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.

摘要奖项：DMS-0404766 首席研究员：H. Blaine Lawson, Jr. 该项目涉及循环、留数、边界和微分特征的研究。 该提案有几个相互关联的部分。 第一个涉及射影簇 $X$ 上的代数环和余环群。 目的是将这些群体与 $X$ 的全局结构联系起来。 研究者与其他人基于循环空间的同伦群建立了代数簇的同调型理论。该理论将用于研究有关代数空间的具体问题。 将探讨对实代数几何的影响。 还将研究先前研究中出现的与拓扑中普遍结构的显着联系。该提案的第二部分涉及将复杂子类型限制在射影流形中的循环。 一些新的猜想将这些循环与近似理论、多能理论和巴纳赫分级代数的射影谱联系起来。 该提案的第三个领域涉及奇点和特征形式的研究。本主题包括 Chern-Weil 理论的推广，该理论给出了丛图奇点和特征形式之间的规范同源性。它包括一个有用的分析工具——几何原子性——将被研究，并且它产生了莫尔斯理论的新方法。将奇点与全局几何相关的应用仍有待研究。该提案的第四部分涉及火花和火花复合体。这个最近开发的微分特征研究框架产生了有趣的概括，扩展了德利涅上同调和算术 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。提出了该理论的进一步发展及其在循环研究中的应用。第五个领域涉及几何中的特殊循环，特别是 Calabi-Yaumanifolds 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凯莱循环。后面这些主题涉及镜像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。该项目还将关注学生的发展，包括旨在培养数学独立性和开发交互式环境的本科生教育工作。几何学中最重要的概念是“循环”。 在代数几何中，循环对应于多项式方程组的联立解。微分几何它们以多种方式出现：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。 具有特定几何形状的循环在现代物理理论中也发挥着重要作用。该提案涉及跨广泛范围的循环研究。在代数环境中，循环与其周围空间的基本大规模几何相关。这一发现揭示了代数环空间与代数拓扑基本构造之间令人惊讶且重要的关系，并为这两个领域带来了新的见解。 这项工作将继续进行。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循环。 它们代表了分析中经典边值问题的非线性版本。此类问题在许多情况下都会出现。最近，提议者提出了将此类循环的某些重要类别与逼近论和巴拿赫代数中的问题相关联的猜想。成功的解决方案应该会在几个数学领域产生重大的新见解。第三个研究领域涉及提议者开发的数学装置，用于检测周期与其所居住的空间的整体结构之间的微妙关系。该装置包含一些历史上开发的最有效的工具这个目的，而且更为普遍。该理论及其应用的进一步发展将被追求。第四个研究领域涉及几何中的特殊循环：卡拉比-丘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凯莱循环。这些后面的科目涉及物理学中的规范场理论和引力，该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。 还将开展本科生教育工作，旨在培养数学独立性和开发互动环境。