Required in stage
Optional in stage
First term
Second term
Both terms
Not organised
This year
Alternating years
External
Prerequisites
Taught by
Language of instruction
Duration
Identical coursesSymplectic Geometry (B-KUL-G0B11A)
This course is not taught this academic year, but will be taught next year.Aims
The aim of the course is to give a introduction to the field of symplectic geometry. Symplectic geometry arose as the mathematical framework to describe classical mechanics, and nowaways is a rich subject which bears connections with other fields, including Riemannian geometry, complex geometry, and Lie group theory. A symplectic structure is given by a suitable differential form. In many ways it behaves differently from Riemannian geometry: symplectic manifolds have no local invariants such as curvature, hence the global geometry is more interesting than the local one, and there are topological obstructions to the existence of symplectic structures on given manifold. Further, Lie algebras play a fundamental role in the study of symplectic geometry.The students will get familiarized with all the above mentioned features of symplectic geometry.
The course will have an emphasis on symmetries - i.e. group actions - in symplectic geometry. They are described by so-called moment maps, which possess surprisingly nice global geometric properties that the students will learn both at the conceptual level and studying examples.
Previous knowledge
Some basic knowledge of differential geometry, in particular the notion of differential manifold and tangent bundle, as well as the notion of Lie group, is required. Familiarity with differential forms is recommended.
Is included in these courses of study
Activities
6 ects. Symplectic Geometry (B-KUL-G0B11a)
Content
PART 1:
- Symplectic linear algebra.
- Symplectic manifolds. The physical motivation of symplectic geometry: classical mechanics.
- Lagrangian submanifolds, coisotropic submanifolds. Normal form theorems: Darboux's, Weinstein's and Gotay's theorems.
PART 2:
- Lie algebra cohomology and representations.
- Hamiltonian actions and moment maps. Existence and uniqueness theorems.
- The Marsden-Weinstein symplectic reduction theorem. The convexity theorem of Atiyah and Guillemin-Sternberg.
Course material
- Ana Cannas da Silva, "Lectures on symplectic geometry", Springer Verlag. Available at http://www.math.ethz.ch/~acannas/Papers/lsg.pdf
- Eckhart Meinrenken, "Symplectic geometry", lecture notes available from http://www.math.toronto.edu/mein/teaching/lectures.html
Language of instruction: more information
The course will be held in English
Evaluation
Evaluation: Symplectic Geometry (B-KUL-G2B11a)
Explanation
There will be Take Home Tasks during the course, each consisting of exercises/problems related to certain parts of the course. The student will write solutions (in English) of the Take Home Tasks, and the solutions will be graded. For the solutions it is allowed to use notes of the lecture and of the exercise sessions, and the portion of the recommended literature corresponding to the material treated in class. Students are allowed to discuss the problems with each other, but are supposed to write their solutions individually. Solutions that are literally copied from peers will not be tolerated and every reference that is used should be quoted.
30% of the final grade will be based on the take home tasks, and 70% on the final exam (January exam or September exam).
In particular, a student who does not take the final exam can obtain at most 6/20 as grade for the course.
The final grade is meant to reflect to what extent the student assimilated the basic notions of symplectic geometry, and is able to work with them and apply them in concrete situations.