SFB 1060: The Mathematics of Emergent Effects

合作研究中心（CRC）的中心目标是了解在更大尺度上从许多单位在较小尺度上的相互作用中产生的新效应。它建立在这样的观察之上，即在小尺度上相当不同的相互作用可以在大尺度上导致相同的“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，并在具体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物质和材料科学随机系统与连续统极限。几何结构和高维问题项目组A的项目研究经典力学和量子力学中许多粒子系统的集体效应。主题包括经典和量子玻尔兹曼方程，多体Schrödinger方程的有效多层近似，固体的统计力学，以及复杂材料行为的有效描述。项目组B侧重于随机系统和连续体极限。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细节和小尺度的相互作用。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逼近。在头两个供资期间，中心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、概率和数字之间的联系和协同作用。在此基础上，CRC将在第三个资助期解决新的挑战并开发新的相互作用，特别是在最优输运和Wasserstein梯度流，高维问题的学习和模型降阶，普适性，重归一化和自相似性的研究，多线性谐波分析的前沿，以及远程相互作用的研究。

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.