Course Plan: Singular Symplectic Manifolds

Lecturer: Eva Miranda

Session 1 — Motivation: Why Singular Symplectic Geometry?

• Motivating examples: The space of geodesics on the Lorenz plane and the regularized restricted three-body problem.

• Singular symplectic structures as natural phase spaces.

• Introduction to Poisson manifolds: definition and first examples

• Symplectic manifolds as special cases

• Overview of the lectures

Session 2 — From Symplectic to Poisson manifolds.

• Definition of symplectic manifold. Examples.

• Moser trick. Darboux theorem. Classification of compact symplectic surfaces.

• Symplectic and Hamiltonian vector fields. Examples and obstructions.

• Poisson bracket of a symplectic structure.

• Integrable systems. The Arnold-Liouville-Mineur theorem.

• Hamiltonian actions and the proof of Arnold-Liouville-Mineur theorem.

Session 3. Problem session and brackets.

 Session 4- Structure of Poisson Manifolds and b-Poisson manifolds

•  General Poisson brackets. Examples.b-Poisson manifolds. Definition and some examples.

• Hamiltonian vector fields. The symplectic foliation.

Session 5- From local to global Poisson manifolds. Weinstein theorem and Poisson cohomology.

• Weinstein splitting theorem. Local normal forms and linearization on Poisson geometry: Weinstein splitting theorem and Conn's linearization theorem.

Session 6 (October 28)

• Bivector fields as Poisson structures. Bivector fields calculus. 

• The Schouten bracket

• The Poisson cohomology of a Poisson manifold.

•  b-Poisson structure: Associated Poisson submanifolds: The critical hypersurface of a b-Poisson manifold.

Session 7 (November 10) — A dual language: b-Poisson Manifolds and b-symplectic Forms

• Foliated cohomology invariants of the regular symplectic foliation on the critical set

• The Serre-Swann Theorem

Session 8 (November 11) — Normal Forms and First Applications

• The b-tangent bundle and b-symplectic forms

• The b-complex

• The Mazzeo-Melrose theorem

• Introduction to b-contact structures

• Further examples (b^m, c-Poisson), motivation for singular symplectic geometry

Session 9—(November 18) Normal forms and Applications: Action angle-coordinates and toric symplectic manifolds in the b-world. 

• Moser theorem and b-Darboux theorem (Guillemin–Miranda–Pires).

• Application 1: Classification of 2-dimensional b-symplectic manifolds (Radko)

•  Application 2: Action–angle theorem for b-symplectic manifolds. Aplications to KAM theorem.

• Application 3: The b-Delzant Theorem

Special Problem session scheduled for Monday 24: The session will have two parts:

Part I. Discussion of Hofer norms for b-symplectic manifolds (by Adrian Dawid).

Part II. Resolution of problems in Set 4. 

Session 10 (November 25)— b^m-symplectic Manifolds: Topology and Obstructions

• Examples and counterexamples.

• Topological properties and obstructions

• Folded symplectic manifolds

• The desingularization (Deblogging) Theorem. Applications.

On Thursday 27 we will have a special Nachdiplom Kolloquium by Alberto Ibort (I am attaching a small poster below).

13:15–15:00 Alberto Ibort, 

Universidad Carlos III de Madrid

Witten-Floer with symmetry

Abstract

After revisiting the approach by Witten and Floer to Morse theory, that is, the construction of a cochain complex, using inspiration from classical and quantum mechanics, whose cohomological invariants provide relevant information about the supporting spaces, we will address the problem of incorporating a symmetry group to the picture and the study of the corresponding equivariant theory. This is part of a research program started in collaboration with E. Miranda.

Session 11 (December 2)— The (singular) Weinstein conjecture.

• Periodic orbits in the b-setting

• The singular Weinstein conjecture and generalizations of the Weinstein conjecture.

• Escape orbits and exoplanets.

Session 12 (December 9)- The singular Arnold conjecture.

• The Arnold conjecture and b-Floer Homology. Approach 1- Following Brugués-Miranda-Oms).

• Approach 2: Following Witten and Ibort-Miranda. Defining Floer homology of more general Poisson manifolds.

• The case of convex symplectic manifolds.

Session 13 (December 16) — Beyond b: E-symplectic Manifolds and Poisson Limits. Open problems.

• E-symplectic geometry and its relation to b- and b^m-structures

• A Hironaka desingularization for transversally linearizable Poisson structures

• Limits to Poisson geometry and broader generalizations

• Outlook and Open Problems

Problem sessions:

Problem sessions will be held on:

1. October 27 at 10:15. The first part of the session will be lead by Baptiste and he will address the exercises on this list  exercisebaptise.pdf

The second part of the session will be addressed to solving exercises 2/3, 4, 6, 8, 10 in the list set3.pdf

2. November 24 at 10:15- Special session with two parts. Part I  from 10:15 to 11. dedicated to discussions on Hofer norms for singular symplectic manifolds (by Adrian Dawid) Part II from 11:15 to 1200 will be dedicated to solving exercises on Set 4.
   
   

All extra sessions will take place in the same room.

Please note that there will be no class on November 4. This session will instead be held on November 10 at 10:15 a.m., also in the same room.