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IRTG "Geometry and Analysis of Symmetries"

Study Program

Organizing an International Research Training Group for people with diverse backgrounds in two relatively distant places requires new forms of teaching.

This school will rely on the following main elements:

1.Weekly  research seminars at both universities.
In Metz, the LMAM (UMR 7122 - CNRS ) finances the seminar AGA (attended by all scientists proposed to participate in the Metz group of the IRTG), which invites 30 - 40 speakers each academic year.
2.Weekly "working group" seminars at both universities, in which students, researchers, and guests discuss their research projects and report either on their own results or on interesting new developments in the field. Typically these working groups will be organized by the graduate students and post-docs.
3.Joint workshops between the groups in Metz and in Paderborn, where the participants study topics of common interest:
  • The members select a topic for the seminar. An organizer defines the seminar plan (possibly with the help of a specialist external to the two groups) and  assigns the talks.
  • The students and the researchers coordinate the preparation of their talks (i.e., they inform the other speakers which prerequisites are needed for their exposition and which background knowledge will be provided by their own talk. Communication among the different participant is either verbal or via e-mail).
  • Each year two to three such one-week meetings will take place, either in Metz or in Paderborn or in an appropriate institution (MFO Oberwolfach, CIRM Luminy etc.). During these meetings the research groups  also discuss in detail the progress of their projects.
  • Occasionally, a set of lectures by an external specialist can be delivered during these meetings.
4.Ph.D. students and post-docs have the possibility to work at the partner university either for repeated short stays or for longer stays up to a total of 18 months during three years. Such stays are strongly encouraged and require only the consent of the speakers and no formal change of the affiliation.
5.Each graduate student has a Ph.D. advisor at her/his home institution and a mentor at the partner university. Apart from monitoring the scientific progress the mentors counsel the student on professional aspects such as preparation of presentations, scientific writing and participation in congresses and summer schools. Furthermore they help to integrate the student in national and international scientific networks. At the end of the first two years of Ph.D. studies the mentors report on the students progress and prospects. This report serves as a base for the decision on the student's admission to her/his third year.
6.Regular courses given by the senior members of the International Research Training Group:
Their are two types of courses: A modular sequence of introductory mini-courses (explaining important principles in special cases not requiring the full machinery needed in general) and more traditional topical courses.
The advanced courses are given by each lecturer at her/his home university. She/he provides detailed course material (in English) for both universities (preferably online). The lecturer gives a compact version of her/his course at the partner university, typically in the form of an intensive one- or two-week course.
7.Local mini-courses given by external specialists upon invitation:
This part of the course program is meant to give the young scientists the opportunity of learning new developments by leaders in the fields.
The choice of the topics and the organization of the mini-courses will be partly decided with the students and post-docs. It is expected that the lecturers furnish course material and lecture notes and make them available to the entire group.
8.Professional qualification:
The graduate program offers to their participants the possibility of attending  special courses for the development of their professional qualifications. At the University of Metz, these courses are offered in the form of "Modules scientifiques" and "Modules professionels" of the Ecole Doctorale IAEM. For instance, during the academic year 2003-2004, the courses offered included "Scientific Communication in English", the series of lectures "Recherche et entreprise", and an introduction to the typesetting system TeX.
Each Ph.D. student in the Paderborn branch of the IRTG is entitled to participate in three seminars (2 or 3 days) on general professional qualifications. The topics of these seminars are to be chosen from a list including "Rhetorics", "Presentations", "Application Training", "Marketing", etc.
The seminars will be conducted by outside professionals.

The course work will be supported by a special mathematical database software designed to compile coherent study material with prescribed content.
It can be used in particular to produce texts on the background material necessary for advanced or mini-courses.

The course program is decided in accordance with the plenary assembly which meets twice a year during the joint one-week workshops.
Depending on demand and resources, additional courses may be offered.
The separate seminar programs will be organized locally.
Students and post-docs are encouraged to help design and organize mini-courses and seminars.

Formal requirements

Attendance at the regular courses is not compulsory for students and post-docs. A guided choice among these courses allows them to start their research projects.
On the other hand, it is expected that all students and post-docs participate actively in the seminar program and report at least once a semester on their progress:

  • There are biannual two-day self-evaluation sessions at each branch, where all Ph.D. students have to report on their progress.
  • Each Ph.D. student has to give a talk in the joint seminar at least once a year.

It is  expected that all members of the  International Research Training Group take part in the joint meetings.

Course program

Each semester, a cycle of introductory mini-courses and one to three advanced courses on specialized subjects will be delivered.
Both types of courses are typically formatted on the base of 48 hours, divided into instructional lessons and exercise sessions.
Specialized courses take place in the second semester.

Introductory mini-courses

Many of the modern research subjects proposed in this project were first formulated in a "compact setting" (compact Lie group actions; compact topological spaces, compact manifolds etc.) and  a good understanding of these cases remains an important guideline in present day research. The International Research Training Group therefore proposes  a modular set of mini-courses illustrating the geometry and the analysis of symmetries in the compact case, as well as their natural generalizations to non-compact situations. They are complemented  by introductions to neighboring subjects, relevant to the theme of the International Research Training Group. More precisely, the topics to be presented are selected among the following:

  • Symplectic and Kähler geometry;
  • General representation theory of compact groups on topological vector spaces;
  • Cartan-Weyl-theory of representations with highest weight;
  • Harmonic analysis on homogeneous spaces with compact symmetry group;
  • Geometric quantization and orbit method;
  • Realization of representations in homogeneous vector bundles and cohomology spaces;
  • Character and multiplicity formulas, branching laws;
  • Spherical functions;
  • Banach algebras;
  • Introduction to homological algebra.

The specific choice among the above topics will depend on the lecturers and on the students' interests. All these topics will be chosen with a focus on the research program of the International Research Training Group. 

Advanced courses (listed in alphabetical order)

  • Causal symmetric spaces
  • Fréchet and DF-spaces
  • Groups and Lie algebras of unimodular vector fields
  • Homological methods in representation theory
  • Introduction to non-commutative geometry
  • Introduction to Wess-Zumino-Witten models
  • Microlocal analysis
  • Representation theory of loop groups
  • Special functions associated with root systems
  • Topological tensor products and nuclear spaces

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