课程题目：Geometryand Dynamics of Singular Symplectic Manifolds
主 讲 人：prof. Dr. Eva Miranda
单 位：Universitat Politècnica de Catalunya and ParisSciences et Lettres
We will describe a novelgeometrical approach to classical problems in Celestial Mechanics concerning collisions. The upshot of our methods is that the singularities (collisions, infinity line) are included in the geometrical techniques(as b-symplectic manifolds, b-contact manifolds). We will focus on the geometry and Dynamics of these manifolds and describe several techniques such as desingularization, normal forms, action-angle coordinates and perturbation theory used in this study.
九月7日19:00：Introductionto the course. Basic definitions in Symplectic Geometry and motivation for b-symplectic geometry. B-symplectic manifolds as Poisson manifolds.
九月9日19:00：Melroselanguage of b-forms. b-symplectic forms on b-Poisson manifolds. The geometry of the critical set. More degenerate forms b^m-symplectic forms and b^m contact forms. Desingularization of b^m-forms.
九月14日19:00：The pathmethod for b^m-symplectic structures. Local normal form (b^m-Darboux theorem) and extension theorems. b^m-Structures to the test: Examples in Fluid Dynamics and Celestial Mechanics. The b-symplectic and b-contact geometry of the restricted three body problem and of Beltrami fields. Application: Finding periodic orbits for trajectories of a satellite in the restricted three body problem.
九月21日19:00：Someclassical problems for b^m-symplectic and b^m-contact manifolds: The (singular) Weinstein conjecture. Connection to escape orbits in Celestial Mechanics.
九月23日19:00：Moresymmetries: Toric actions, action-angle coordinates and Integrable systems on b^m-symplectic manifolds. Applications: ons Perturbations of integrable systems and KAM theory.
九月30日19:00：Finale: Open problems including Arnold conjecture and Floer homology of Singular SymplecticManifolds.