# CRC 701: Spectral Structures and Topological Methods in Mathematics

The CRC 701 pursues the vision of reinforcing and building bridges between various branches of theoretical and applied mathematics. The guiding principles in this undertaking are the investigation of spectral structures and the development and application of topological methods throughout mathematics and related sciences.

Spectral structures are omnipresent in mathematics and many of its application areas. In mathematics, they occur in different appearances such as eigenvalues of differential operators, critical points of transfer operators in dynamical systems, or prime ideals in a ring. In the natural sciences, they serve to comprehend diverse phenomena such as the frequency range of oscillations, diffraction and scattering behaviour, or energy levels of quantum systems. Topological methods form a unifying tool for organising these different spectral structures as well as for analysing the dynamics of an underlying deterministic or stochastic system.

Many significant developments in mathematics are connected with spectral structures and topological methods, and have their origins in applied fields, for example in new concepts of mathematical physics or in fluid dynamics, crystallography and material sciences. We mention the Seiberg-Witten invariants in topology and quantum groups in algebra, mathematical quasicrystals and their ramifications for harmonic analysis and its connections with dynamical systems and topology, universal spectral distributions from quantum physics and their appearance in number theory, and concepts from quantum field theory applied to moduli spaces in geometry and topology.

The research within the CRC 701 establishes strong connections at the interface between theoretical and applied mathematics: algebraic geometry and dynamical systems, representation theory and probability theory, stochastic analysis and numerics, harmonic analysis connecting nonlinear partial differential equations, stochastics and analytical number theory. The special added value of the CRC 701 is to realise the full potential of the mathematical theories around these interfaces and to incorporate new developments into a coherent research environment.

## Key Figures

Projects: | 21 |

Investigators: | 23 |

Scientists: | more than 80 |

Guests: | more than 750 |

Publications/Preprints: | more than 900 |

Talks: | more than 1420 |

Annual Budget: | ca. 2,1 Mio. € |