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

**Lectures:
** MWF 10-10:50am, APM B412

** Office hours: ** M 1:10-2pm, F 12:10-1pm APM 7444

**Tentative outline: **
This is an advanced course for graduate
students interested in differential geometry and its interactions with
analysis and mathematical physics. The course requires background on
graduate differential geometry (see for instance Lee's book in the
references
below), functional
analysis, topology, and differential equations (for instance
as
covered at UC San Diego first year graduate courses on these
subjects). Topics in the course are expected to cover classical and
current research topics in differential geometry, in particular in
symplectic geometry (for instance group actions on symplectic
manifolds, finite dimensional integrable systems, etc) and its
interactions with
spectral theory (inverse spectral problems about quantum integrable
systems). Other possible topics include important problems of current
interest
in symplectic topology, such as those concerning symplectic embeddings and symplectic
capacities as pioneered in the work of Gromov, Hofer, McDuff and others.

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

Examples of Presentations

More Examples (See Survey
Articles Written by Students)

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

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

D. McDuff and D. Salamon: Introduction to Symplectic Topology

__Suggestions for papers__:

L. Charles, A. Pelayo, S. Vu Ngoc: Isospectrality for quantum toric
integrable systems (dedicated to Peter Sarnak on his 60th Birthday),
Annales Sci. Ec. Norm. Sup. 43 (2013) 815-849

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: Fourier integral operators II, Acta Math. 128** **(1972), no. 3-4, 183-269.

T. Delzant: Hamiltoniens periodiques et images convexes de l'application moment, Bull. Soc. Math. France 116 (1988), no. 3, 315–339

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

R. Hind and E. Kerman: New obstructions to symplectic embeddings, Invent. Math. Volume 196 (2014) Issue 2, pp 383-452

R. Hind: Some optimal embeddings of symplectic ellipsoids, arXiv:1409.5110 (2014)

D. McDuff: Three lectures on symplectic topology today, AMS Meeting January 2014, available here

J. Palmer: Metrics and convergence in the moduli spaces of maps, arXiv:1406.4181 (2014)

A. Pelayo, A.R. Pires, T. Ratiu, S. Sabatini: Moduli spaces of toric manifolds, Geometriae Dedicata 169 (2014) 323-341

A. Pelayo and S. Vu Ngoc: Constructing integrable systems of
semitoric type, Acta Math. 206 (2011) 93-125

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.