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Summer School 2013 in Cologne

Scientific Programme

Morning Lectures and Afternoon Working Groups

The academic programme consists of a series of morning lectures and working groups in the afternoon.
We will have three series of lectures introducing the main subjects of the week. The goal of the lectures will be to communicate the fundamental motivating questions in each field, the tools used to address them, and the important results.

Action minimizing periodic orbits in the N-body problem. Series of four lectures, by Prof. Jacques Féjoz
  1. Central configurations
  2. The Lagrangian action and its minima
  3. Marchal's theorem
  4. The example of the P12 family

Twist maps with non-periodic angles.
Series of four lectures, by Prof. Rafael Ortega

Consider an annulus A with coordinates (q, r), q + 2p q, r [a, b]. An area-preserving map (q, r) (q1, r1) is twist if it satisfies q1 /r>0. Twist maps have been extensively studied and they are useful to understand the dynamics of autonomous or periodic Hamiltonian systems of low dimension. In this course we study twist maps without assuming periodicity on q. In other words, the annulus A is replaced by a strip S = R x [a, b]. This new class of twist maps can be applied to the study of ping-pong models when the motion of the racket is not periodic.

Closed orbits of classical Hamiltonian systems in cotangent bundles. Series of four lectures, by Prof. Felix Schlenk

The search for closed orbits in celestial mechanics was a driving force for the development of Hamiltonian systems and symplectic geometry. (One wants to know whether a meteorite, or the moon, eventually falls onto the earth, or stays on its periodic (???) orbit.) The most natural phase spaces are cotangent bundles. We shall look at classical dynamical systems on theses phase spaces (over a compact configuration space) and shall describe two methods for proving the existence of a closed orbit on a given energy level: The classical minimax method, and the action selector method. For the latter method, we shall describe a recent construction of Alberto Abbondandolo and myself that does not use Floer homology, but "only" the tools in Floer's proof of the degenerate Arnold conjecture.
  1. Configuration spaces and phase spaces; every closed manifold naturally arises as a configuration space; classical Hamiltonian systems on phase spaces (= cotangent bundles): geodesic flows, scalar potentials, electro-magnetic potentials, magnetic monopoles; examples (of geodesic flows, the horocycle flow).
  2. The minimax principle, with proof. Closed orbits at high energies (the Jacobi-metric trick, the non-simply connected case, and the simply connected case).
  3. The action spectrum of a Hamiltonian; definition of an action selector; finiteness of the Hofer-Zehnder capacity, almost existence of closed orbits at small energies in a magnetic field.
  4. Construction of an action selector, without Floer homology; sketch of the J-holomorphic tools used in the construction; an application to Hofer geometry (unboundedness of Ham).

All lectures are supposed to be of introductory nature; they will be geared towards an audience that has some background, but is not necessarily familiar with the subjects taught. We aim to convey enough material so that the participants can then, with a reasonable amount of work, read a current research paper or go to a research talk, and be able to get something useful out of it.

Afternoon Working Groups

In the afternoon, the participants will break up into groups, and work with a mentor on problems that will give them a hands-on feel for the methods of the field. The afternoon groups will be related to the topics discussed in morning lectures. Participants choose their working group in advance when they are accepted for participation.

In these groups, the mentors explain ideas and set problems, which the participants then discuss, try to understand and work out. Throughout the afternoon, the mentors shall lecture for no more than 30-40 minutes total. In the remaining time, the participants will be discussing in smaller groups, work out examples and details of proofs, and present the results to each other. The mentor will be present to guide the discussion, help the subgroups and explain material that isn't clear. Ideally, the subgroups should be getting together in the evenings to continue the discussion, or to prepare a presentation for the next day.

Design 2005-2013 Jiř Hork, Photos courtesy of Dierk Lrbke
last changed: 01 March 2013