Cycles, characters and global geometry

项目负责人：H. Blaine Lawson, jr .本项目主要研究循环、残数、边界和微分特征。这项建议有几个相互关联的部分。第一类是关于射影变量X上的代数环和环的群。目的是将这些群体与$X$的全球结构联系起来。在环空间的同伦群的基础上，建立了代数变量的同伦型理论。这一理论将用于研究代数空间的具体问题。对实代数几何的影响将被探讨。在先前的研究中出现的突出连接和拓扑中的普遍结构也将被研究。该建议的第二部分涉及在投影流形中约束复子变种的环。一些新的猜想将这些环与近似理论、多势理论和Banach梯度代数的投影谱联系起来。该提议的第三个领域涉及对奇点和特征形式的研究。这一主题包括陈-魏尔理论的推广，它给出了束映射的奇点和特征形式之间的正则同调。它包括一个有用的分析工具——几何原子性——这将被研究，它产生了一个新的方法来莫尔斯理论。奇点在全局几何中的应用还有待研究。提案的第四部分涉及火花和火花复合物。这个最近发展的研究微分特征的框架已经产生了有趣的推广，它扩展了Deligne上同调和算术群。它们本质上是介于循环和平滑数据之间的次级不变量。提出了该理论的进一步发展及其在周期研究中的应用。第五个领域涉及几何中的特殊环，特别是calabi - yau流形中的特殊拉格朗日环，以及$G_2$和$ spin $_7$空间中的关联环和Cayley环。后面这些科目涉及到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互动环境的本科教育努力。几何中一个非常重要的概念是“循环”。在代数几何中，一个循环对应于一个多项式方程组的联解。在微分几何中，它们以多种方式出现：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。具有特定几何形状的循环在现代物理理论中也起着重要作用。本提案涉及对这种广谱循环的研究。在代数背景下，循环与它们周围空间的基本大尺度几何有关。这一发现揭示了代数循环空间与代数拓扑基本结构之间惊人而重要的关系，并在这两个领域带来了新的见解。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循环。它们代表了分析中经典边值问题的非线性版本。在许多情况下都会出现这样的问题。最近，作者提出了与近似理论和巴拿赫代数中的问题有关的某些重要类循环的猜想。成功的解决方案应该在数学的几个领域产生重要的新见解。第三个研究领域涉及由提议者开发的一种数学装置，用于检测循环与它们所处空间的整体结构之间的微妙关系。这个仪器包含了历史上为此目的开发的一些最有效的工具，而且它更加通用。这一理论及其应用将得到进一步的发展。第四个研究领域涉及几何中的特殊环：Calabi-Yau流形中的特殊拉格朗日环，G(2)和Spin(7)空间中的结合环和Cayley环。后两门课程涉及物理学中的规范场论和重力。本项目也将涉及研究生的培养。学生将成为研究小组的一员。也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。

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.