Combining ‘the entire spectrum of length scales’, the analysis of criticality is closely interrelated with the investigation of irregularity, long-range interactions and (random) fluctuations. It is often at criticality, at low regularity and in their fluctuations around equilibrium that models from the sciences reveal their most exciting, yet also most challenging features. This includes the formation of singularities, such as defects, interfaces, free boundaries and free discontinuities, as well as energy landscapes and dynamics which are governed by highly irregular objects, such as the Gaussian free field. The lack of regularity requires the development of robust and efficient mathematical tools, such as careful regularization, renormalization and tailor-made computational techniques.

The projects in this research area are

• A01 - Free boundary problems with nonlocal operators (Niethammer, Velázquez)
• A02 - Oscillatory integrals in random matrix models (Disertori)
• A03 - Modelling and simulation of cohesive fracture in solids (Conti, Schweitzer)
• A04 - Adaptive computations on polygonal meshes (Gedicke)
• A05 - Random geometry, renormalization and curvature (Sturm)