Cycles, Differential Characters and Global Problems in Geometry

DMS - 0102525 （Blaine Lawson）的摘要：这个项目是关于几何中的全局问题，特别是关于循环残数和微分特征的研究。它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。特别有趣的是代数循环和与映射的奇点或几何结构的高阶接触有关的循环。这些对象本身就很重要，而且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些物质。这项建议有几个相互关联的部分。第一类是关于射影簇上的代数环和环的群。提出了基于环的同型理论。它将用于研究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惊人联系。对实际代数几何的影响将被探索，四元数的类似物将被研究。该建议的第二部分涉及微分特征，即介于循环和平滑数据之间的对象，并导致重要的几何不变量。最近关于它们的发现——例如，一个基本对偶定理的存在。提出了该理论的进一步发展。在这一领域中，将运用变分法来寻求几何结果。提案的第三个领域涉及对奇点和特征形式的研究。本课题包括陈氏理论的推广，该理论给出了束映射的奇点与特征形式之间的正则同调。本文将研究奇异点的全局几何的许多应用，以及奇异点与特征类和微分特征的关系。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)空间中的结合环和Cayley循环。后面这些科目涉及到物理中的规场理论和重力，以及几何和代数的许多领域。几何中一个非常重要的概念是“循环”。在代数几何中，一个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尺度解，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。这个建议是关于在几何学中出现的某些重要的环类的研究。研究的一部分目的是将它们与周围空间的基本大尺度几何结构联系起来。在代数的情况下，这导致在代数循环空间和代数拓扑的基本结构之间建立了令人惊讶和重要的关系，这些关系在这两个领域都带来了新的见解。这项工作将继续进行，以获得进一步的具体应用。该方案的第二部分涉及差分特征，即介于循环和平滑数据之间的对象。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。提出者最近有了一些关于字符的发现，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。另一个研究领域是关于由连接产生的循环和几何之间的关系。在数学中，联系是基本的，它们构成了微分定律；在物理学中，联系在经典层面上代表了自然的基本力量。这位研究者发展了一种奇异连接理论，它涵盖了许多以前不相关的现象，并应用于许多几何领域。提案将继续这项工作，重点放在应用程序上。然而，该建议的另一个领域涉及几何中的非常特殊的循环，这些循环与物理中的规范场论和引力以及几何和代数的许多领域有关。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。还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。

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.