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Summer School 2013 in Cologne
Scientific Programme
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Morning Lectures and Afternoon Working Groups
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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
- Central configurations
- The Lagrangian action and its minima
- Marchal's theorem
- 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.
- 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).
- The minimax principle, with proof.
Closed orbits at high energies (the Jacobi-metric trick,
the non-simply connected case, and the simply connected case).
- 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.
- 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.
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Afternoon Working Groups
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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.
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