Mathematics of emergent behaviour and applications

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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.*****