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Summer School 2007 in Cologne

Scientific Programme

Morning Lectures

The academic 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 senior speaker.
Billiards and Teichmüller curves. Series of three lectures, by Prof. Möller
Teichmüller 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.
Although 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.
Variational methods, 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.

Dynamics and geometry of billiards. Series of two or three lectures by Prof. Tabachnikov.
  1. Integrable billiards: conics and quadrics
  2. Around Birkhoff's conjecture
  3. Periodic trajectories and Morse theory
  4. Billiards and sub-Riemannian geometry
  5. 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 place in 2005.

Design © 2005-2006 Jiří Horák, Photos courtesy of Dierk Lürbke
last changed01. Feb, 2007