SFB 1720:  Analysis of criticality: from complex phenomena to models and estimates

• Subject Area: Mathematics
• Term: since 2025
• Website: Homepage
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 539309657

Criticality is at the origin of a ubiquity of complex phenomena in nature, the sciences and mathematics, including, for instance, random growth processes, cell polarization, structure formation in multiphase materials and indirect measurement processes. However, the underlying mathematical structures are still poorly understood, which limits the potential of simulations and the transfer to applications in the sciences. Thus, it is our objective to pursue a systematic and comprehensive approach combining complementary analytical, numerical and stochastic perspectives to study carefully selected model problems and to extract key features of criticality. To this end, we focus on three central, closely connected, complementary facets. In project area A. Criticality, irregularity and long-range interactions, we investigate models which display a wide -- often infinite -- range of strongly interacting scales of equal strength, giving rise to irregularity. In project area B. Criticality, asymptotics and scaling limits, we focus on identifying and isolating the core features of complex phenomena arising from criticality as manifested in the strong coupling of competing effects. In project area C. Criticality, ill-posedness and efficient rep-resentations, we explore the effects of ill-posedness and the need to deduce effective representa-tions for systems at criticality. The challenges of universality, high-dimensionality, the multiple effects of nonlocality and of taming ill-posedness constitute unifying themes. Important tools in our approach are robust geometric perspectives, the design of tailor-made function spaces, and a thorough analysis of scattering trans-forms and renormalization methods. In particular, these perspectives will be developed to address challenges associated with low regularity frameworks and with the absence of integrable structures. The individual projects cover a wide range of models inspired by the sciences in which criticality leads to rich mathematical structures, strongly coupled effects and complex phenomena. Our approach is designed to result in strong, synergistic interaction within the CRC and important progress for the field.

• DFG Programme: Collaborative Research Centres

Current projects

• A01 - Free boundary problems with nonlocal operators (Project Heads Niethammer, Barbara ;  Velázquez, Juan José López )
• A02 - Oscillatory integrals in random matrix models (Project Head Disertori, Margherita )
• A03 - Modelling and simulation of cohesive fracture in solids (Project Heads Conti, Sergio ;  Schweitzer, Marc Alexander )
• A04 - Adaptive computations on polygonal meshes (Project Head Gedicke, Joscha )
• A05 - Random geometry, renormalization and curvature (Project Head Sturm, Karl-Theodor )
• B01 - Random growth and strongly correlated systems (Project Head Ferrari, Patrik L. )
• B02 - Optimal design of curved folding structures in thin shells (Project Heads Conti, Sergio ;  Rumpf, Martin )
• B03 - Geometry and materials: rigidity, flexibility and scaling in some problems from materials science (Project Heads Müller, Stefan ;  Rüland, Angkana )
• B04 - Numerical homogenization of multiscale problems with coupled scales (Project Head Verfürth, Barbara )
• B05 - Convergence acceleration by non-reversibility and degenerate noise (Project Head Eberle, Andreas )
• B06 - Kinetic models in inhomogeneous settings (Project Heads Niethammer, Barbara ;  Velázquez, Juan José López )
• B07 - Avoiding critical slowing down in lattice methods by means of tensor methods (Project Heads Dölz, Jürgen ;  Griebel, Michael )
• B08 - Random matrices and anticoncentration (Project Head Sauermann, Ph.D., Lisa )
• C01 - Inverse problems for genuinely nonlocal elliptic operators: uniqueness, stability and reconstruction (Project Head Rüland, Angkana )
• C02 - Reconstructing cardiac electrical activation with learned spatiotemporal regularization on unstruc-tured grids (Project Head Effland, Alexander )
• C04 - Multilevel operator sampling in natural function spaces (Project Head Dölz, Jürgen )
• C05 - Inverse problems and nonlinear PDEs (Project Heads Koch, Herbert ;  Rüland, Angkana )
• C06 - Variational estimates in multi- and nonlinear harmonic analysis (Project Head Thiele, Christoph )
• Z - Central Task of the Collaborative Research Center (Project Head Rüland, Angkana )

• Applicant Institution: Rheinische Friedrich-Wilhelms-Universität Bonn
• Spokesperson: Professorin Dr. Angkana Rüland