Description.  In recent years there has been a growing interest in the science of complex systems. A complex systems is loosely defined as one whose collective dynamical behavior cannot be readily deduced by a reductive study of its individual components.  For example, it is difficult to predict where traffic jams will occur by studying the behavior of individual drivers, or to understand turbulence in water by studying the behavior of water molecules.

Important questions are how coherent collective behavior emerges in seemingly random systems, how complex systems undergo change, what makes certain behavior more or less stable.  In this course we will study mechanisms for emergence, including synchronization and pattern formation, and mechanisms for transitions between system regimes, with an emphasis on analytical and computational methods. We will ask, what are the mathematical foundations of complexity science?  What aspects of complex systems can we model successfully with mathematics, and where do we fall short?

One property commonly associated to complex systems is “self-organization”, and we will try to implement that in this seminar:  in the first meeting, we will study the wikipedia pages (complexity science, emergence, self-organization, …), prepare a list of topics with references, and assign each student an article to present in a subsequent meeting.  The goal is to develop a comprehensive mathematical basis for complexity and build from there. 

Prerequisites.  Differential equations, dynamical systems, numerical analysis.  Experience with stochastic and partial differential equations is useful, but not required.

Format.  At the beginning of the semester, class meetings will follow the format:  presentation of an article by a student, discussion with the whole class, preparation/assignment of new topic for the following week.  We will leave room for changes of course, based on questions that arise during the class.  

Towards the end of the semester, each student will carry out a small project related to one of the topics discussed.  These projects, involving numerical simulations, will provide an opportunity to get some hands‐on experience on the topic. The course concludes with presentations by students on their projects. Course grades will be based on article presentation, project and active participation during discussions.

Learning goals with assessment weighting:

Some background reading:

May, R.M., Stability and Complexity in Model Ecosystems, Princeton University Press, 1973.

Scheffer, M., Critical Transition in Nature and Society, Princeton University Press, 2009.

Johnson, N. Simple Complexity: A Clear Guide to Complexity Theory. Oneworld Publications, 2009.

Boccara, N., Modeling Complex Systems, Springer, 2010. 

Remco van der Hofstad, Lecture Notes Random Graphs and Complex Networks, (http://www.win.tue.nl/~rhofstad/NotesRGCN.html).

M.J. Newman, Networks, An Introduction, Oxford University Press, 2010.