*** Revision supervision ***

May 4, from 4 to 5pm, in room MR13.

We will go over the 2012 exam (skipping 3 because it's solely bookwork), and then also add some parts of questions 1, 2, 3 from the 2013 exam. And I'll answer questions that you might have.

Part III Symplectic flavored exams from past years:

(Not all questions / parts of questions are on material that we covered, see comments in parenthesis below. Because I didn't solve all other problems carefully, there may be some that are not flagged but which use things we did not cover either, my apologies.)

• 2014 (not covered: 6)
• 2012 (we didn't talk about contact structures, but if you read the definition in the book you should be ok attempting 2)
• 2003 (not covered: last part of 1, 4(a)-(d), last part of 5, 6)
• 2002 (not covered: 2, 3, 5)

• A proof of the contractibility of J(M,omega) (more precise than the one in class)
• A note about the proof of "Claim" in lecture 7.

Examples classes and examples sheets:

• First class: 8 / 9 February in MR13 :: Sheet 1, problems for marking due February 6 / 7
• Second class: 20 / 22 February in MR13 :: Sheet 2*, problems for marking due 19 (or 21) / 20 February

* updated on February 21, to fix imprecisions in exercise 6.

• Third class: 7 / 9 March in MR13 :: Sheet 3, problems for marking due 5 / 7 March
• Fourth class: 17 / 24 April (see below) :: Sheet 4, (typo in 5b corrected, 2n --> n) problems for marking due 17/20 April (see just below)

Charlotte: Wed April 18, 5-6pm in MR13, problems due for marking at 6pm of April 17

Brunella: Tue April 24, 4-5pm and 5-6pm (might be only one time slot, email Brunella to confirm) in MR15, problems due for marking on Fri April 20

• Revision examples class: (by the lecturer) TBA

Office hours: Feel free to show up and ask any questions about the lectures or problem sets. No appointment is necessary.

• Friday February 2, 4-5pm at Pavilion D ground floor lounge.
• Friday February 16, 4-5pm at Pavilion D ground floor lounge.
• Friday March 2, 4-5pm at Pavilion D ground floor lounge.

More details about the schedule of examples classes and handing in work for marking:

Brunella will hold exercises classes on Thursday, 4-5pm and 5-6pm; Charlotte will hold exercise classes on Friday 5-6pm (actually, Tuesday Feb 20), any exceptions will be noted above. I will circulate a sign-up sheet during the week before the first exercise class.

I will indicate in the exercise sheets which problems may be handed in for marking. For a Thursday exercise class, hand in your problems for marking at Dr. A.R. Pires' CMS pigeonhole (put work inside the folder) by 4pm of the previous Tuesday. For a Friday exercise class, hand in your problems for marking at BL.33, opposite the main entrance to the Part III room (put work in the folder) by 4pm the previous Wednesday.

Lectures: Monday, Wednesday, Friday, 10:00-11:00 at MR4

Lecturer: Ana Rita Pires (arp75@cam...)

TAs: Brunella Torricelli (bct27@cam...) and Charlotte Kirchhoff-Lukat (csk34@cam...)

Material covered in lectures:

• Week 0: Symplectic linear algebra, symplectic manifolds, examples (Chapter 1)
• Week 1: Moser and Darboux theorems, cotangent bundle, lagrangian submanifolds, symplectomorphism vs. lagrangians, symplectomorphism vs. generating functions (Chapters 7, 8.1, 2, 3, 4.1, 4.2. Also review of differential geometry in Chapter 6 - handout.)
• Week 2: Fixed points of symplectomorphisms (and billiards), lagrangian neighbourhood theorem, Weinstein theorem, Arnold conjecture (Chapters 4.3, 5, 8, 9)
• Week 3: Compatible almost complex structures, compatible triples, integrable almost complex structures, complex manifolds, Dolbeault theory, Kähler manifolds (Chapters 12, 13, 14, 15, 16.1-16.3)
• Week 4: Hodge theory for real manifolds and Kähler manifolds, topological consequences, hamiltonian and symplectic vector fields, hamiltonian functions, Poisson bracket (Chapters 16.4, 17, 18.1, 18.3)
• Week 5: Integrable systems, action-angle coordinates, classical mechanics, hamiltonian and lagragian formulations, Legendre transform, hamiltonian actions, moment maps, coadjoint orbits (Chapters 18.4, 19.1-19.4, 21, 22.1)
• Week 6: Symplectic quotients, orbit spaces, Marsden-Weinstein-Meyer theorem, examples of moment maps and symplectic quotients, convexity theorem (Chapters 22.2-3, 23, 25.1-4, 27.1)
• Week 7: Proof of the convexity theorem, Schur-Horn theorem, effective actions (Chapters 27.1-27.3)
• Week 8: Symplectic toric manifolds, Delzant classification theorem, 15 minutes of embeddings and capacities (Chapter 28)

References:

• (primary) "Lectures in Symplectic Geometry", Ana Cannas da Silva, Springer (available in the library, also as a pdf)
• "Introduction to Symplectic Topology", Dusa McDuff and Dietmar Salamon, Oxford (available in the library)

Plan

The fist part of the course will be an overview of the basic structures of symplectic geometry, including symplectic linear algebra, symplectic manifolds, symplectomorphisms, Darboux theorem, cotangent bundles, Lagrangian submanifolds, and Hamiltonian systems. The course will then go further into one or two topics, as time permits: the first one is moment maps and toric symplectic manifolds, the second one is capacities and symplectic embedding problems.

Prerequisites

Some familiarity with basic notions from Differential Geometry (in particular differential forms and vectors fields on manifolds) and Algebraic Topology will be assumed. I will provide references for other key facts necessary, for example from Lie theory.