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SFB 1060:  The Mathematics of Emergent Effects

Subject Area Mathematics
Term from 2013 to 2024
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Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 211504053
 
The central aim of the Collaborative Research Center (CRC) is to understand the emergence of new effects at larger scales from the interactions of many units at a smaller scale. It builds on the observation that rather different interactions at the small scale can lead to the same 'universal' behavior at larger scales. The CRC develops new rigorous mathematical concepts and tools to address this phenomenon and sharpens and tests these tools in specific situations. The research in the CRC is organized in three interrelated project groups.A. From quantum mechanics to condensed matter and materials scienceB. Stochastic systems and continuum limitsC. Geometric structures and high-dimensional problemsThe projects in project group A study collective effects of many particle systems in classical and quantum mechanics. Topics include the classical and quantum Boltzmann equation, efficient multilevel approximations of the many-body Schrödinger equation, the statistical mechanics of solids, and the effective description of complex material behavior.Project group B is focused on stochastic systems and continuum limits. In many problems in science the effective behavior usually does not depend on the details of the randomness on the microscopic level and on small scale interaction. Instead, interesting effects such as phase transitions, ageing, or universality emerge in the limit of infinite or continuous systems. Project group C addresses in a systematic way the analytic, probabilistic, and algorithmic aspects of high-dimensional problems, and deals with the identification and effective approximation of emerging geometric structures. In the first two funding periods the CRC has obtained new rigorous mathematical insights and, in the process, strengthened the ties and synergies between analysis, probability, and numerics in Bonn. Building on this, the CRC will address in the third funding period new challenges and exploit new interactions in particular in optimal transport and Wasserstein gradient flows, in learning and model order reduction for high-dimensional problems, in the study of universality, renormalization and self-similarity, at the frontier of multilinear harmonic analysis, and in the investigation of long-range interactions.
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