Mastermath Course on Symplectic Geometry - 8EC, spring semester, 2018/ 2019

Schedule

General information

What is symplectic geometry?

A symplectic structure is a closed and nondegenerate 2-form on a smooth manifold. Such a form is similar to a Riemannian metric. However, while a Riemannian metric measures distances and angles, a symplectic structure measures areas. The closedness condition is an analogue of the notion of flatness for a metric. 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.

Many problems in symplectic geometry are either flexible or rigid. In the flexible case methods from differential topology, such as Gromov's h-principle, can be applied to construct objects. In the rigid case partial differential equations can be used to define symplectic invariants. As an example, holomorphic curves (solutions of the Cauchy-Riemann equations) are used to define the so-called Gromov-Witten invariants.

Apart from classical mechanics, symplectic structures appear in a few other fields, for example in:

• Algebraic geometry: Every smooth algebraic subvariety of the complex projective space carries a canonical symplectic form.
• Gauge theory: The space of Yang-Mills instantons over a product of two real surfaces is closely related to the space of gauge equivalence classes of flat connections over a surface, which carries a canonical symplectic form.
• Differential topology: Certain invariants of smooth real 4-manifolds (the Seiberg-Witten invariants) are closely related to certain symplectic invariants (the Gromov-Witten invariants).

Contents of this course

Some highlights of this course will be the following:

• A normal form theorem for a submanifold of a symplectic manifold. A special case of this is Darboux's theorem, which states that locally, all symplectic manifolds look the same.
• Symplectic reduction for a Hamiltonian Lie group action. This corresponds to the reduction of the degrees of freedom of a mechanical system. It gives rise to many examples of symplectic manifolds.
• A construction of symplectic forms on open manifolds, which is based on Gromov's h-principle.

Here is a more complete list of topics that we will cover:

• linear symplectic geometry
• symplectic manifolds
• canonical symplectic form on a cotangent bundle
• symplectomorphisms, Hamiltonian diffeomorphisms
• Poisson bracket
• Moser's isotopy method
• symplectic, (co-)isotropic and Lagrangian submanifolds of a symplectic manifold
• normal form theorem for a submanifold of a symplectic manifold
• Darboux's theorem
• Weinstein's neighbourhood theorem for a Lagrangian submanifold
• Hamiltonian Lie group actions, momentum maps
• symplectic reduction, Marsden-Weinstein quotient
• coadjoint orbits
• Gromov's h-principle and the construction of symplectic forms on open manifolds

We will also explain connections to classical mechanics, such as Noether's theorem and the reduction of degrees of freedom. Furthermore, we will develop the basics of contact geometry, which is a field that is closely related to symplectic geometry.

If time permits, we will also cover one or more of the following topics:

• Delzant's classification of toric symplectic manifolds
• Atiyah-Guillemin-Sternberg convexity theorem

The last lecture 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.

Some mathematicians whom we will encounter in this course, are the following:

Literature

Prerequisites

• manifold
• smooth map
• immersion
• submersion
• tangent vector
• tangent bundle
• differential form
• the flow of a vector field
• Lie derivative along a vector field
• de Rham cohomology