Mathematics of emergent behaviour and applications

Nature is replete with examples where individuals (or particles) following simple rules can lead to complex and beautiful overall patterns, often given a vague name of "emergent behaviour". Some examples include biological swarms (from bacterial aggregation to school of fish, murmuration of starlings...), Bose-Einstein Condensates at atomic level, autonomous robots, and many others. Amazingly, these complex and beautiful patterns are often the result of relatively simple physical or chemical laws at a microscopic level. Oftentimes, these patterns are self-assembled from constituent components that have a relatively simple description. The overall system behaviour results from complex nonlinear interactions of these individual components. The goal of my research is to study the fundamentals of emergent behaviour, from microscopic rules to macroscopic emergence. This includes: understanding how to derive the macroscopic description from the rules at a microscopic level; developing new and extending existing models; applying and extending mathematical techniques and computer tools; collaborating with other scientists to apply these ideas to problems in mathematics, physics and biology. There is a very strong potential for cross-fertilization between the different disciplines. I bring a unique perspective to this field, with extensive expertise in both PDE methods (asymptotics, localized patterns...) as well as multi-particle dynamical systems. The main subtopics are (1) Emergent behaviour in the context of Partial Differential Equations; (2) Noise-driven particle models and their macroscopic limits; and (3) Applications to biology and autonomous systems.