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

复杂系统或复杂性科学这一术语在从社会学和经济学到生物学和物理学的一系列不断扩大的科学中得到了非常广泛的使用。虽然已经发展了定量的数学方法来描述和分析复杂系统，但想要研究复杂系统的人往往不会深入了解这些方法。这是一个严重的限制，可能会阻止研究人员超越纯粹的定性描述和分析。我们在这里提出了一门课程，强调复杂系统科学中的数学定量技术。在本课程的第一部分中，介绍了与复杂系统一般相关的基本数学概念。第二部分从物理、地球物理、材料科学和生物学等方面举例说明了这些方法的应用。总的主题是复杂系统的动力学，重点是涌现现象，如时空模式。复杂系统的特点是在不同的时间尺度和不同的长度尺度上发生不同的现象。可以使用特定的数学技术来分析在一个尺度上发生的现象与不同尺度上的现象之间的关系。通常，一个人从微观的、经过中观的、到宏观层面的描述。每个层面都会出现新的互动和新的行为。本课程的重点是在前五天的课堂和问题解决课程中介绍和应用用于研究复杂性的基本量化概念和工具。之后是一天的项目工作，项目工作报告和考试。最后两天将集中在当代研究领域，并邀请研讨会演讲者。讲师和研讨会演讲者将在学校呆上几天，以确保有足够的时间进行师生互动。此外，著名主讲人的当代回顾讲座安排在晚上。

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.