# Research Program

HCM channels and expands the productivity of its participating institutes.

The research program of HCM is based on the unique constellation of mathematics in Bonn (see also Participating Institutes) that has been developed in a systematic way over many years, and that was strongly enhanced by the foundation of our cluster of excellence.

As a result, Bonn has strong and internationally leading groups in a remarkably broad range: pure mathematics, with foci on arithmetic and algebraic geometry, representation theory, global and harmonic analysis, differential geometry and topology, is strongly represented at the Mathematical Institute and the Max Planck Institute for Mathematics. Applied mathematics is represented in applied and stochastic analysis at the eponymous institute. Numerical analysis and scientific computing are very strong fields at the Institute for Numerical Simulation and the Research Institute for Discrete Mathematics is strongly application oriented. Moreover, the Institute for Economic and Social Sciences works on game theory, econometrics, and mathematical finance. The measures taken since the foundation of HCM have reinforced and broadened this basis.

It is probably fair to say that no other single location in Germany offers such a complete coverage of the entire range of mathematics from the abstract core to its concrete applications. This is the basis for an ambitious research program that attacks some of the most challenging problems both in mathematics and its applications.

HCM is dedicated to exploring links and exploiting synergies between different areas of mathematics. Examples include:

- new links between quantum field theory and fundamental questions in arithmetic and algebraic geometry;
- a growing interaction of analysis, geometry, algebraic K-theory, and topology;
- the relation between efficient simulation algorithms for virtual material design and a better theoretical understanding of effective governing laws of quantum many-particle systems;
- the use of the mathematical theory of optimal transport in theoretical economics.