Degenerate Microlocal Methods in Geometric Analysis

摘要奖：DMS-9974395首席研究员：安德鲁·索洛作为1999年活动的一部分，纪念拉乌尔·博特在哈佛的正式教学生涯的结束，以及Atiyah-Bott或伍兹霍尔定理发表和证明35周年，会议将于1999年10月8-10日在伍兹霍尔海洋研究所举行。Shimura最初的猜想是1964年伍兹霍尔代数几何夏季研究所宣布的关于全纯映射和射影簇的Lefschetz不动点定理的推广。博特和阿提亚意识到，应该有一个通用的公式来保持任何椭圆算子，或者实际上是任何椭圆复数，这个定理就形成了。他们的结果和指数定理奠定了几何、算术和物理之间现代关系的基础。拉乌尔·博特将阐述证明这一定理的方法和推动当前研究的概括。其他讲座将讨论指数论中拟微分算子的发展，这使得分析和问题几何之间的桥梁成为可能。例子将包括非经典的，如接触流形上的拟微分算子的海森堡代数。得到了某些Fredholm型算子的Atiyah-Singer公式，以及由切丛的齐次辛自同构定义的一类傅立叶积分算子的Atiyah-Singer公式。其他讲座将解释如何使用初等莫尔斯理论来推导模空间上的自然莫尔斯函数，Atiyah-Bott定理与物理的联系，以及Lefschetz公式方法在群表示的Harish-Chandra理论中的应用。在本世纪之前，物理和数学之间没有明显的界限。这两个领域之间的分歧加速了，到了50年代的S，它们几乎没有共同点，尽管物理学家们已经变得越来越善于使用他们的理论所需的数学知识。自20世纪70年代S以来，数学和物理在空间和物质的基本理论上出现了非凡的融合。Atiyah-Bott不动点定理，或称Wood Hole定理，是上世纪60年代S提出的一个里程碑式的数学成果，它在数学问题中得到了广泛的应用，是一个活跃的研究领域。为了考虑目前的工作和Atiyah-Bott定理提出的未来可能性，计划在1999年秋季召开一次会议，以庆祝这项工作和Raul Bott的照顾者。我们将探讨这项研究对物理和数学中常见问题的影响。

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.