MATH 291A: Topics in Topology:
Symplectic Geometry and Topology (Fall 2016)


Instructor: Alvaro Pelayo, alpelayo@math.ucsd.edu

Lectures: Tu Th 9:30-10:50 APM 5402

Office hours: Tu 1-2pm and Th 2-3pm APM 5240

Tentative outline: This Ph.D. course assumes background on topology and differential geometry for instance as covered in UC San Diego first year graduate courses on these subjects. Topics are expected to cover classical and current research topics in symplectic geometry and topology, for instance some of the following:
- symplectic manifolds and their submanifolds, Darboux's theorem, Moser's stability, Weinstein's Lagrangian neighborhood theorem;
- geometry and topology of symplectic group actions with symplectic or Lagrangian orbits;
- integrable Hamiltonian systems, singularities, singular fibrations, normal forms, toric and semitoric systems, classifications, etc;
- noncommutative integrable Hamiltonian systems;
- symplectic embeddings, Gromov's Nonsqueezing Theorem, symplectic capacities, symplectic fibrations.

Presentations: the instructor will give lectures on a number of topics covered. The plan is for each student to give some lectures. Here are some examples of how to prepare the written version of your presentation:
http://www.math.ucsd.edu/~alpelayo/GroupActionsSeminar.html
http://math.berkeley.edu/~alanw/
It would be helpful if you discuss and practice the presentation with someone else before the in class presentation to help with timing etc. Prepare written notes of your presentation (hand written notes are OK) to share with the audience.

Reading and presentation materials: here are some references where you may find a particular topic which interests you.
You can discuss with me what choice of topic you would like to present. Feel free to suggest other papers or books.

Suggestions for books:

A. Cannas da Silva: Lectures on symplectic geometry. Lecture Notes in Mathematics 1764. Springer-Verlag, Berlin, 2001. xii+217 pp.
V. Guillemin, Moment Maps and Combinatorial Invariants of Hamiltonian T^n Spaces, Birkhauser 1994.
E. Lerman, V. Guillemin, and S. Sternberg, Symplectic Fibrations and Multiplicity Diagrams. Cambridge University Press 2009
H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Modern Birkhauser Classics 2011
D. McDuff and D. Salamon, Introduction to Symplectic Topology. Oxford University Press 1998

Suggestions for papers by general topic:

symplectic group actions:
M.F. Atiyah and R. Bott, The moment map and equivariant cohomology. Topology 23 (1984), no. 1, 1-28.
J.J. Duistermaat and A. Pelayo, Symplectic torus actions with coisotropic principal orbits. Ann. Inst. Fourier (Grenoble) 57 (2007) 2239-2327.

geometry of symplectic and volume forms:
R. Greene and S. Shiohama: Diffeomorphisms and volume preserving embeddings of noncompact manifolds. Trans. Amer. Math. Soc 255, 403-414, 1979.
L. Guth, Symplectic embeddings of polydisks. Invent. Math. (2008) 477-489
K. Cielieback, H. Hofer, J. Latschev, and F. Schlenk, Quantitative symplectic geometry. Math. Sci. Res. Inst. Publ 54, 1-44 Cambridge Univ. Press 2007.
J. Moser: On the volume elements on a manifold. Trans. Amer. Math. Soc. 286-294, 1965.
A. Pelayo and S. Vu Ngoc, Hofer's question on intermediate symplectic capacities. Proc. Lond. Math. Soc. 110 (2015) 787-804

general theory of integrable systems:
J.J. Duistermaat, On global action-angle coordinates. Comm. Pure Appl. Math. 33 (1980) 687-706.
J-P. Dufour, P. Molino, Pierre, and A. Toulet, Classification des systemes integrables en dimension 2 et invariants des modeles de Fomenko. C. R. Acad. Sci. Paris Ser. I Math. 318 (1994) 949-952.
R. L. Fernandes, C. Laurent-Gengoux, P. Vanhaecke: Global Action-Angle Variables for Non-Commutative Integrable Systems, arXiv:1503.00084
A. Pelayo and S. Vu Ngoc, Symplectic theory of completely integrable systems, Bull. Amer. Math. Soc 48 (2011) 409-455
S. Vu Ngoc, On semi-global invariants for focus-focus singularities, Topology 42(2) 365-380, 2003.
N.T. Zung, A note on focus-focus singularities, Diff. Geom. Appl. 7(2) 123-130, 1997.

toric or semitoric systems:
N.C. Leung and M. Symington, Almost toric symplectic four-manifolds. J. Symplectic Geom. 8 (2010) 143-187.
J. Palmer: Moduli spaces of semitoric systems, arXiv:1502.07296
A. Pelayo and S. Vu Ngoc: Semitoric integrable systems on symplectic 4-manifolds. Invent. Math. 177 (2009) 571-597.
M. Symington, Four dimensions from two in symplectic topology. Topology and geometry of manifolds (Athens, GA, 2001), 153-208, Proc. Sympos. Pure Math., 71, Amer. Math. Soc., 2003.