**Instructor:**
Alvaro Pelayo, apelayo@math.wustl.edu

**Lectures:
** Monday 3-4pm, Wednesday 3-5pm Room 199 Cupples I

** Office hours: ** Monday 4:30-5:30pm, Wednesday 10-11:30am, Cupples 212

**Tentative outline: ** This is an advanced Ph.D. course assuming background on graduate differential geometry, functional analysis, topology, and partial differential equations (for instance as covered in Washington University first year graduate courses on these subjects). Topics in the course are expected to cover classical and current research topics in geometry and analysis, primarily on microlocal analysis, semiclassical analysis/spectral theory, and its interactions with symplectic geometry. We will also discuss some important problems in symplectic topology, also from a semiclassical view point.

**Presentations:** the instructor will give introductory lectures on most of the topics covered. The plan is for each student to give about two lectures..
Here are some examples of how to prepare the written version of your presentation:

http://www.math.wustl.edu/~apelayo/GroupActionsSeminarWinter08.html

http://math.berkeley.edu/~alanw/

It would be helpful if you could discuss and practice the material with someone else before the in class presentation to help with with timing etc.

**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.

J.J. Duistermaat and J.A.C. Kolk:
Lie groups. Universitext. *Springer-Verlag, Berlin,* 2000. viii+344 pp.

V. Guillemin and S. Sternberg:
Symplectic techniques in physics. *Cambridge University Press, Cambridge,* 1990. xii+468 pp

L. Hormander: The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Second edition. *Springer-Verlag, Berlin,* 1990. xii+440 pp.

T. Kato: Perturbation theory for linear operators. *Springer-Verlag New York, Inc., New York* 1966 xix+592 pp.

J. M. Lee: Introduction to smooth manifolds. Graduate Texts in Mathematics 218. *Springer-Verlag, New York,* 2003. xviii+628 pp.

M. Zworski: Semiclassical Analysis, Graduate Studies in Mathematics 138, AMS

__Suggestions for papers__:

L. Charles, A. Pelayo, S. Vu Ngoc: Isospectrality for quantum toric integrable systems, arxiv 111.5985

K. Cielieback, H. Hofer, J. Latschev, F. Schlenk: Quantitative symplectic geometry. Dynamics, ergodic theory, and geometry, 1-44,

Math. Sci. Res. Inst. Publ 54, Cambridge Univ. Press, Cambridge, 2007.

J.J. Duistermaat and L. Hormander*: *Acta Math. 128** **(1972), no. 3-4, 183-269.

L. Guth: Symplectic embeddings of polydisks. Invent. Math. (2008) 477-489

A. Pelayo and S. Vu Ngoc: Semitoric integrable systems on symplectic 4-manifolds, Invent. Math. 177 (2009) 571-597

A. Pelayo and S. Vu Ngoc: Symplectic theory of completely integrable systems, Bull. Amer. Math. Soc 48 (2011) 409-455

J. Sjostrand: Singularites analytiques microlocales. Asterisque 95, 1-166, Soc. Math. France, Paris, 1982.