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首席研究员：安德鲁·索洛作为1999年活动的一部分，标志着拉乌尔博特在哈佛正式教学生涯的结束，以及阿蒂亚-博特定理或伍兹霍尔定理的陈述和证明35周年，会议将于1999年10月8日至10日在伍兹霍尔海洋研究所举行。志村的原始猜想是一个扩展的莱夫谢茨不动点定理的holomorphicmaps和投影品种，宣布在1964年夏季研究所代数几何在伍兹霍尔。博特和阿蒂亚意识到，应该有一个通用的公式来保存任何椭圆算子，或者任何椭圆复形，定理形成了。他们的结果和指数定理巩固了现代几何、算术和物理之间的关系。拉乌尔博特将给出一个方法forprove定理和推广，激励当前的研究博览会。其他讲座将涵盖指数理论中伪微分算子的发展，这使得问题的分析和几何之间的桥梁成为可能。例子将包括非经典的，如接触流形上伪微分算子的海森伯代数。对某些Fredholm算子和一类由正切丛的齐次辛自同构定义的Fourier积分算子，得到了Atiyah-Singer公式.其他讲座将解释使用一个基本的莫尔斯理论，以获得自然莫尔斯函数的模空间，连接的阿蒂亚-博特定理与物理，并应用莱夫谢茨公式的方法来theHarish-Chandra理论的群表示。在此之前，世纪有没有明确的界限之间的物理和数学。 这两个领域之间的分歧加速了，到了20世纪50年代，几乎没有什么共同之处，尽管物理学家们越来越善于使用他们的理论所必需的数学。 自20世纪70年代以来，在空间和物质的基本理论方面，数学和物理学出现了惊人的融合。 Atiyah-Bott不动点定理，或称伍兹霍尔定理，是20世纪60年代数学中的一个里程碑式的结果。这一定理在数学问题中有大量的应用，是一个活跃的研究领域。考虑目前的工作和未来的可能性所建议的阿蒂亚-博特定理，会议计划于1999年秋季，在庆祝这项工作和事业的拉乌尔博特。本研究对物理和数学共同问题的影响将被探讨。