Symplectic and Poisson Geometry
| Title | ΣΥΜΠΛΕΚΤΙΚΗ ΚΑΙ POISSON ΓΕΩΜΕΤΡΙΑ / Symplectic and Poisson Geometry |
| Code | 0675 |
| Faculty | Sciences |
| School | Mathematics |
| Cycle / Level | 2nd / Postgraduate |
| Teaching Period | Spring |
| Coordinator | Fani Petalidou |
| Common | Yes |
| Status | Active |
| Course ID | 600014677 |
Programme of Study: PMS Tmīmatos Mathīmatikṓn (2018-sīmera)
Registered students: 8
| Orientation | Attendance Type | Semester | Year | ECTS |
|---|---|---|---|---|
| THEŌRĪTIKA MATHĪMATIKA | Core Courses A3 | 2 | 1 | 10 |
| Academic Year | 2025 – 2026 |
| Class Period | Spring |
| Faculty Instructors | |
| Weekly Hours | 3 |
| Total Hours | 39 |
| Class ID | 600291533
|
Type Of Offer
- Disciplinary Course
Course Type 2021
Specialization / Direction
Mode of Delivery
- Face to face
- Distance learning
Digital Course Content
- e-Study Guide https://qa.auth.gr/en/class/1/600291533
- eLearning:
Language of Instruction
- Greek (Instruction, Examination)
Prerequisites
Required Courses
- 0658 Differenrial Manifolds
General Prerequisites
Very good understanding of all theoretical courses of the Undergraduate Program. Specifically, of the courses: Linear Algebra, Theory of groups, Differentiable manifolds. Also, of the course Theory of Differentiable manifolds of Graduate Program.
Learning Outcomes
1. Familiarity with a modern branch of Differential Geometry.
2. Know the historical development of Symplectic Geometry.
3. Know of the basic calculus on the Symplectic manifolds.
4. Know of the basic theorems of Symplectic manifolds.
Understanding of complex theoretical concepts.
Enlargement their horizons for further third cycle studies.
General Competences
- Adapt to new situations
- Work autonomously
- Work in teams
- Work in an international context
- Work in an interdisciplinary team
- Generate new research ideas
- Be critical and self-critical
- Advance free, creative and causative thinking
Course Content (Syllabus)
Symplectic Vector Spaces. Symplectic manifolds. Kahler manifolds. Symplectomorphisms. Reduction Theorem. Darboux-Weinstein theorems. Lagrangian submanifolds. Hamiltonian Machanics. Moment map. Action of a Group - Marsden-Weinstein reduction theorem. Poisson brackets and Poisson manifolds.
Keywords
Symplectic form, Hamiltonian vector field, Symplectomorphism, Reduction theorem, Moment map
Educational Material Types
- Notes
- Book
Use of Information and Communication Technologies
Use of ICT
- Use of ICT in Course Teaching
- Use of ICT in Communication with Students
- Use of ICT in Student Assessment
Course Organization
| Activities | Workload | ECTS | Individual | Teamwork | Erasmus |
|---|---|---|---|---|---|
| Lectures | 45 | 1.5 | |||
| Seminars | 15 | 0.5 | |||
| Reading Assigment | 90 | 3 | |||
| Tutorial | 45 | 1.5 | |||
| Written assigments | 105 | 3.5 | |||
| Total | 300 | 10 |
Student Assessment
Description
Homeworks - Exercises - Presentations.
A mini thesis for the final evaluation of the students. Analysis, reduction and presentation of a advanced subject of the course.
Student Assessment methods
- Written Assignment (Formative, Summative)
- Oral Exams (Formative, Summative)
- Performance / Staging (Formative, Summative)
Bibliography
Additional bibliography for study
1. R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd edition, Benjamin/Cummings, Reading, 1978.
2. R. Berndt, An Introduction to Symplectic Geometry, GSM 26, American Mathematical Society Providence, 2001.
3. A. Cannas da Silva, Lectures on Symplectic Geometry, LNM 1764, Springer, 2001. New edition 2008.
4. V. Guillemin and Sh. Sternberg, Symplectic techniques in physics, Cambridge University Press, 1984.
5. J.-L. Koszul and Yi M. Zou, Introduction to Symplectic Geometry, Science Press Beijing, Springer, 2019.
6. P. Libermann and Ch.-M. Marle, Symplectic Geometry and Analytical Mechanics, D. Reidel Publishing Company, Dordrecht, 1987. (Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers.)
7. Ch.-M. Marle, Géométrie Symplectique et Géométrie de Poisson, Calvage & Mounet, 2018.
8. D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford graduate texts in mathematics 27, Oxford University Press, 2017.
Last Update
12-05-2025
