MATH 257A: Topics in Differential Geometry: Moduli spaces, integrable systems, and Moser stability in symplectic geometry (Spring 2016)


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

Lectures: Tu Th 11am-12:20p

Office hours: Tu 3-4pm, Th 12:30-1:30pm, APM 7444

Tentative outline: This Ph.D. course assumes background on differential geometry, for instance as covered in UC San Diego first year graduate courses on these subjects. Topics in the course are expected to cover classical and current research topics in symplectic and spectral geometry, for instance several of the following will be covered:
- Atiyah-Guillemin-Sternberg convexity theorem, classifications of symplectic and Hamiltonian torus actions
- Classical integrable Systems, toric and semitoric systems, invariants and moduli spaces of integrable systems
- Quantum integrable systems and their spectral geometry
- Symplectic embeddings, Gromov nonsqueezing and Ekeland-Hofer symplectic capacities, symplectic fibrations and their deformations
- Volume preserving and symplectic diffeomorphisms, Moser and Greene-Shiohama stability theorems
- Symplectic dynamics on non-compact manifolds

Presentations: the instructor will give introductory lectures on most of the 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.
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
L. Hormander: The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis. Second edition. Springer-Verlag, Berlin, 1990. xii+440 pp.

Suggestions for papers:

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.J. Duistermaat and L. Hormander: Fourier Integral Operators II, Acta Math. 128 (1972), no. 3-4, 183-269.
A. Figalli, J. Palmer, and A. Pelayo: Symplectic G-capacities and integrable systems, arXiv:1511.04499
Y. Le Floch and A. Pelayo: Spectral asymptotics of semiclassical unitary operators, arXiv:1506.02873
L. Guth: Symplectic embeddings of polydisks. Invent. Math. (2008) 477-489
J. Moser: On the volume elements on a manifold. Trans. Amer. Math. Soc. 286-294, 1965.
R. Greene and S. Shiohama: Diffeomorphisms and volume preserving embeddings of noncompact manifolds. Trans. Amer. Math. Soc 255, 403-414, 1979.
J. Palmer: Moduli spaces of semitoric systems, arXiv:1502.07296
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.