The science of complexity and the role of mathematics.

In the middle of the second decade of the 21st century, Complexity Science has reached a turning point. Its rapid advancement over the last 30 years has led to remarkable new concepts, methods and techniques, whose applications to complex systems of the physical, biological and social sciences has produced a great number of exciting results. The approach has so far depended almost exclusively on the solution of a wide variety of mathematical models by sophisticated numerical techniques and extensive simulations that have inspired a new generation of researchers interested in complex systems. Still, the impact of Complexity beyond the natural sciences, its applications to Medicine, Technology, Economics, Society and Policy are only now beginning to be explored. Furthermore, its basic principles and methods have so far remained within the realm of high level research institutions, out of reach of society's urgent need for practical applications. To address these issues, evaluate the current situation and bring Complexity Science closer to university students, a series of Ph.D. Schools on Mathematical Modeling of Complex Systems was launched, starting in July 2011 at the University of Patras, Greece (see ). These Schools lasted two weeks each and included, beyond introductory lectures, a conference component of 2-3 days where students were exposed to recent results mainly presented by young researchers. The Ph.D. Schools continued successfully, the 2nd one taking place at Pescara, Italy (2012), (see ), the 3d one at Heraklion, Crete, Greece (2013) (see ) and the 4th one in Athens, Greece (2014) (see ) (2014). This Special Theme volume contains the proceedings of the 5th Ph.D. School-Conference of this series held at the University of Patras, Greece, 20 30 July, 2015 (see ) and includes many of the introductory lectures and research talks presented at this event. The primary concern of all those that participated in these events was to emphasize the role of mathematics, modeling and numerical simulation, which are indispensable for understanding what we call complex behavior of physical, biological, technological and socio - economical systems. In the discussions that took place, a great number of participants expressed the need to formulate a unifying theory of complex systems based on the main conclusions that have been reached so far in the science of Complexity. As Guest Editors of this volume, we also feel that it is important to reach some fundamental conclusions concerning common phenomena, theories and methodologies that arise in Complexity. We should all work to explore common rules and approaches, particularly in view of the remarkable challenges that face us all regarding complex social problems that threaten present day society and civilization as we know them. [ABSTRACT FROM AUTHOR]