## Mastermath Course on Symplectic Geometry - 8EC, spring semester, 2016/ 2017

### General information

Lecturers
F. Ziltener (UU)
F. Pasquotto (VU)

Assistant
Dušan Joksimović, d.joksimovic@uu.nl

Lectures
Thursdays, 14:00-15:45, Universiteit Utrecht, HFG 611

Tutorials
Thursdays, 16:00-16:45, Universiteit Utrecht, HFG 611

Hand-in assignments
These need to be handed in by Thursday one week later before the lecture. You may put them into Dušan's mailbox or send them to him by e-mail.

Exam: The exam will be a written exam of 3 hours and take place on June 15.

The material will be everything treated in the lectures and the assignments except for the material treated in the extra lecture on April 25 (proof of the symplectic reduction theorem and coadjoint orbits). (The example of the Grassmannian, in particular of complex projective space, was mentioned in that lecture, but also used in exercises. It may show up in the exam.)

The exam will consists of some questions that were already assigned as homework exercises, a few new questions that are comparable to the homework questions, and one more challenging last question.

At the exam you are not allowed to use any book or lecture notes. However, you are allowed to use 1 sheet of hand-written notes (A4 format, written on both sides).

Retake: tba

The final grade will be based on the hand-in assignments and presentations of these (30 %) and a final written or oral examination (70 %).

### Description and aims of the course

Symplectic geometry has its roots in the Hamiltonian formulation of classical mechanics. The canonical symplectic form on phase space occurs in Hamilton's equation. Symplectic geometry studies local and global properties of symplectic forms and Hamiltonian systems. A famous conjecture by Arnol'd, for instance, gives a lower bound on the number of periodic orbits of a Hamiltonian system. There are deep connections between symplectic geometry and the theory of dynamical systems, algebraic geometry and modern physics, for example string theory.

This course will focus on the foundations of symplectic geometry:
• linear symplectic geometry
• canonical symplectic form on a cotangent bundle
• symplectic manifolds, (co-)isotropic and Lagrangian submanifolds
• Moser's isotopy method
• Darboux's theorem
• Weinstein's neighbourhood theorem for a submanifold of symplectic manifold
If time permits, we will also cover one or more of the following topics:
• Hamiltonian group actions, momentum maps and symplectic reduction
• Delzant's and the Atiyah-Guillemin-Sternberg convexity theorem
• constructions of symplectic manifolds, e.g. symplectic fibrations and blow-ups
The last two lectures will be reserved for a panorama of recent results in the field of symplectic geometry, for instance the existence of symplectic capacities and the Arnol'd conjecture.

Some mathematicians whom we will encounter in this course, are the following:
 Jean-Gaston Darboux, 1842 - 1917 Emmy Noether, 1882 - 1935 Jürgen Moser, 1928 - 1999 Alan Weinstein, 1943 -

### Literature

A. Cannas da Silva, Lectures on symplectic geometry, Lecture Notes in Mathematics, 1764, Springer-Verlag, Berlin, 2001 and 2008 (corrected printing).

D. McDuff and D.A. Salamon, Introduction to symplectic topology, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998.

### Prerequisites

The standard notions taught in a first course on differential geometry, such as: Test yourself! If you do not know one of these notions then quickly consult a book on this subject, for example:

J. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, Springer, 2002.

Some knowledge of classical mechanics can be useful in understanding the context and some examples.

And now comes the ultimate question: Is every closed differential two-form on the two-sphere $$S^2$$ exact?
Yes No I don't know.