[113] An Introduction to Equivariant Cohomology

Chapter
Part of the Contemporary Mathematicians book series (CM)

Abstract

It gives me great pleasure to start off this “anniversary” meeting, even if by default.

The 1970 summer school here at Les Houches was a very memorable event for me and my whole family and it is quite wonderful and magical to see, among the many young and eager faces, so many friends of old. In the nearly 30 years since 1970, I am very pleased to report that the dialogue between physiCs and mathematics has increased dramatically, and, on the mathematical side at least, this interaction has been highly productive. The Yang-Mills theory spawned the Donaldson invariants of four-manifolds, later to be augmented by the Seiberg- Witten invariants. Current algebras engendered a new and beautiful representation theory of Loop groups and Kac- Moody algebras. Knot theory has been reinvigorated by the “ChernSimons Theory,” while 19th century questions in algebraic geometry have been solved by methods initiated by “String Theory.” And, on a personal note, I might add that nowadays it is usually the Physics graduate students who are the stars in my geometry courses.

References

  1. Atiyah, M.F. and Bott, R. ( 1984). The moment map and equivariant cohomology, Topology, 23, 1–28.MathSciNetCrossRefMATHGoogle Scholar
  2. Berline, N., Getzler, E. and Vergne M. Heat Kernels and Dirac Operators. Springer-Verlag, 1992.Google Scholar
  3. Berline, N. and Vergne, M. (1982), Classes caractéristiques équivariantes. Formules de localisation en cohomologie équivariante. C.R. Acad. Sci. Paris, 295, 539–541.MathSciNetMATHGoogle Scholar
  4. Bott, R. (1967). Vector fields and characteristic numbers, Mich. Math. Journal, 14, 231–244.MathSciNetCrossRefMATHGoogle Scholar
  5. Bott, R. (1967). A residue formula for holomorphic vector fields, J. of Differential Geometry, 4, 311–312.MathSciNetCrossRefMATHGoogle Scholar
  6. Cartan, H.(1950). La transgression dans un groupe de Lie et dans un fibreé principal. In Colloque de Topologie, C.B.R.M., Brussels, 57–71.Google Scholar
  7. Danilov, V.I. (1978). The geometry of toric varieties, Russian Math. Surveys, 33, 2, 97–154.MathSciNetCrossRefMATHGoogle Scholar
  8. Duflo, M., Heckman, G. and Vergne, M.(1984). Projection d’orbites, formule de Kirillov et formule de Blattner, Mém. Soc. Math. France, 15, 65–128, suppl. au. Bull. Soc. Math. France, 112.Google Scholar
  9. Duistermaat, J.J. and Heckmann, G.J. (1982). On the variation in the cohomology of the symplectic form of the reduced phase space, Invet. Math., 69, 259–268.MathSciNetCrossRefMATHGoogle Scholar
  10. Duistermaat, J.J. and Heckman, G.J. (1983). Addendum to“On the variation in the cohomology of the symplectic form of the reduced space,”Invent. Math., 72, 153–158.Google Scholar
  11. Ginzburg, V.A. (1987). Equivariant cohomology and Kahler geometry, Funct. Anal. and its Appl., 21, 271–283.CrossRefMATHGoogle Scholar
  12. Gotay, M.J. On coisotropic embeddings of presymplectic manifolds, Proc. Amer. Math. Soc., 84, 111–114.Google Scholar
  13. Guillemin, V., Lerman, E. and Sternberg S. On the Kostant multiplicity formula, J. Geom. Phys., 5, 721–750.Google Scholar
  14. Guillemin, V. and Prato, E. (1990). Heckman, Kostant and Steinberg formulas for symplectic manifolds, Advances in Math., 82, 160–179.MathSciNetCrossRefMATHGoogle Scholar
  15. Cloward, R. A., & Ohlin, L. E. (1960). Delinquency and opportunity: A theory of delinquent gangs. Glencoe: Free Press.Google Scholar
  16. Harish-Chandra. (1956). Invariant differential operators on a semi-simple Lie algebra, Proc. Nat. Acad. Sci. U.S.A., 42, 252–253. Collected Papers II, 231-232.Google Scholar
  17. Jeffrey, L.C. and Kirwan, F.C. (1993). Localization for nonabelian group actions. Technical report, Balliol College, Oxford.Google Scholar
  18. Kalkman, J. (1993). Cohomology rings of symplectic quotients. Technical Report 795, Mathematisch Instituut, Universiteit Utrecht.Google Scholar
  19. Kirwan, F.C. (1984). Cohomology of Quotients in Symplectic and Algebraic Geometry. Princeton University Press.Google Scholar
  20. Marie, C.-M. (1985). Modele d’action hamiltonienne d’un groupe de Lie sur une variété symplectique. Rendiconti del Seminario Matematico, Université e Politechnico, Torino, 43 227–251.MathSciNetGoogle Scholar
  21. Ness, L. (1984). A stratification of the null cone via the moment map, Amer. J. Math., 106, 1281–1329.MathSciNetCrossRefMATHGoogle Scholar
  22. Satake, I. (1956). On a generalization of the notion of manifold. Proc. Nat. Acad. Sci., 42, 359–363.MathSciNetCrossRefMATHGoogle Scholar
  23. Witten, E. (1992). Two-dimensional gauge theories revisited, J. Geom. Phys, 9, 303–368.MathSciNetCrossRefMATHGoogle Scholar
  24. Wu, S. (1992). An integration formula for the square of moment maps of circle actions. Technical Report hep-th/9212071, Department of Mathematics, Columbia University.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  • R. Bott
    • 1
  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

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