Symplectic Geometry - M2 - 8EC

Prerequisites

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. 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 our 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 we 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
  • coadjoint orbits

We will also explain connections to classical mechanics, such as Noether's theorem and the reduction of degrees of freedom. 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 for the image of the momentum map

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.

Lecturer

Fabian Ziltener (UU)

Heinz Hanßmann (UU)