Scientific Series

Synchronization phenomena are abundant in nature, science, engineering and social life. Synchronization was first recognized by Christiaan Huygens in 1665 for coupled pendulum clocks; this was the beginning of nonlinear sciences. First, several examples of synchronization in complex systems are presented, such as in organ pipes, fireflies, epilepsy and even in the (in)stability of large mechanical systems as bridges. These examples illustrate that, literally speaking, subsystems are able to synchronize due to interaction if they are able to communicate. Second, general physical mechanisms for synchronization and de-synchronization phenomena in coupled complex systems are presented and conditions for synchronizability are discussed. It is explained that diffusion properties give a crucial insight into this problem. I will show that the general concepts of curvature and recurrence are helpful to uncover complex synchronization. Third, applications of these new techniques are given. They range from El Nino – Monsoon interactions via electrochemical oscillators and lasers to cognitive processes during reading and to neuroscience.

Many systems take the form of networks: the Internet, the World Wide Web, social networks, distribution networks, citation networks, food webs, and neural networks are just a few examples. I will show some recent empirical results on the structure of these and other networks, particularly emphasizing degree sequences, clustering, and vertex-vertex correlations. I will also discuss some graph theoretical models of networks that incorporate these features, and give examples of how both empirical measurements and models can lead to interesting and useful predictions about the real world.

Noncommutative geometry is a more general formulation of geometry that does not require coordinates to commute. As such it unifies quantum theory and geometry and should appear in any effective theory of quantum gravity. In this general talk we present quantum groups as a microcosm of this unification in the same way that Lie groups are a microcosm of usual geometry, and give a flavour of some of the deeper insights they provide. One of them is the ability to interchange the roles of quantum theory and gravity by `arrow reversal'. Another is that noncommutative spaces typically carry a canonical 1-parameter evolution or intrinsic time created from the fundamental conflict between noncommuting coordinates and differential calculus. In physical terms one could say that quantising space typically has an anomaly for the spatial translation group and this forces the system to evolve. We give an example where we derive Schroedinger's equation in this way.

The causal set -- mathematically a finitary partial order -- is a candidate discrete substratum for spacetime. I will introduce this idea and describe some aspects of causal set kinematics, dynamics, and phenomenology, including, as time permits, a notion of fractal dimension, a (classical) dynamics of stochastic growth, and an idea for explaining some of the puzzling large numbers of cosmology. I will also mention some general insights that have emerged from the study of causal sets, the most recent one concerning the role of intermediate length-scales in discrete spacetime theories.