Mathematics of emergent behaviour and applications

自然界充满了这样的例子：遵循简单规则的个体（或粒子）可以导致复杂而美丽的整体模式，通常被赋予一个模糊的名称“涌现行为”。一些例子包括生物群（从细菌聚集到鱼群，椋鸟的低语.），原子水平的玻色-爱因斯坦凝聚，自主机器人，以及许多其他。令人惊讶的是，这些复杂而美丽的图案往往是微观层面上相对简单的物理或化学定律的结果。通常，这些模式是由描述相对简单的组成组件自组装而成的。整个系统的行为是由这些单独组件的复杂非线性相互作用引起的。我的研究目标是研究涌现行为的基本原理，从微观规则到宏观涌现。这包括：理解如何从微观层次的规则中获得宏观描述;开发新的和扩展现有的模型;应用和扩展数学技术和计算机工具;与其他科学家合作，将这些想法应用于数学，物理和生物学问题。不同学科之间有很大的相互促进的潜力。我为这个领域带来了独特的视角，在PDE方法（渐近，局部模式......）方面拥有广泛的专业知识。以及多粒子动力学系统。 主要的副主题是（1）偏微分方程背景下的涌现行为;（2）噪声驱动的粒子模型及其宏观极限;（3）在生物学和自治系统中的应用。

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.