Summer School 2007 in Cologne
programme consists of a series of morning lectures, afternoon working
groups and a few contributed talks.
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. We plan to have two lectures
each morning. Prof. McMullen will give one or two special lectures as a
Teichmüller curves. Series of three lectures, by Prof.
curves are algebraic curves in the moduli space of curves that are
geodesic for the Teichmüller metric. They arise from billiard
tables with a lot of symmetries and special dynamical behaviour. As an
introduction, examples of Teichmüller curves and their relatives,
Shimura curves, will be presented.
Teichmüller curves are almost never Shimura curves, they can be
characterized by their variation of Hodge structures, just as Shimura
curves can. This characterization will be the starting result for the
lectures. We study its consequences for the classification of
Teichmüller curves and use it to construct special billiard tables.
Variational methods in dynamical systems and billiards.
Series of three lectures, by Prof. Siburg.
in finite as well as in infinite dimensions, have proven to be very
effective in the study of dynamical systems, e.g., for finding periodic
orbits of Hamiltonian systems in classical mechanics. These lectures
will introduce various general principles from the calculus of
variations and apply them to the model case of mathematical billiards.
geometry of billiards.
Series of two or three lectures by Prof. Tabachnikov.
- Integrable billiards: conics and quadrics
- Around Birkhoff's conjecture
- Periodic trajectories and Morse theory
- Billiards and sub-Riemannian geometry
- Exotic billiards: Lorentz, Finsler,
magnetic and outer billiards
Dynamics over moduli
space. One lecture by senior speaker Prof. McMullen
The lectures are supposed to be of introductory nature; they will be
geared towards an audience that has some background in geometry, 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.
The lectures of Profs. Siburg and Tabachnikov as well as the lectures
of Prof. Möller and McMullen will be
coordinated and related.
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.
We will also have a few contributed talks in the afternoon by
participants whose research is related to our main topics.
The format of the academic program will be loosely modeled on two
hugely successful weeks, the Snowbird conference
in Algebraic Geometry 2004, and the Graduate Student Warmup Week to
the AMS Summer Research Institute in Algebraic
Geometry that took