Mastermath Course on Symplectic Geometry - 8EC, spring semester, 2020/ 2021

Schedule

General information

Lecturer
Fabian Ziltener (UU), f.ziltener special sign uu.nl

Assistant
Wilmer Smilde, w.t.smilde special sign uu.nl

Lectures
The lectures take place on Thursdays from 10:15 to 12:00 via zoom. You can find the meeting ID and passcode on the elo-website for the course, at the top.

Tutorials
The tutorials take place on Thursdays from 12:00 to 13:00. The assistant will inform you about how to join them online.

Hand-in assignments
These need to be handed in by Thursday before the lecture, one week after being assigned. The assistant will let you know how to hand them in.
You may work on the exercises together with your fellow students. However, you need to hand in your own version of the solutions, which reflects your understanding of the material. You may not copy or paraphrase solutions by others.

Homework and exam
You will be asked to hand-in solutions to exercises every week. At the end of the course there will be a written exam at which you may not use any book, course material, or calculator. However, you may use one sheet of paper with hand-written notes (A4 format, both sides). There will be a retake exam (oral or written), which every student may take.

If the grade for the exam is at least 5 then the final grade is computed as 0.3*(grade for hand-in exercises) + 0.7*(grade for exam). Otherwise the final grade equals the grade for the exam. In case of a retake, the same formula applies to the retake exam.

What is symplectic geometry?

A symplectic structure is a closed and nondegenerate 2-form on a smooth manifold. Such a form is similar to a Riemannian metric. However, while a Riemannian metric measures distances and angles, a symplectic structure measures areas. The closedness condition is an analogue of the notion of flatness for a metric. 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.

Apart from classical mechanics, symplectic structures appear in a few other fields, for example in:

Contents of this course

Some highlights of this course will be the following:

Here is a more complete list of topics that we will cover:

We will also explain connections to classical mechanics, such as Noether's theorem and the reduction of degrees of freedom. The last lecture 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

Some prerequisites for this course are the notions taught in a first course on differential geometry, such as: Basic understanding of Lie groups and Lie algebras will also be useful, but not strictly necessary.

In addition, we will use the notion of a smooth vector bundle over a manifold and some basic operations involving vector bundles, such as dualization and direct sum. The theory of vector bundles is treated in a mastermath course on differential geometry. However, we will only use a small part of this theory in our course. In particular, we will not use any characteristic classes.

A suitable reference for differential geometry is:

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

The relevant chapters from this book are: 1-5,7-12,14-17,19,21. Some of the material covered in these chapters, in particular the one involving Lie groups, will be recalled in our lecture course.

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

The ultimate question (with thanks to Arjen Baarsma for programming this)

And now comes the ultimate question: Is every closed differential two-form on the two-sphere \(S^2\) exact?

yes, of course no, of course not What in God's name is meant by “differential form”?