Program

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



Abstracts

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



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