Course teacher(s)
Mélanie BERTELSON (Coordinator)ECTS credits
5
Language(s) of instruction
french
Course content
The notions introduced in this course are:
- Symplectic manifold and symplectomorphism.
- Symplectic topology.
- Darboux and Moser's Theorems.
- Symplectic capacity.
- Almost complex structure.
- Pseudo-holomorphic curve.
- Gromov's non-squeezing theorem (if time permits).
Objectives (and/or specific learning outcomes)
The goal of this course is to introduce the student to symplectic geometry, a field of research that stems from classical mechanics and that has deep connections with a large number of other fields, such as algebraic geometry, the theory of dynamical systems, low-dimensional topology, Kählerian geometry, to mention just a few. We will specifically explore topological aspects of symplectic geometry such as Gromov's non squeezing theorem or the existence of symplectic capacities.
Prerequisites and Corequisites
Required and Corequired knowledge and skills
To be able to follow this course, it is necessary to be familiar with the following concepts :
- intrinsic manifold
- smooth map between smooth manifolds
- tangent vector and tangent space at a point in a manifold
- vector field and its flow
- differential of a smooth map
- differential forms and de Rham cohomology.
Cours co-requis
Teaching methods and learning activities
Theoretical courses (24 hrs) taught in French and homeworks.
Contribution to the teaching profile
This course allows the student to discover an active field of contemporary research.
References, bibliography, and recommended reading
- Holomorphic curves in symplectic geometry. Michèle Audin et Jacques Lafontaine éditeurs. Progress in Mathematics 117. Birkhäuser Verlag, 1994.
- Cannas da Silva, Ana Lectures on symplectic geometry. Lecture Notes in Mathematics, 1764. Springer-Verlag, Berlin, 2001.
- Eliashberg, Yakov, Mishachev, Introduction to the h-principle. Grad. Stud. Math., 48, American Mathematical Society, Providence, RI, 2002. xviii+206 pp.
- Hatcher, Allen, Algebraic topology. Cambridge University Press, Cambridge, 2002.
- Hofer, Helmut; Zehnder, Eduard, Symplectic invariants and Hamiltonian Dynamics, Birkhäuser Advanced Texts, 1994.
- McDuff, Dusa; Salamon, Dietmar Introduction to symplectic topology. Third edition. Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, 2017.
- Dusa McDuff, Dietmar Salamon, J-holomorphic curves and symplectic topology.
- Milnor, J.~W., Stasheff, J.~D. Characteristic classes. Princeton University Press and University of Tokyo Press. Princeton, New Jersey, 1974.
Course notes
- Syllabus
- Université virtuelle
Other information
Contacts
Mélanie Bertelson (2.O7.111) - Melanie.Bertelson@ulb.be - 02 650 58 28.
Campus
Plaine
Evaluation
Method(s) of evaluation
- Personal work
- written examination
Personal work
written examination
Written exam and homeworks which contribute for 1/4th of the final grade.
Mark calculation method (including weighting of intermediary marks)
Final note based on the note for the written exam Ne and the note for the homeworks Nh according to the rule : 3/4 Ne + 1/4 Nh.
Language(s) of evaluation
- french