Investigation of algebraic properties of gauge field theories on noncommutative spaces

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The physics of space-time at very short distances is not understood. Up to now a satisfactory theory of quantum gravitation has not bee found. Infinities at short distances lead to problems for quantum field theories. A natural possibility to resolve these singularities are noncommutative spaces, since in some sense they do not possess points, or - in other words - the physical system "lives" on a lattice. The development of the last years, in particular coming also from string theory, has shown how important is an understanding of gauge theories on these spaces. Gauge theories are physical theories, which are invariant under certain symmetry transformations; the standard model and also gravitation can be formulated in this way. Objective of this research project is the investigation of such theories. Especially one should try to understand noncommutative systems with invariance under general coordinate transformations.