Degenerate Microlocal Methods in Geometric Analysis

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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.