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.