Symplectic geometry 2022/2023

Schedule

Lectures (João Nuno Mestre):
Mondays 11:00 - 12:30 and 14:30 - 16:00

Problem Sessions (Lennart Obster):
Thursdays 14:30 - 16:00

Where: Room 3.1

Bibliography

The main reference we are using is:  Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer-Verlag, 2001 and 2008 (corrected printing).

Both the book, and an errata, are on the author's webpage. For convenience, here is the errata, in two versions: for the Springer 2008 printed text and for the author's 2006 website text (updated July 2021).

Complementary references

For symplectic topology (e.g. recurrence, fixed points of symplectomorphisms):  Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 1997 (The version in the library of DMUC).

For Hamiltonian systems (e.g. Arnold-Liouville Theorem, many examples): V. I. Arnold, Mathematical Methods of Classical Mechanics, second edition, 1989.
https://link.springer.com/book/10.1007/978-1-4757-2063-1

For historical context (e.g. the history of least action principles): M. Spivak, Physics for Mathematicans: Mechanics I, 2010.

For gauge theory and the moduli space of flat connections:

V. Hoskins, On algebraic aspects of the moduli space of flat connections. 

D. Michiels, Moduli spaces of flat connections, Master Thesis Leuven 2013.

J. Dupont, Fibre bundles and Chern-Weil theory, Aarhus Universitet Lecture Notes Series no. 69, 2003.

Other resources, mainly expository papers:

A. Weinstein (1981). "Symplectic Geometry" (PDF). Bulletin of the American Mathematical Society. 5 (1): 1–13.

A Conversation with Alan Weinstein, Notices of the AMS, Jan. 2023

M. J. Gotay, J. A. Isenberg (1992), The symplectization of science. Gazette des Mathématiciens, vol. 54, 1992, pp. 59–79. Original version in French.

J. N. Mestre, O princípio do camelo simpléctico, book chapter in "Números, Cirurgias e Nós de Gravata", IST Press, Lisbon 2013; (In Portuguese).

Lectures

13/02

Lecture 1: Introduction and motivation; symplectic linear algebra.

Lecture 2: The tautological 1-form and the and canonical 2-form on the cotangent bundle of a manifold. 

20/02

Lecture 3: Lagrangian submanifolds; symplectomorphisms

Lecture 4: Generating functions for symplectomorphisms

27/02

Lecture 5: Recurrence

Lecture 6: Preparation for the local theory (of normal forms)

06/03

Lecture 7: Moser Theorems

Lecture 8: Darboux-Moser-Weinstein Theory

13/03

Lecture 9: Weinstein Tubular neighborhood Theorem

Lecture 10: Hamiltonian vector fields

20/03

Lecture 11: Variational principles

Lecture 12: Legendre Transform

27/03

Lecture 13: Actions

17/04

Lecture 14: Legendre transform (continuation)

Lecture 15: Hamiltonian Actions

24/04

Lecture 16: Marsden-Weinstein-Meyer Theorem

Lecture 17: Reduction

04/05

Lecture 18: Moment Map in Gauge Theory

29/05

Lecture 19: Almost Complex Structures

Lecture 20: Compatible triples

12/06

Lecture 21: Dolbeault Theory

Lecture 22: Complex manifolds

15/06

Lecture 23: Kähler forms

Lecture 24: Compact Kähler Manifolds