SFB 1060:  The Mathematics of Emergent Effects

• Subject Area: Mathematics
• Term: from 2013 to 2024
• Website: Homepage
• 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.

• DFG Programme: Collaborative Research Centres

Current projects

• A02 - Classical and quantum kinetic equations (Project Head Velázquez, Juan José López )
• A04 - Statistical mechanics for lattice models of elasticity (Project Heads Bovier, Anton ;  Müller, Stefan )
• A05 - From pair potentials to macroscopic plasticity (Project Heads Conti, Sergio ;  Müller, Stefan ;  Ortiz, Ph.D., Michael )
• A06 - Hysteresis and microstructure in shape memory alloys (Project Heads Conti, Sergio ;  Knüpfer, Hans ;  Zwicknagl, Barbara )
• A07 - Sparse grid methods for the electronic Schrödinger equation and mesoscopic material models (Project Head Griebel, Michael )
• A08 - Nonlinear sigma models (Project Head Disertori, Margherita )
• B01 - Metastability (Project Heads Bovier, Anton ;  Müller, Stefan )
• B02 - Ageing and slow dynamics (Project Head Bovier, Anton )
• B03 - Optimal transport and random measures (Project Heads Huesmann, Martin ;  Sturm, Karl-Theodor )
• B04 - Random matrices and random surfaces (Project Heads Ferrari, Patrik L. ;  Peltola, Eveliina )
• B05 - Dynamics of coagulation-fragmentation equations (Project Heads Niethammer, Barbara ;  Velázquez, Juan José López )
• B08 - Screening and long-range interactions in problems in materials science (Project Heads Niethammer, Barbara ;  Velázquez, Juan José López )
• C01 - Wasserstein geometry, Ricci curvature and geometric analysis (Project Heads Erbar, Matthias ;  Kopfer, Eva ;  Maas, Jan ;  Sturm, Karl-Theodor )
• C03 - Nonlinear dispersive equations (Project Head Koch, Herbert )
• C04 - Multilevel sparse tensor product approximation for manifolds and for functions and operators on manifolds (Project Head Griebel, Michael )
• C05 - Discrete Riemannian calculus on shape space (Project Heads Rumpf, Martin ;  Sturm, Karl-Theodor )
• C06 - Numerical optimization of shape microstructures (Project Heads Conti, Sergio ;  Rumpf, Martin )
• C08 - Multilinear estimates in geometric Fourier analysis (Project Head Thiele, Christoph )
• C10 - Sparse controls in optimization of quasilinear partial differential equations (Project Head Neitzel, Ira )
• Z - Central Tasks of the Collaborative Research Center (Project Head Müller, Stefan )

Completed projects

• A01 - Effective theories in quantum physics (Project Head Schlein, Benjamin )
• A03 - Semirelativistic Schrödinger equations, ground states and dynamics (Project Heads Koch, Herbert ;  Schlein, Benjamin )
• B06 - Macroscopic limits of reinforced random walks (Project Head Velázquez, Juan José López )
• B07 - Low-rank approximation for PDEs with uncertainties (Project Heads Bebendorf, Mario ;  Beuchler, Sven )
• B09 - Large scale modeling of non-linear microscopic dynamics via singular SPDEs (Project Head Gubinelli, Massimiliano )
• C02 - Invertibility of random matrices (Project Heads Rauhut, Holger ;  Schlein, Benjamin )
• C07 - Generalized finite element methods for composite materials (Project Head Schweitzer, Marc Alexander )
• C09 - Adaptive methods for high-dimensional eigenvalue problems and their computational complexity (Project Head Bachmayr, Markus )

• Applicant Institution: Rheinische Friedrich-Wilhelms-Universität Bonn
• Participating Institution: Rheinische Friedrich-Wilhelms-Universität Bonn; Institut für Angewandte Mathematik; Humboldt-Universität zu Berlin; Institut für Mathematik; Rheinische Friedrich-Wilhelms-Universität Bonn; Institut für Numerische Simulation; Rheinische Friedrich-Wilhelms-Universität Bonn; Mathematisches Institut
• Spokesperson: Professor Dr. Stefan Müller