Why do surprisingly complex patterns sometimes emerge from simple rules - for example in the growth of crystals, the structure of cell tissue or phenomena in modern materials? The answer often lies in a special state: criticality. This is a regime in which systems react particularly sensitively and the smallest influences can trigger major changes.
In the new Collaborative Research Center CRC 1720 'Analysis of Criticality: From Complex Phenomena to Models and Estimates', the mathematicians involved are dedicated to the task of better understanding the underlying mathematical structures of criticality - and thus paving the way for more accurate simulations and new applications in the natural and engineering sciences.
“Criticality often occurs in systems in which many different processes occur simultaneously and on very different length scales - and influence each other in the process,” explains Angkana Rüland from the Institute of Applied Mathematics at the University of Bonn. “This complexity leads to singular structures, strong interactions and exciting phase transitions between different states. Classical mathematical methods often reach their limits here.”
In order to overcome these hurdles, the researchers in the CRC are pursuing an approach that combines various mathematical perspectives and sub-disciplines. Among other things, the research team is investigating models in which strong interactions across many scales lead to complex patterns, such as those that occur in biological or physical systems. Another area deals with the question of which key properties make up critical systems and how these can be filtered out from a multitude of competing effects. As many questions in this field are difficult to grasp mathematically, new tools are also to be developed in order to systematically tackle so-called “ill-posed problems”.