There will be a 6-hours precourse by
Otto van Koert and Felix Schlenk. Then there will be five more
specialized courses, by Youngjin Bae, Urs Frauenfelder, Otto van
Koert, Seonhee Lim, Ana Rechtman and Felix Schlenk.
Each
specialized course will have two 1-hour sessions of exercises.
All
lectures are in English.
Here is a tentaive schedule (PC =
Precourse, C = Course, Ex = Exercices):
Monday 25 May |
Tuesday 26 May |
Wednesday 27 May |
Thursday 28 May |
Friday 29 May |
Saturday 30 May |
Sunday 31 May |
Monday 1 June |
Tuesday 2 June |
Wednesday 3 June |
|
8:45-9:45 |
PC Van Koert |
C Lim |
C Bae |
|
C Rechtman |
C Frauenfelder |
C Frauenfelder |
C Frauenfelder |
C Van Koert |
|
10:00-11:00 |
PC Van Koert |
PC Van Koert |
C Lim |
C Rechtman |
|
C Rechtman |
C Frauenfelder |
C Frauenfelder |
C Frauenfelder |
C Schlenk |
11:15-12:15 |
PC Van Koert |
C Lim |
C Bae |
C Rechtman |
Excursion |
C Schlenk |
C Van Koert |
C Van Koert |
C Van Koert |
C Schlenk |
12:30-14:00 |
lunch |
lunch |
lunch |
lunch |
|
lunch |
lunch |
lunch |
lunch |
lunch |
14:00-15:00 |
PC Schlenk |
C Bae |
C Lim |
Ex Bae |
|
Ex Rechtman |
Ex Frauenfelder |
Ex Van Koert |
Ex Van Koert |
|
15:15-16:15 |
PC Schlenk |
Ex Lim |
Ex Bae |
Ex Rechtman |
Ex Schlenk |
Ex Van Koert |
Ex Frauenfelder |
Ex Frauenfelder |
|
|
16:30-17:30 |
Ex Bae |
Precourse Otto van Koert and Felix Schlenk
Definition of geodesics, geodesics on round spheres and real projective space, the flat torus, surfaces of revolution (the Clairaut Integral), Kepler's 2-body problem. Configuration spaces; Definition of the billiard map, billiards in rectangles (unfolding, each orbit is either closed or uniformly filling), billiards in the circle (each orbit is either closed or with bounce points uniformly distributed on the boundary)
Examples Seonhee Lim
The geodesic flow, a non-round metric on S^2 all of whose geodesics are closed, magnetic flows, magnetic monopoles, Arnold's example on T^4, closed geodesics on hyperbolic surfaces are dense in phase space, the horocycle flow has no closed orbits, the rotating 3-body problem
Billiards Youngjin Bae
The billiard map is measure preserving, billiards in ellipses, two applications (a light trap and a non-illuminatable room), The Poincaré recurrence theorem, non-existence for light-traps for non-parallel rays, Caustics, existence of caustics in smooth stricly convex billiards via KAM, Geodesics on the 3-ellipsoid, and relation to billiard orbits. Billiards.Tabachnikov and Omnibus
Variational Methods Felix Schlenk
The mountain pass theorem in R^n, broken geodesics, Existence of a geodesic on S^2 for any Riemannian metric, existence of closed orbits in a convex billiard (for any number p of bounces, with rotation number q < p/2), relation to Poincaré's Last Geometric theorem. Notes
Flows without periodic orbits and plugs Ana Rechtman
The problem of determining when the flow of a non-singular vector field on a closed 3-manifold has a periodic orbit has a long history. We will study examples of vector fields whose flow has no periodic orbits on any closed 3-manifold. These were first constructed by P. A. Schweitzer for C^1 vector fields, and then by K. Kuperberg in the smooth and real analytic categories. Schweitzer's construction was then achieved in the volume preserving category by G. Kuperberg, giving C^1 volume preserving vector fields without periodic orbits. An open question is whether a volume preserving flow on a closed 3-manifold must have periodic orbits. The construction of these flows are all based on the use of plugs: a devise that allows to destroy periodic orbits. In the course I will present the constructions and main applications. Ghys and Matsumoto
The restricted 3-body problem Urs Frauenfelder
The restricted 3-body problem in inertial coordinates. The restrichted 3-body problem in rotating coordinates and the Jacobi integral. Hill's lunar theory. Regularization. Periodic orbits in the restricted three body problem: -The direct and retrograde period orbit. -The Lyapunov periodic orbit. -The antisymplectic involution and symmetric periodic orbits. -Birkhoff's shooting method. Global surfaces of section -Existence of global surfaces of section according to Poincarée, Conley, and McGehee. -Area preserving disk and annulus maps. Results of Birkhoff and Franks-Handel. Homoclinic and Heteroclinic periodic orbits in the restricted three body problem and its applications to space mission design. Outlook on holomorphic curve techniques in the restricted three body problem and the work of Hofer, Wysocki, and Zehnder. Resticted 3-Body Problem