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

1. Homework 1

References

1. M. Audin, "Torus actions on symplectic manifolds", Progress in Mathematics Vol. 93, Birkhauser - Second Edition

2. D. McDuff, D. Salamon, "Introduction to symplectic topology", Oxford Mathematical Monographs

3. A. Cannas da Silva, "Lectures on symplectic geometry", Lecture Notes in Mathematics, Springer Verlag

4. B. Hoffman, "Toric symplectic stacks", Advances in Mathematics 368 (15), 2020