# Research Areas

**Background**

Fundamental progress in key areas of mathematics has often been driven by problems in physics, and similarly progress in basic physical research only became possible by the development of new mathematical theories. A good example for this mutual stimulation is the development of quantum physics in Göttingen in the early twentieth century, in close collaboration between mathematicians and physicists. This tradition also helps us to attract excellent students and established researchers.

The need for a joint effort by mathematicians and physicists on a wide range of problems has become more urgent recently because of newly discovered connections to algebraic topology and geometry, which yield conceptual insights in both mathematics and physics. At the same time, interaction with physics remains an important theme in mathematical areas such as functional analysis, differential geometry, and partial differential equations that have been shaped by this interaction for a long time. Accordingly, our research programme covers connections to physics in several areas of mathematics.

One focus of our RTG are topological and geometric notions related to quantum field theory and string theory. This includes Calabi–Yau varieties and twisted K-theory, which describe inner degrees of freedom in string theory; invariants of von Neumann algebras and their inclusions, which have proven to be a powerful tool in algebraic quantum field theory; non-commutative geometry, which may help to model the small-scale structure of spacetime; and coarse geometry, which may explain how the classical large-scale structure can emerge.

Another focus is the construction of quantum field theories, both perturbatively and non-perturbatively, and the study of their physical properties. This requires various mathematical tools, mostly from harmonic analysis, micro-local analysis, and the theory of von Neumann algebras. The study of hyperbolic differential equations and their singularities is a prerequisite to construct quantum fields propagating on singular spacetimes.

**Research areas**

Coarse geometry allows us to study global properties of metric spaces while disregarding their local small-scale structure. Such an approach may help to understand how classical notions from geometry can apply to spacetime despite its delicate small-scale structure. Non-commutative topology studies topological properties of generalised spaces, which are usually described by C*-algebras. Given the established usefulness of operator algebras in quantum physics, such algebras provide good candidates to model the small-scale structure of spacetime. The most interesting topological invariant in this context is K-theory; twisted K-theory occurs in the description of charges in string theory and is close enough to ordinary homotopy theory to be studied with classical topological means. Another classical area that has recently been linked to physics is the study of special algebraic varieties such as Calabi–Yau varieties, which describe inner degrees of freedom in string theory.

Topological invariants of von Neumann algebras and their inclusions are powerful tools in algebraic quantum field theory and are also relevant to several problems in pure mathematics like the classification of free group factors. Besides the theory of von Neumann algebras, the construction of quantum field theories and the study of their physical properties also requires ideas from other areas of mathematics such as harmonic analysis, scattering theory, and micro-local analysis. In particular, this includes the study of solutions of hyperbolic differential equations with suitable singularities and of diffraction effects in the context of hyperbolic differential equations on manifolds with singularities.