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CRC 1283: Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications

In the last decades, the two major mathematical fields of analysis and probability theory have become more and more intertwined, both with respect to aims and achievements and with respect to methodology and techniques. A considerable number of significant results in either field was recognized to be relevant for the other. As always in mathematics when there are rapid new developments on the conceptual and theoretical side, a vast number of new applications emerged in other sciences, in particular in economics, physics, and biology. For example, the developments in stochastics provided the perfect language to formulate a rigorous theory of financial markets; the theoretical insight created new markets, which in turn led to new mathematical questions and methods. Likewise, random matrix theory has led to a detailed understanding of physical systems, for instance in a non-perturbative approach to quantum field theory. The other way round, random matrices opened up new mathematical questions and challenges in analysis and probability theory. Also, progress in point process theory around determinantal and related systems proved helpful for the further development and understanding of random spatial arrangements. In biology, the mutually beneficial cross-fertilization between diffusion theory and population genetics has led to far-reaching insight into the processes of evolution.

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.

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