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Moment Maps Working Seminar

Spring 2019


Organised by Ana Rita Pires and Johan Martens.

The aim of the working seminar is to discuss various results related to moment maps in symplectic geometry. All are welcome to attend - we assume a familiarity with the basics of differential geometry (manifolds and smooth actions of Lie groups), but nothing more.

The seminar meets weekly on Wednesday from 10am till 11am. Talks are given by various participants. If you wish to be kept informed please contact Johan Martens to be added to the mailing list.

Tentative Seminar Schedule (this is subject to change):

DateTopicSpeaker Location
January 23Organization and Introduction to Moment Maps Johan Martens Notes Appleton Tower 2.14
January 30Equivariant CohomologyJosé Figueroa-O'Farrill Notes Bayes 5.02
February 6Localization, Duistermaat-Heckman-Berline-Vergne Theorem Johan Martens Notes Bayes 5.45
February 13Convexity of moment maps Ana Rita Pires Bayes 5.02
February 27 (?) Symplectic Reducution and GITCarlos Zapata-Carratalá
March 6 (?) Toric symplectic manifolds and the Delzant Construction
Symplectic Blowing Up & Down
Group-valued Moment Maps & Quasi-Hamiltonian ReductionAlexander Shapiro
Moduli Spaces of Flat Unitary Connections on a Closed Surface

Suggested References:

Generalities about moment mapsAna Canas da Silva, Lectures on Symplectic Geometry.
Equivariant CohomogyAtiyah, M. F. and Bott, R. The moment map and equivariant cohomology. Topology 23 (1984), no. 1, 1–28.
Meinrenken, E. Equivariant Cohomology and the Cartan Model, in Encyclopedia of Mathematical Physics
Guillemin, Victor W. and Sternberg, Shlomo. Supersymmetry and equivariant de Rham theory. With an appendix containing two reprints by Henri Cartan. Mathematics Past and Present. Springer-Verlag, Berlin, 1999.
LocalizationDuistermaat, J. J. and Heckman, G. J. On the variation in the cohomology of the symplectic form of the reduced phase space.
Berline, Nicole and Vergne, Michèle. Classes caractéristiques équivariantes. Formule de localisation en cohomologie équivariante. C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 9, 539–541.
ConvexityAtiyah, M. F. Convexity and commuting Hamiltonians. Bull. London Math. Soc. 14 (1982), no. 1, 1–15.
Guillemin, V. and Sternberg, S. Convexity properties of the moment mapping. Invent. Math. 67 (1982), no. 3, 491–513.
Kirwan, Frances. Convexity properties of the moment mapping. III. Invent. Math. 77 (1984), no. 3, 547–552.
Guillemin, Victor and Sjamaar, Reyer. Convexity properties of Hamiltonian group actions. CRM Monograph Series, 26. American Mathematical Society, Providence, RI, 2005. iv+82 pp. ISBN: 0-8218-3918-7
Symplectic Reduction and Geometric Invariant Theory Kirwan, Frances Clare. Cohomology of quotients in symplectic and algebraic geometry.
Thomas, R. P. Notes on GIT and symplectic reduction for bundles and varieties. Surveys in differential geometry. Vol. X, 221–273, Surv. Differ. Geom., 10, Int. Press, Somerville, MA, 2006.
Toric Symplectic Manifolds Cannas da Silva, Ana. Symplectic toric manifolds.
Symplectic Blowing Up and DownMcDuff, Dusa and Salamon, Dietmar. Introduction to symplectic topology.
Group-Valued Moment Maps Alekseev, A., Malkin, Anton and Meinrenken, Eckhard. Lie group valued moment maps.
Page last modified on Monday February 11, 2019 10:15:01 UTC