Mastermath Course on Symplectic Geometry - 8EC, spring semester, 2018/ 2019


General information

Fabian Ziltener (UU)
Álvaro del Pino Gómez

Dušan Joksimović,

Thursdays, 10:15-13:00, Universiteit Utrecht, HFG 611

Hand-in assignments
These need to be handed in by the first Monday after the lecture. You may put them into Fabian's mailbox in the Hans Freudenthalgebouw or send them to Dušan by e-mail.

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. The final grade will be given by the following formula:

0.15*grade for the hand-in exercises + 0.85*max(grade for the exam, grade for the retake exam)

The maximum of the grade for the exam and the grade for the retake exam needs to be at least 5 in order to pass the course.

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.

Many problems in symplectic geometry are either flexible or rigid. In the flexible case methods from differential topology, such as Gromov's h-principle, can be applied to construct objects. In the rigid case partial differential equations can be used to define symplectic invariants. As an example, holomorphic curves (solutions of the Cauchy-Riemann equations) are used to define the so-called Gromov-Witten invariants.

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

Symplectic geometry also has connections to the theory of dynamical systems and string theory.

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. Furthermore, we will develop the basics of contact geometry, which is a field that is closely related to symplectic geometry.

If time permits, we will also cover one or more of the following topics:

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 -


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.


The 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. 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”?