### Symplectic Geometry - M2 - 8EC

Prerequisites
Some prerequisites for this course are the 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.

Basic understanding of Lie groups and Lie algebras will also be useful, but not strictly necessary. The mastermath course ''Lie groups'' covers considerably more material from Lie theory than what we will use in this course.

In addition, we will use the notion of a smooth vector bundle over a manifold and some basic operations involving vector bundles, such as dualization and direct sum. The mastermath course ''Differential Geometry'' covers considerably more material from differential geometry than what we will use in this course.

A suitable reference for differential geometry is:

J. Lee, Introduction to Smooth Manifolds, second edition Graduate Texts in Mathematics, Springer, 2002.

The relevant chapters from this book are: 1-5,7-12,14-17,19,21. Some of the material covered in these chapters, in particular the one involving Lie groups, will be recalled in the lecture course.

Some knowledge of classical mechanics can be useful in understanding the context and some examples.

Aims and contents of this course
A symplectic structure is a closed and nondegenerate 2-form. 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.

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 moduli space of Yang-Mills instantons over a product of two real surfaces carries a canonical symplectic form.

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.

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

• linear symplectic geometry
• canonical symplectic form on a cotangent bundle
• symplectic manifolds, 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