Degenerate Microlocal Methods in Geometric Analysis

几何分析中的简并微局部方法

• 批准号: 9971975
• 负责人: Rafe Mazzeo
• 金额: $ 20.29万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971975&HistoricalAwards=false
• 关键词: Degenerate; Microlocal; Methods; Geometric; Analysis

AbstractAward: DMS-9974395Principal Investigator: Andrew SolowAs part of the activities in 1999 marking the end of Raoul Bott'sformal teaching career at Harvard and the 35th anniversary of thestatement and proof of the Atiyah-Bott, or Woods Hole theorem, ameeting will be held October 8-10, 1999 at the Woods HoleOceanographic Institution. Shimura's original conjecture was anextension of the Lefschetz fixed point theorem for holomorphicmaps and projective varieties, announced during the 1964 SummerInstitute in Algebraic Geometry at Woods Hole. Bott and Atiyahrealized that there should be a general formula for mapspreserving any elliptic operator, or indeed any elliptic complex,and the theorem took shape. Their result and the index theoremunderpin the modern relationship between geometry, arithmetic,and physics. Raoul Bott will give an exposition of approaches forproving the theorem and generalizations that motivate currentresearch. Other lectures will cover the development ofpseudodifferential operators in index theory, which made possiblea bridge between analysis and the geometry of the problem.Examples will include non-classical ones, like the Heisenbergalgebra of pseudodifferential operators on a contactmanifold. Atiyah-Singer formulas are obtained for certainFredholm operators, and also for a class of Fourier integraloperators defined by homogeneous symplectic automorphisms of acotangent bundle. Other lectures will explain the use of anelementary Morse theory to derive natural Morse functions onmoduli spaces, connections of the Atiyah-Bott theorem withphysics, and the application of Lefschetz formula methods to theHarish-Chandra theory of group representations.Before this century there was no sharp line between physics andmathematics. The divergence between the two fields accelerated,and by the 1950's there was little in common, though physicistshad become increasingly adept at using what mathematics wasnecessary for their theories. Since the 1970's there has been anextraordinary convergence of mathematics and physics in the basictheories of space and matter. The Atiyah-Bott fixed pointtheorem, or Woods Hole theorem, is a landmark mathematical resultfrom the 1960's. This theorem has led to a great number ofapplications to problems in mathematics, and is an active fieldof research. To consider the present work and the futurepossibilities suggested by the Atiyah-Bott theorem, a meeting isplanned for the fall of 1999, in celebration of this work and thecareer of Raoul Bott. The implications of this research on theproblems common to both physics and mathematics will be explored.

Abstractaward：DMS-9974395原理研究者：安德鲁·索洛瓦斯（Andrew Solowas）在1999年的活动的一部分，标志着哈佛大学的拉乌尔·博特（Raoul Bott）形式教学生涯的终结，以及atiyah-bott的第35周年纪念日，Atiyah-Bott的证明，或者Atiyah-Bott的证明，或者Woods Hole eorem定理，Ameet定理将于10月8日至10日在1999年在Woods of woods of Woodsearcophing Insticaphitiation。 Shimura的原始猜想是Lefschetz固定点定理用于Holomorphicmaps和射影品种的固定点定理，该定理在伍兹霍尔（Woods Hole）的1964年Summerinstitute上宣布。 Bott and Atiyahrealate认为，应该有一个通用公式，以提供任何椭圆形算子或实际上任何椭圆形复合物，并且定理成形。他们的结果和指数定理定理在几何，算术和物理学之间的现代关系。 Raoul Bott将提供有关促进当前研究的定理和概括的方法。其他讲座将涵盖索引理论中的分类算子的发展，这使得分析与问题的几何形状之间的桥梁将包括非古典主义者，例如pseudodifferential operators of ContactManifold上的伪差异操作员的Heisenbergalgebra。对于某些Fredholm运算符，也可以为Atiyah-Singer公式，以及由Acotangent Bundle的均匀符号自动形态定义的一类傅立叶积分器。其他讲座将解释使用肛门摩尔斯理论来推导自然摩尔斯的功能onmoduli空间，atiyah-bott定理与物理学的联系，以及将lefschetz公式方法应用于群体代表理论。 尽管物理学家越来越熟练地使用数学是其理论所必需的，但这两个领域之间的差异加速了，到1950年代几乎没有共同点。 自1970年代以来，在空间和物质的基本理论中，数学和物理学一直存在着非方面的融合。 Atiyah-Bott固定点理论或伍兹孔定理是1960年代的具有里程碑意义的数学结果。该定理导致了数学问题的大量应用，并且是一项活跃的研究。考虑一下当前的工作和未来的占地，即1999年秋季的本计划，以庆祝这项工作和Raoul Bott的表演者。将探讨这项研究对物理和数学共有的问题的含义。