We investigate the mathematical structure of physical theories. Especially in high-energy physics, but also in quantum gravity, where experiments are not always feasible, mathematical consistency plays an increasingly important role.
Conformal field theory and string theory form two prominent classes of theories which are largely determined by the condition of mathematical self-consistency. On this basis we investigate the physical consequences of string theory and quantum gravity in particle physics and cosmology. An important field of application is the quantum structure of black holes. Overall, we try to gain new insights into the quantum and symmetry structure of space and time by considering duality symmetries and by formulating quantum geometries motivated by string theory, such as non-associative geometry.
We are also developing new mathematical methods to simplify concrete calculations in established theories, for example the strong interaction, and to extend their scope. This approach to physics does also provide valuable impulses for pure mathematics, which are then further developed in their own way. String topology is a prominent example of this.