# Symplectic Geometry (2014, Mastermath, UU)

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## Course Description

Symplectic geometry is the study of manifolds equipped with a closed nondegenerate two form. It is the modern language for classical mechanics, notably in the Hamilton formalism.
The following subjects will be discussed in the lectures:

• Symplectic linear algebra
• Symplectic manifolds
• Hamilton formalism
• Integrable systems
• Moment map geometry

## Prerequisites

This is a basic course for students in the first year of the master. We only assume some familiarity with the concept of manifold. Moreover, we shall review some manifold theory and algebraic topology, and subsequently review the De Rham theorem. These concepts are needed for the symplectic geometry, notably in Moser's proof of the Darboux theorem.

## Examination

There is a written exam. Time: Wednesday 14 January 2015, 15:15-18:15 hrs. Location: Utrecht University, Ruppert Blauw Building. The exam is not open book. However you are allowed to make a summary of at most two pages A4, which you can use to remember the various formulas. I will check these at the beginning of the exam. More than two pages is not allowed! For those who fail or missed the exam of 14 January 2015 there is a possibility of an oral reexamaniation, to be held in Nijmegen. Please, contact me for an appointment, if you are interested. An old exam as example can be found here.

Biweekly the second part of the lecture is an exercise class, given by the assistent Lei Zhao (l.zhao@rug.nl) and by me. Some exercises are given as homework, and will be graded as follows: Good=10, sufficient=7 and insufficient=4. Your final grade for the course SG will be the highest of the two marks 0.75E+0.25H and E, where E is the (written) exam grade and H the average of your homework (after deletion of the worst mark). The homework needs to be handed in a week later in the next lecture, or by mail to Lei Zhao.

## Literature

Lecture notes for this course can be found here. I intend to improve and slightly adapt them during this year course. In addition there are supplementary lecture notes on symplectic geometry by Hans Duistermaat from a spring school in 2004 here. I took the freedom to put these on my website after the sudden death of Hans in 2010.

## Locations, material covered and exercises

Location: Utrecht University, Wednesday afternoon, MIN211 (Minnaert Building), Time: 14-16:45
• September 10, No lecture on symplectic geometry, but this first week tuesday and wednesday will be used for a crash course on category theory.
• September 17, Lecture: Sections 1.1, 1.2 and 1.3. No excercise section this day.
• September 24, Lecture: Sections 1.4, 2.1 and 2.2 up to the introduction of the Lie derivative on page 16, Homework exercise: 1.6
• October 1, Lecture: Sections 2.2, 2.3, 2.4 and 2.5. No excercise section this day.
• October 8, Lecture: Sections 2.6 and 2.7, Homework exercise: 2.4 (Be careful: the numbering has changed This is the exercise about the details of the proof of the Cartan formula).
• October 15, Lecture: Sections 3.1, 3.2 and 3.3. No excercise section this day.
• October 22, Lecture: Sections 3.4 and half of 3.5. Homework exercise: 2.12 and 2.14
• October 29, Lecture: Sections 3.5 and part of 3.6. Homework exercise: 3.6
• November 5, Lecture: Sections 3.6, 3.7. No excercise section this day.
• November 12, Lecture: Sections 4.1, 4.2. Homework exercise: 3.12 and 3.13
• November 19, Lecture: Sections 4.3, begin 4.4. No excercise section this day.
• November 26, Lecture: Sections 4.4, 4.5. Homework exercise: 4.1
• December 3, Lecture: Sections 5.1, 5.2.
• December 10, Lecture: Sections 5.2, 5.3.
• December 17, Lecture: Sections 5.5.