Sonderforschungsbereiche

Diskrete Strukturen spielen in der Mathematik und ihren Anwendungen eine zentrale Rolle. Die grundsätzliche Tendenz menschlichen Denkens, komplexe Sachverhalte durch Begriffe zu strukturieren, zeigt sich in der Mathematik in der fortschreitenden Axiomatisierung, Formalisierung und Algebraisierung.

Diskrete und kontinuierliche Methoden befinden sich in der Mathematik in einem lebendigen Wechselspiel: Die Entwicklung der homologischen Methoden in der algebraischen Topologie diente dazu, topologische, also deformierbare Objekte mit Hilfe algebraischer Strukturen zu beschreiben. Differentialgleichungen lassen sich approximativ lösen, indem man den Konfigurationsraum diskretisiert. Umgekehrt werden kombinatorische Probleme durch Übergang zur großen Zahl analytisch zugänglich. Diese Dialektik spiegelt sich auch auf einer anderen Ebene wider. Unser sprachliches, lineares Denken drängt zur Algebraisierung. Andererseits ermöglichen es gerade die modernen, schnellen, auf Diskretisierung beruhenden Rechensysteme, geometrische Objekte nicht nur durch Graphiken, Bilder und Prozeßabläufe zu veranschaulichen, sondern auch interaktiv zu manipulieren.

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.

Randomness (or noise) is a common characteristic of many mathematical models in the above fields; the underlying structures (e.g. of the underlying state spaces or the observables) are of low regularity. The CRC is devoted to the deep analysis of both the “bad” and the “good” features of randomness and noise, and to exploiting the underlying (albeit) low regularity structures. Usually, noise comes into a model as a nuisance, destroying, for example, the possibility of an exact (that is, deterministic) description of the dynamics of a system. In this case, noise is treated as a random term subsuming influences that are generically impossible to quantify, for example because they are caused by strong disorder or by the many-particle nature of the problem.

Another example for “bad” randomness is the lack of information about probability distributions when modeling financial markets; such model “uncertainty” is considered generic and has to be “tamed”. On the other hand, randomness does have definite advantages, and techniques to exploit low regularity are being developed. In the presence of “sufficient” noise for example, ill-defined deterministic dynamics turn into well-defined mathematical objects when reformulated in probabilistic terms. The CRC focuses on “taming uncertainty” and “profiting from randomness and low regularity”. It aims at developing the underlying concepts and theory, and at their application to long-standing open problems in economics and the sciences.