Geometry and Dynamics of Singular Symplectic Manifolds

IHP, Paris, November-December 2017, Salle 314

This course is done in the framework of Eva Miranda's Chaire d'Excellence de la Fondation Sciences Mathématiques de Paris



Official webpage:

Annonce: Exceptionnellement, mardi 12 Décembre le cours aura lieu à la Salle 201 de l'IHP



Summary: b-Calculus was introduced by Richard Melrose when considering pseudodifferential operators on manifolds with boundary. Later on, Ryszard Nest and Boris Tsygan applied these ideas to study the deformation quantization of symplectic manifolds with boundary.

The purpose of this minicourse is to unravel the geometrical structures (b-symplectic structures) behind this picture and describe some applications to Dynamical systems. b-Symplectic manifolds are Poisson manifolds which are symplectic away from an hypersurface and satisfy some transversality condition. b-Symplectic manifolds lie "close enough" to the symplectic category and indeed their study can be addressed using an "extended" De Rham complex. In particular many peculiarities from Symplectic manifolds are shared with b-symplectic manifolds. Using these ideas, we will study normal form theorems, action-angle theorems, toric actions and applications to KAM theory. At the end of the minicourse we present other singular symplectic structures such as folded symplectic structures and b^m-symplectic structures (for which the transversality condition is relaxed) and explain how they are related to b-symplectic and symplectic structures.

We will give a general overview of the theory using some examples in celestial mechanics as leitmotiv. For some of them (like double collision), we can even construct b^m-symplectic structures and m-folded structures. This apparent "duality" will be used as an excuse to closely explore the relation between the $b^m$-symplectic category with the symplectic and folded symplectic category. This relation depends surprisingly on the parity of m and is given by a desingularization procedure called deblogging. Time permitting, several applications of deblogging to dynamics and quantization will be presented.

Syllabus/Scheme of the lectures

  1. Lecture 1: Introduction. Poisson manifolds: First Examples.

  2. Lecture 2: Classical examples. The language of bivector fields. Multivector calculus and the Schouten bracket. Symplectic foliations.

  3. Lecture 3:  Weinstein's splitting theorem and normal forms. Conn's linearization theorem. Poisson Cohomology computation kit.

  4. Lecture 4: Poisson cohomology and integrable systems on Poisson manifolds.

  5. Lecture 5:  Part I: Proof of existence of action-angle coordinates in Poisson Geometry. Part II: Introduction to b-Poisson Geometry.

  6. Lecture 6:  Part I: Examples in b-Poisson Geometry. b-Darboux theorem. Modular vector fields in Poisson Geometry. Part II: Codimension one symplectic foliations and unimodular Poisson structures.

  7. Lecture 7:  Part I Poisson Geometry of the critical hypersurface of a b-Poisson manifold. Part II. Changing the glasses: A dual language for b-Poisson manifolds.

  8. Lecture 8:  Part I: Correction of assignments Part II. b-Symplectic structures. The b-complex. Mazzeo-Melrose formula. b-Cohomology and Poisson Cohomology.

  9. Lecture 9: (Salle 201-IHP) Part I: The path method in b-Symplectic Geometry: Relative and global. Applications to classification theorems in b-symplectic geometry (including Radkos' theorem, Delzant theorem and action-angle coordinates for $b$-symplectic manifolds) PartII: Other singular symplectic forms. Introduction to Deblogging: The magic stick. How to convert a $b^m$-symplectic manifold into a (folded) symplectic one.

  10. Lecture 10: Applications of Deblogging to Geometry and Topology of b-manifolds, to Dynamics and Quantization. Open problems (and open problems for the Working group).


Videos of the course:

Available in the webpage of the course at FSMP:


Complementary material:

Related activities of the Chair: