Cycles, Residues & Global Problems in Geometry

摘要提案：DMS 9802054主要研究者：H. B。小劳森M.- L.这个多部分的项目是关于几何中的循环和剩余的研究。 第一部分是关于射影簇X上的代数圈群和上圈群，目的是将这些群与X的整体结构联系起来。 建立了基于循环空间同伦群的代数簇的同调型理论，并将用于研究代数空间的具体问题。 该建议的第二部分涉及投射空间中的圈，它与代数拓扑中的基本结构有着密切的联系。 由此产生的一些问题涉及空间的真实的和四元数周期有关的特征类和代表性理论。 另一些则涉及有限群作用下的圈。 在这里，空间导致了新的等变上同调理论，其发展和应用将被探讨。 第三个研究领域涉及奇异联系和特征流，这是经典陈-韦尔理论的推广，该理论以规范分析的方式将映射的奇异性与特征形式联系起来;该理论的应用和发展包括对莫尔斯理论的新方法。 后半部分涉及几何学中的校准循环：Calabi-Yau流形中的特殊拉格朗日循环，与p-Kaehler空间存在性相关的循环，以及M-膜理论中出现的循环。 这个项目也关注研究生的发展，特别是研究生之间在研究水平上的互动。这个项目关注几何中的全局结构问题，有几个相互关联的部分。 第一个目的是为了促进我们对空间的理解，这些空间是作为代数方程组（所谓的“代数圈”）的解而出现的。 这些空间有着悠久的历史，并在数学、应用数学和物理学的许多领域发挥着核心作用。 过去十年的突破性进展使人们对这一主题有了新的认识，并形成了一个结构丰富的理论。 该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问题。 研究的另一个领域是关于循环和几何之间的关系，它们是由连接产生的。 联系在数学中是基本的，在数学中，它们构成微分定律;在物理学中，它们代表经典层次上的基本自然力。 研究人员已经发展出一种奇异联系理论，它包含了许多以前不相关的现象，并已应用于几何学的几个领域。 本项目将继续这项工作，重点是应用。 在研究最小面积问题时，一位研究者发展了一种标定周期理论，它在物理理论中起着重要的作用. 这种新的关系提出了一些重要的问题和建议，将进行研究。

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.