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

This CRC has set out from the fact that 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. 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 provide 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. Likewise, random matrix theory has led to a detailed understanding of many physical systems on the one hand, and to new challenges in analysis and probability theory on the other hand. In biology, the mutually beneficial cross-fertilization between diffusion theory and population genetics has led to far-reaching insights 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. state spaces or 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. On the one hand, noise comes into a model as a nuisance, destroying the possibility of a deterministic description of the dynamics.

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. E.g. in the presence of “sufficient” noise, ill-defined deterministic dynamics turn into well-defined mathematical objects.

The focal points for future research will be: Rough state spaces and diverse geometry, restricted versus unrestricted uniqueness and regularization by noise, robust economics and strategic aspects of investment under uncertainty, fluctuating hydrodynamics and stochastic fluid dynamics, and universality in higher dimensions.

Thus, the CRC will focus 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 natural sciences.

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