇点和特征形式的研究。这门课包括陈-韦伊理论的推广，它给出了丛映射的奇点和特征形式之间的规范同调。 许多应用有关的整体几何的奇点，以及它的关系，特征类和微分特征，将被调查。 第四个领域涉及几何中的特殊圈：Calabi-Yau流形中的特殊拉格朗日圈，G（2）和Spin（7）空间中的结合圈和凯莱圈。这些后面的科目涉及规范场理论和重力在物理学以及许多领域的几何和代数。 几何学中一个重要的概念是“圈”。在代数几何中，圈相当于多项式方程组的同时求解。微分几何中 周期以多种方式出现：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。 这个建议是关于研究几何中出现的某些重要的圈类。 部分研究旨在将它们与周围空间的基本大尺度几何学联系起来。在代数的情况下，这导致了代数圈空间和代数拓扑中的基本结构之间的联系的建立，从而导致了这两个领域的新见解. 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，即在周期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的“镜像对称猜想”的讨论中。这位提出者最近有一些关于特征的发现，包括一个基本的对偶定理。提出了理论和应用的进一步发展。 研究的另一个领域是关于由连接引起的循环和几何之间的关系。 联系在数学中是基本的，它们构成了微分定律，在物理学中，它们代表了经典水平上的基本自然力。 这位研究者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，并可应用于几何学的任何领域。 该提案将继续这项工作，重点是应用。然而，另一个领域的建议是关注非常特殊的周期几何关系到规范场理论和重力在物理学以及许多领域的几何和代数。这个项目也将关注研究生的发展。学生将成为研究团队的一部分。 也将有一个本科教育的努力，旨在培养学生的独立性和发展互动的环境。

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.