Symplectic geometry has its roots in the Hamiltonian formulation of classical mechanics. The canonical symplectic form on phase space occurs in Hamilton's equation. Symplectic geometry studies local and global properties of symplectic forms and Hamiltonian systems. A famous conjecture by Arnol'd, for instance, gives a lower bound on the number of periodic orbits of a Hamiltonian system.There are deep connections between symplectic geometry and the theory of dynamical systems, algebraic geometry and modern physics, for example string theory.
This course will focus on the foundations of symplectic geometry:
- linear symplectic geometry
- canonical symplectic form on a cotangent bundle
- symplectic manifolds, (co-)isotropic and Lagrangian submanifolds
- Moser's isotopy method
- Darboux's theorem
- Weinstein's neighbourhood theorem for a submanifold of symplectic manifold
If time permits, we will also cover one or more of the following topics:
- Hamiltonian group actions, momentum maps and symplectic reduction
- Delzant's and the Atiyah-Guillemin-Sternberg convexity theorem
- constructions of symplectic manifolds, e.g. symplectic fibrations and blow-ups
The last two lectures will be reserved for a panorama of recent results in the field of symplectic geometry, for instance the existence of symplectic capacities and the Arnol'd conjecture.
The standard notions taught in a first course on differential geometry, such as: manifold, smooth map, immersion, submersion, tangent vector, Lie derivative along a vector field, the flow of a vector field, tangent bundle, differential form, de Rham cohomology. A suitable reference for this is:
J. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, Springer, 2002.
Some knowledge of classical mechanics can be useful in understanding the context and some examples.