SFB 1060: The Mathematics of Emergent Effects

协作研究中心（CRC）的中心目标是了解在更大规模的新效应的出现，从许多单位在一个较小的规模的相互作用。它建立在这样的观察基础上：在小尺度上相当不同的相互作用可以导致在更大尺度上相同的“普遍”行为。CRC开发了新的严格的数学概念和工具来解决这一现象，并在特定情况下改进和测试这些工具。CRC的研究分为三个相互关联的项目组。从量子力学到凝聚态和材料科学。随机系统与连续极限C.几何结构和高维问题项目组A的项目研究经典和量子力学中许多粒子系统的集体效应。课题包括经典和量子玻尔兹曼方程，多体薛定谔方程的有效多级近似，固体的统计力学，以及复杂材料行为的有效描述。项目组B专注于随机系统和连续极限。在许多科学问题中，有效行为通常不依赖于微观水平上的随机性和小尺度相互作用的细节。相反，有趣的效应，如相变，老化或普遍性出现在无限或连续系统的极限。项目组C以系统的方式处理高维问题的分析、概率和算法方面，并处理新兴几何结构的识别和有效近似。在最初的两个资助期内，CRC获得了新的严格的数学见解，并在此过程中加强了波恩分析，概率和数值之间的联系和协同作用。在此基础上，CRC将在第三个资助期内应对新的挑战，并利用新的相互作用，特别是在最佳运输和Wasserstein梯度流，高维问题的学习和模型降阶，普遍性，重整化和自相似性的研究，在多线性谐波分析的前沿，以及长距离相互作用的研究中。

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.