Symplectic Geometry
What is this course about?
A symplectic manifold is a smooth manifold endowed with a 2-form which is closed and nondegenerate. Symplectic geometry has its roots in the Hamiltonian formalism of classical mechanics. In suitable coordinates, a symplectic manifold locally looks like the phase space of Hamilton's equations. In this course we are interested in the geometry of symplectic manifolds and their symmetries. We plan to cover the following topics:
Part 1: Symplectic Manifolds
Definition and main examples
Submanifolds
Local normal forms
Part 2: Hamiltonian actions
Lie group actions and their infinitesimal description
Symplectic actions
Moment maps and Marsden-Weinstein Reduction
Toric symplectic manifolds and the Atiyah-Guillemin-Sternberg Convexity Theorem
Part 3: Singular symplectic spaces
Symplectic orbifolds
Lie groupoids and their stacks
0-symplectic groupoids
Lie 2-groups acting on 0-symplectic groupoids
Toric symplectic stacks
When and Where?
Tuesday and Thursday 10:00-11:30. Room 241 Building A.
Assessments
There will be homework assignments and a written exam. Also, students are required to write a paper and present it as seminar.
Written Exam: May 16
Deadlines: The paper will be due by June 30. Presentations will be in class (July 06, 11 and 13).
Suggestions to develop your paper can be found here.
Assignments
References
M. Audin, "Torus actions on symplectic manifolds", Progress in Mathematics Vol. 93, Birkhauser - Second Edition
D. McDuff, D. Salamon, "Introduction to symplectic topology", Oxford Mathematical Monographs
A. Cannas da Silva, "Lectures on symplectic geometry", Lecture Notes in Mathematics, Springer Verlag
B. Hoffman, "Toric symplectic stacks", Advances in Mathematics 368 (15), 2020