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?
Topics. Emergence, synchronization, entropy, large deviations, resilience of complex systems, critical transitions. Applications in biology, climate science, economics, sociology, innovation science.|
Prerequisites. Familiarity with elementary concepts from linear algebra (eigen values), dynamical systems and probability. Programming in mathematical software (Matlab, Mathematica).
Format. Lectures will alternate between introductions to concepts and theory of complex systems and guest lectures by researchers from other disciplines. Each student will submit two research reports involving computer simulations. There will be a final exam.
Learning goals with assessment weighting:
read and demonstrate (in class discussions) understanding of multidisciplinary literature (20%)
understand and be able to apply mathematical analysis and methods to carry out and write two project reports involving computer simulation (50%)
demonstrate understanding of theoryetical concepts on final exam (30%)