Mastermath Course on Symplectic Geometry - 8EC, spring semester, 2021/ 2022
Fabian Ziltener (UU), f.ziltener special sign uu.nl
Federica Pasquotto (Universiteit Leiden), f.pasquotto special sign math.leidenuniv.nl
Nikolaos Adaloglou, n.adaloglou special sign math.leidenuniv.nl
Yann Guggisberg, y.b.guggisberg special sign uu.nl
The lectures take place on Thursdays from 10:15 to 12:00 at Utrecht University. The room will be announced on the elo-website. We may also live-stream the lecture.
The tutorials take place on Thursdays from 12:00 to 13:00.
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 0.2*(grade for hand-in exercises) + 0.8*(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:
- Algebraic geometry: Every smooth algebraic subvariety of the complex projective space carries a canonical symplectic form.
- Gauge theory: The space of Yang-Mills instantons over a product of two real surfaces is closely related to the space of gauge equivalence classes of flat connections over a surface, which carries a canonical symplectic form.
Contents of this course
Some highlights of this course will be the following:
- A normal form theorem for a submanifold of a symplectic manifold. A special case of this is Darboux's theorem, which states that locally, all symplectic manifolds look the same.
- Symplectic reduction for a Hamiltonian Lie group action. This corresponds to the reduction of the degrees of freedom of a mechanical system. It gives rise to many examples of symplectic manifolds.
Here is a more complete list of topics that we will cover:
- linear symplectic geometry
- symplectic manifolds
- canonical symplectic form on a cotangent bundle
- symplectomorphisms, Hamiltonian diffeomorphisms
- Poisson bracket
- Moser's isotopy method
- symplectic, (co-)isotropic and Lagrangian submanifolds of a symplectic manifold
- normal form theorem for a submanifold of a symplectic manifold
- Darboux's theorem
- Weinstein's neighbourhood theorem for a Lagrangian submanifold
- Hamiltonian Lie group actions, momentum maps
- symplectic reduction, Marsden-Weinstein quotient
- contact geometry and its interactions with symplectic geometry
We will also explain connections to classical mechanics, such as Noether's theorem and the reduction of degrees of freedom. If time allows, we will also discuss coadjoint orbits, as an important source of examples of symplectic manifolds with Hamiltonian Lie group actions, or the algebraic structure of the symplectomorphism group and the symplectic flux homomorphism.
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:
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.
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.