MATH 257A: Intro/Topics in Differential Geometry (Fall 2014)


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.