Cycles, Residues & Global Problems in Geometry

循环、残留

• 批准号: 9802054
• 负责人: H. Blaine Lawson
• 金额: $ 22.82万
• 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: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9802054&HistoricalAwards=false
• 关键词: Cycles; Residues; & Global; Problems

AbstractProposal: DMS 9802054Principal Investigators: H. B. Lawson, Jr. and M.-L. MichelsohnThis multi-part project is concerned with the study of cycles andresidues in geometry. One part concerns the groups of algebraiccycles and cocycles on a projective variety X and aims to relate thesegroups to the global structure of X. A theory of homology type foralgebraic varieties based on homotopy groups of cycle spaces has beendeveloped, and will be used to study concrete questions aboutalgebraic spaces. A second part of the proposal concerns cycles inprojective space, which have surprizing connections to fundamentalconstructions in algebraic topology. Some of the resulting questionsconcern spaces of real and quaternionic cycles related tocharacteristic classes and representation theory. Others concerncycles under the action of a finite group. Here the spaces have ledto new equivariant cohomology theories whose development andapplication will be explored. A third area of investigation concernssingular connections and characteristic currents, a generalization ofclassical Chern-Weil theory which relate singularities of mappings tocharacteristic forms in a canonical analytic way; applications anddevelopments of the theory include a new approach to Morse Theory. Afourth area concerns calibrated cycles in geometry: special Lagrangiancycles in Calabi-Yau manifolds, cycles related to existence ofp-Kaehler spaces, and cycles appearing in M-brane theory. Thisproject is also concerned with graduate student development,especially interaction at the research level among graduate students.This project concerns questions of global structure in geometry andhas several interrelated parts. The first aims at furthering ourunderstanding of the spaces which arise as solutions of systems ofalgebraic equations (so called ``algebraic cycles''). These spaceshave a long history and play a central role in many areas ofmathematics, applied mathematics and physics. Breakthroughs over thepast ten years have given fresh insights into the subject and a richlystructured theory has emerged. The proposed research will forge newlinks between algebra and geometry/topology, and lead towards settlingsome important conjectures in the field. Another area ofinvestigation is concerned with relations between cycles and geometrywhich arise from connections. Connections are fundamental inmathematics, where they constitute differentiation laws, and inphysics, where they represent the fundamental forces of nature at theclassical level. The investigators have developed a theory ofsingular connections which encompasses many previously unrelatedphenomena and has applications to several areas of geometry. Thisproject will continue this work with emphasis on applications. Instudying the least area problem one of the investigators developed atheory of calibrated cycles which currently plays an important role inphysical theories. This new relationship has raised some importantquestions and conjectures that will be studied.

摘要建议：DMS 9802054主要调查人员：H.B.劳森，Jr.和M.-L.米歇尔逊这个由多个部分组成的项目涉及几何中的循环和剩余的研究。第一部分是关于射影簇X上的代数循环群和余循环群，目的是将这些群与X的整体结构联系起来。基于循环空间的同伦群，建立了代数簇的同调型理论，并将用它来研究代数空间的具体问题。该建议的第二部分涉及射影空间中的圈，它与代数拓扑学中的基本构造有着惊人的联系。由此产生的一些问题涉及与特征类和表示理论有关的实数和四元数圈空间。其他人则关注在有限群作用下的循环。这里介绍了新的等变上同调理论，并对其发展和应用进行了探讨。第三个研究领域涉及奇异联系和特征流，这是经典的Chern-Weil理论的推广，它以规范的分析方式将映射的奇点与特征形式联系起来；该理论的应用和发展包括对Morse理论的新方法。第四部分涉及几何中的校准循环：Calabi-Yau流形中的特殊拉格朗日循环，与p-Kaehler空间的存在有关的循环，以及M-膜理论中出现的循环。这个项目还涉及研究生的发展，特别是研究生之间在研究层面上的互动。这个项目涉及几何中的全球结构问题，有几个相互关联的部分。第一个目的是加深我们对作为代数方程组(所谓的‘’代数圈‘’)的解出现的空间的理解。这些空间有着悠久的历史，在数学、应用数学和物理的许多领域发挥着核心作用。过去十年的突破给这一主题带来了新的见解，一个结构丰富的理论已经出现。拟议的研究将在代数和几何/拓扑学之间建立新的联系，并导致解决该领域的一些重要猜想。研究的另一个领域是关于圈和几何之间的关系，这些关系是由联系产生的。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。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，并应用于几何的几个领域。该项目将继续这项工作，重点是应用。在研究最小面积问题时，一位研究人员发展了一种校准循环理论，该理论目前在物理理论中发挥着重要作用。这种新的关系提出了一些重要的问题和猜想，将予以研究。