Cycles, characters and global geometry

循环、字符和全局几何

• 批准号: 0404766
• 负责人: H. Blaine Lawson
• 金额: --
• 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: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2008-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0404766&HistoricalAwards=false
• 关键词: Cycles; characters; global; geometry

AbstractAward: DMS-0404766Principal Investigator: H. Blaine Lawson, Jr.This project is concerned with the study of cycles, residues,boundaries and differential characters. The proposal has severalinterrelated parts. The first concerns the groups of algebraiccycles and cocycles on a projective variety $X$. The aim is torelate these groups to the global structure of $X$. Theinvestigator has, with others, established a theory of homologytype for algebraic varieties based on the homotopy groups ofcycles spaces. This theory will be used to study concretequestions about algebraic spaces. Implications for realalgebraic geometry will be explored. Striking onnections touniversal constructions in topology which emerged in priorresearch will also be investigated. A second part of the proposalconcerns cycles which bound complex subvarieties in a projectivemanifold. Several new conjectures relate these cycles toapproximation theory, pluripotential theory, and the projectivespectrum of Banach graded algebras. A third area of the proposalconcerns the study of singularities and characteristicforms. This subject includes a generalization of Chern-Weiltheory which gives canonical homologies between singularities ofbundle maps and characteristic forms. It includes a usefulanalytic tool -- geometric atomicity -- which will be studied,and it yields a new approach to Morse Theory. Applicationsrelating singularities to global geometry remain to beinvestigated. The forth part of the proposal concerns sparks andspark complexes. This recently developed framework for the studyof differential characters has yielded interestinggeneralizations which extend Deligne cohomology and arithmeticChow groups. They are essentially secondary invariants whichmediate between cycles and smooth data. Further development ofthe theory and its application to the study of cycles isproposed. A fifth area is concerned with special cycles ingeometry, in particular Special Lagrangian cycles in Calabi-Yaumanifolds, and associative and Cayley cycles in $G_2$ andSpin$_7$ spaces. These latter subjects relate to mirror symmetryconjectures and to M-theory in Physics as well as many areas ofgeometry and algebra. This project will also be concerned withstudent development, including an undergraduate educationaleffort aimed at fostering mathematical independence anddeveloping interactive enviornments.A concept of central importance in geometry is that of a``cycle''. In algebraic geometry a cycle corresponds to thesimultaneous solution of a system of polynomial equations. Indifferential geometry they arise in many ways: as the large scalesolutions of certain differential equations, and as the levelsets and singularity sets of differentiable mappings. Curves andsurfaces in space are simple examples. Cycles with a particulargeometry also play a fundamental role in modern physical theoriesThis proposal is concerned with the study of cycles across thisbroad spectrum. In the algebraic setting cycles have been relatedto fundamental large-scale geometry of their surroundingspace. This discovery has revealed surprizing and importantrelationships between spaces of algebraic cycles and fundamentalconstructions in algebraic topology and has led to new insightsin both fields. This work will be continued.Another area of investigation concerns cycles which form theboundary of subsets with special geometric structure. Theyrepresent non-linear versions of classical boundary valueproblems in analysis. Such questions arise in many contexts.Recently the proposer has formulated conjectures relating certainimportant classes of such cycles to questions in approximationtheory and Banach algebras. Successful resolution should producesignificant new insights in several fields of mathematics.A third area of study concerns a mathematical apparatus developedby the proposer to detect subtle relationships between cycles andthe global structure of the space they live in. This apparatusencompasses some of the most effective tools historicallydeveloped for this purpose, and it is much more general. Furtherdevelopment of this theory and its applications will be persued.A fourth domain of investigation is concerned with special cyclesin 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 gauge field theory and gravity inPhysicsThis project will also be concerned with graduate studentdevelopment. Students will be part of the research team. Therewill also be an undergraduate educational effort aimed atfostering mathematical independence and developing interactiveenvironments.

摘要奖：DMS-0404766首席研究员：H·布莱恩·劳森，Jr。这个项目涉及循环、残基、边界和微分特征的研究。该提案有几个相互关联的部分。第一个是关于射影簇$X$上的代数圈群和上圈群。其目的是将这些集团与美元X$的全球结构联系起来。作者与其他人建立了基于圈空间同伦群的代数簇的同调类型理论。这个理论将被用来研究关于代数空间的具体问题。我们将探讨实代数几何的含义。在先前的研究中出现的拓扑学中的普遍结构的惊人联系也将被调查。该建议的第二部分涉及限制射影流形中复数亚簇的圈。几个新的猜想将这些环与逼近理论、多位势理论和Banach分次代数的投影谱联系起来。该提议的第三个领域涉及奇点和特征形式的研究。这个主题包括了Chern-Weil理论的推广，该理论给出了丛映射的奇点和特征形式之间的典范同调。它包括一个有用的分析工具--几何原子性--将被研究，它产生了一种新的莫尔斯理论的方法。将奇点与整体几何联系起来的应用仍有待研究。提案的第四部分涉及火花和火花复合体。这个最近发展起来的研究差异特征标的框架已经产生了有趣的推广，它扩展了Deligne上同调和算术Chow群。它们本质上是介于循环和平滑数据之间的次要不变量。展望了该理论的进一步发展及其在周期研究中的应用。第五个领域涉及特殊循环几何，特别是Calabi-Yaumanifold中的特殊拉格朗日循环，以及$G_2$和Spin$_7$空间中的结合循环和Cayley循环。后者涉及镜面对称猜想和物理学中的M理论，以及几何和代数的许多领域。这个项目还将关注学生的发展，包括旨在培养数学独立性和发展互动环境的本科教育努力。几何学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，它们以多种方式产生：作为某些微分方程解的大尺度，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。具有特殊几何结构的循环在现代物理理论中也扮演着基本的角色。这一建议涉及到跨这个广谱的循环的研究。在代数环境中，循环与其周围空间的基本大尺度几何有关。这一发现揭示了代数圈空间与代数拓扑学中基本结构之间惊人而重要的关系，并对这两个领域有了新的认识。这项工作将继续下去。另一个研究领域涉及形成具有特殊几何结构的子集边界的圈。它们代表了分析中经典边值问题的非线性版本。最近，提出者提出了一些猜想，将某些重要的循环类与逼近理论和Banach代数中的问题联系起来。成功的解决应该会在几个数学领域产生重要的新见解。第三个研究领域涉及由提出者开发的数学仪器，用于检测周期和它们所居住的空间的全球结构之间的微妙关系。这个装置包含了历史上为此目的开发的一些最有效的工具，而且它要通用得多。第四个研究领域涉及几何学中的特殊循环：Calabi-Yau流形中的特殊拉格朗日循环，以及G(2)和自旋(7)空间中的结合循环和Cayley循环。这些后一学科与物理学中的规范场理论和重力有关。本项目还将与研究生的发展有关。学生将成为研究团队的一部分。还将有一个旨在培养数学独立性和发展互动环境的本科生教育努力。