GRK 1052:  Representation Theory and its Applications in Mathematics and Physics

• Subject Area: Mathematics
• Term: from 2005 to 2009
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 375986

Representation theory is a branch of mathematics that studies objects by investigating their symmetries. These symmetries can be rotations, reflections, translations or abstract generalisations thereof. Such considerations can have far-reaching consequences. For instance, they impose constraints which drastically reduce the number of candidates for various fundamental physical theories. This makes representation theory one of the most important tools in modern theoretical physics. What is more, representation theory also plays an important role in concrete calculations of experimentally observable quantities. By developing beyond the elementary, intuitive concept of geometric symmetry, representation theory has shown itself to be useful in more and more branches of mathematics and theoretical physics.Representation theory provides methods for classifying the complex spectra of atoms and molecules. It is used to analyse systems in both, solid state physics and the physics of elementary particles. One significant modern application is the study of interacting many-body systems via their correlation functions. Each symmetry imposes a constraint on the evolution in time, and the presence of a sufficiently diverse collection of symmetries means that the behaviour of the system is more or less uniquely determined. The possibility of handling quantum mechanical many-body systems in this manner is particularly significant, for the direct numerical approach fails because of its (non-polynomial) complexity. Similar effects occur in applications within mathematics, one example being the interplay between geometry and representation theory. On the one hand one often uses symmetries of well-understood geometrical objects to construct representations, translating geometrical knowledge to knowledge about representation theory. On the other hand, if one understands a representation, then the constraints it imposes provide a lot of information about the geometric object under consideration.Summing up, the scientific aim of the Research Training Group is the development of representation-theoretic methods and their applications in mathematics and theoretical physics.

• DFG Programme: Research Training Groups
• Applicant Institution: Bergische Universität Wuppertal
• Participating Researchers: Professor Dr. Klaus Bongartz; Professor Dr. Walter Borho; Professor Dr. Klaus Fabricius; Professor Dr. David J. Green; Professor Dr. Frank Göhmann; Professor Dr. Roland Huber; Professor Dr. Michael Karbach; Professor Dr. Andreas Kluemper
• Spokesperson: Professor Dr. Markus Reineke, since 12/2006