Collective behavior in complex systems

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Collective behaviour is an ubiquitous natural phenomenon that pervades both the animal world and many inanimate physical systems. In biology, group formation is observed across all levels of the animal kingdom, from bacterial colonies, insect swarms, fish schools and flocks of birds, to complex human population interactions. Collective behaviour also arises in a multitude of physical systems such as granular gases, self-assembly of nanoparticles and DNA buckyball molecules, and crystal lattices in Bose-Einstein Condensates. The emergence of collective group behaviour is often a consequence of individuals (or atoms) following very simple rules, without any external coordination. In recent years, many models of group emergence and behaviour have been proposed. The purpose of this program is to study the fundamentals of emergent behaviour, develop new mathematical and computer tools, and to collaborate with other scientists to apply these ideas to problems in chemistry, physics and biology.In biology, collective behaviour confers certain benefits on individuals or group as a whole, such as predator avoidance for bird flocks, food gathering for ants and heat preservation in penguin huddles. This poses numerous mathematical, biological and even philosophical questions such as: What are the physical/biological mechanisms that cause aggregation? What is the optimal shape (or size) of the swarm? To what extent is swarming an evolutionary adaptation? An example of collective behaviour in physics are Bose-Einstein Condensates (BEC). This is a state of matter, theorized in 1920's by Bose and Einstein, that exists at extremely low temperatures (close to absolute zero), and which consists of concentrations of atoms. It was not until 1990's that BEC's existence was experimentally observed at MIT and JILA (Joint Institute for Laboratory Astrophysics), a work for which these groups were awarded the Nobel prize in Physics (2001). In this experiments, BEC were observed to self-organize into rigidly rotating "vortex crystals", which are very regular hexagonal lattice configurations of vortex-particles, with each vortex representing a concentration of atoms. One of the goals of this proposal is to apply new methods recently developed in studies of biological swarms to gain insight into the structure of these vortex crystals: their shape, stability, and the so-called Tkatchenko vibrational modes. Recently, BECs have been found to have numerous potential applications such as Ramsey spectroscopy. Tantalizingly, BECs hold promise for potential breakthroughs in computing; it was demonstrated experimentally that BECs can potentially be used to factor large numbers and to perform logic operations with qubits in lieu of a quantum computer. The study of swarms and collective behaviour is a very rich topic at the intersection of several mathematical, biological and physical sub-disciplines, and is currently undergoing an explosive development. A progress in any of these problems will have repercussions across multiple disciplines and the formation of lasting collaborations and the cross-pollination across disciplines is expected to be both timely and important. The variety of problems makes this area an ideal source of problems for training HQP at all levels (from undergrad to postdoc) and with different backgrounds.