Sympletic Geometry

Recommended Prerequisites

Differentiable Manifolds.

Teaching Methods

The theoretical lectures are predominantly expository. The formal lectures will be complemented by periods of individual attendance.

Learning Outcomes

Provide knowledge on basic concepts of Symplectic Geometry and promote the capacity of developing research activity in the field. 

Work Placement(s)

No

Syllabus

Symplectic linear algebra. Symplectic manifolds. Symplectomorphisms. Lagrangian submanifolds. Theorems of Darboux and Weinstein  (lagrangian neighborhood). Hamiltonian vector fields. Brief introduction to Poisson manifolds. Hamiltonian actions and moment map. Noether Theorem. Equivariance of the moment map and coadjoint orbits. Convexity Theorem. Symplectic reduction.

Additional topics: Almost complex structures. Kähler manifolds. Contact manifolds. Lie algebroids and Poisson geometry. Toric symplectic manifolds.

Assessment Methods

Assessment 3
Synthesis work: 100.0%

Assessment 1
Exam: 100.0%

Assessment 2
Research work: 100.0%

Bibliography

P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and Its Applications, vol. 35, D. Reidel Publishing Company, Dordrecht, 1987.

A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics, vol. 1764, Springer-Verlag, Berlin,  2001.

A. Weinstein, Lectures on Symplectic Manifolds, Conference Series in Mathematics, vol.  29, AMS, Providence, 1977.

R. Abraham and J. Marsden, Foundations of Mechanics, 2nd edition, Addison-Wesley Publ. Company, Inc., 1978.

D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford Mathematical Monographs, Oxford University Press, New York, 1995.