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):
| Date | Topic | Speaker | Location | |
| January 23 | Organization and Introduction to Moment Maps | Johan Martens | Notes | Appleton Tower 2.14 |
| January 30 | Equivariant Cohomology | José Figueroa-O’Farrill | Notes | Bayes 5.02 |
| February 6 | Localization, Duistermaat-Heckman-Berline-Vergne Theorem | Johan Martens | Notes | Bayes 5.45 |
| February 13 | Convexity of moment maps | Ana Rita Pires | Notes | Bayes 5.02 |
| February 27 | Symplectic Reducution and GIT | Carlos Zapata-Carratalá | Notes | Bayes 5.45 |
| March 6 | Toric symplectic manifolds and the Delzant Construction | Ben Brown | Notes | Bayes 5.02 |
| March 13 | Group-valued Moment Maps & Quasi-Hamiltonian Reduction | Alexander Shapiro | Notes | Bayes 5.45 |
| March 20 | Symplectic Blowing Up & Down, Cutting and Gluing, Implosion | Johan Martens | Bayes 5.02 | |
| March 27 | Quantum Hamiltonian Reduction | Sue Sierra | Notes | Bayes 5.02 |
| April 3 | A Homological Approach to Hamiltonian Reduction | José Figueroa-O’Farrill | Notes | Bayes 5.02 |
Suggested References:
| Generalities about moment maps | Ana Canas da Silva, Lectures on Symplectic Geometry. | |||
| Equivariant Cohomogy | Atiyah, 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. | ||||
| Localization | Duistermaat, 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. | ||||
| Convexity | Atiyah, 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. | |||
| Audin, Michele. Torus Actions on Symplectic Manifolds. | ||||
| Symplectic Blowing Up and Down | McDuff, Dusa and Salamon, Dietmar. Introduction to symplectic topology. | |||
| Group-Valued Moment Maps | Alekseev, A., Malkin, Anton and Meinrenken, Eckhard. Lie group valued moment maps. | |||
| Jiang-Hua Lu and Alan Weinstein. Poisson Lie groups, dressing transformations, and Bruhat decompositions. | ||||