Taught Course on Dynamics of Complex Systems: Emergent Phenomena via Separation of Scales in Time and Space.

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The terms complex systems or complexity science are used very broadly across an ever expanding set of sciences ranging from sociology and economics to biology and physics. Though quantitative mathematical methods have been developed to describe and analyse complex systems these methods are often not known in any depth to people intending to study complex systems. This is a serious limitation, that may prevent the researcher from reaching beyond a purely qualitative description and analysis. We propose here a course that emphasises the mathematical quantitative techniques in the science of complex systems. In the first part of the course basic mathematical concepts of general relevance to complex systems are introduced. The applications of these are illustrated in the second part by examples from physics, geophysics, materials science and biology.The overall theme is the dynamics of complex systems, with an emphasis on emergent phenomena such as spatio-temporal patterns. Complex systems are characterised by different phenomena occurring at different time scales as well as different length scales. Specific mathematical techniques are available to analyse how phenomena occurring at one scale are related to phenomena at different scales. Typically, one moves from microscopic, through mesoscopic, to macroscopic levels of description. New interactions and new behaviours emerge at each level. The course focuses on the introduction and application of the basic quantitative concepts and tools for studying complexity via lectures and problem-solving classes during the first five days. This is followed by a day of project work, project work presentation and exam. The last two days will focus on contemporary research areas with invited seminar speakers. Lecturers and seminar speakers will spend several days at the school to ensure that plenty of time is available for interaction between teachers and students. In addition, contemporary review lectures by distinguished key-speakers are scheduled in the evenings.