Besse and Zoll manifolds
Advanced Master 2 course, UMPA, ENS de Lyon
January-March 2020

ANNOUNCEMENT: due to the COVID-19 (coronavirus) outbreak, the lecture scheduled for March 20th is canceled. The material of the lecture will be available on this webpage by March 19th, and the students are welcome to contact Marco Mazzucchelli (marco.mazzucchelli@ens-lyon.fr) in order to ask questions concerning the course; if needed, a video call can be scheduled.

Instructor

Schedule

  • Friday 9:00am - 12:00pm, room M7411

Exam

  • March 27, 2020.
    Solutions:

Calendar

January 17: Symplectic linear algebra
Symplectic manifolds
Darboux theorem
Symplectomorphisms
Hamiltonian isotopies
Contact manifolds
Reeb flows
January 24: Geodesic flows
Besse contact manifolds
Wadsley theorem
Strata of a Besse contact manifold
January 31: Zoll contact manifolds
Principal circle bundles
Euler class
Gysin sequence
Uniqueness of Zoll contact forms on a given 3-manifold
February 7: Geodesic flow of a sphere of revolution
Classification of Zoll spheres of revolution
February 14: The curve shortening semi-flow
The min-max theorem on the space of embedded loops
February 21: The Lusternik-Schnirelmann theorem on subordinated min-max values
Topology of the space of embedded loops on the 2-sphere
Existence of three simple closed geodesics on any Riemannian 2-sphere
March 13: Spectral characterization of Zoll Riemannian 2-spheres
Open conjectures in Reeb dynamics
Reeb flows on convex contact spheres
Clarke's action functional
March 20: Variational properties of the Clarke's action functional
Proof of the Weinstein conjecture for convex contact spheres
Notes:
UMPA ENS de Lyon Université de Lyon