Final Activity Report Summary - LORENTZ WORKSHOPS I (European science workshops at the Lorentz Center. Computer science and Mathematics) This project provides advanced training in modern topics in mathematics and computer science. The four workshops in this series of events cover collective dynamics and new computational developments. The workshops are run and hosted by the Lorentz Center at Leiden University in the Netherlands, in the open and stimulating spirit for which the Center has earned its reputation. 1: Gossip-based Computer Networking (information technologies, network stability). This workshop brought together the major players in a well-known, but reviving field of distributed computer systems, known as gossip-based networks. These types of distributed systems are characterised by the way information is disseminated among a huge collection of computing nodes. Periodically, each node simply contacts one of its randomly chosen (current) neighbours to exchange new information. The effect of this combination of randomness, periodicity, and information exchange is rapid spreading of data akin to the epidemic behaviour of diseases. Important areas for further research were identified: Applications, application development support, security, probabilistic modelling, theoretical frameworks, gossiping in wireless environments, gossip-based storage, and controlling emergent behaviour. 2: MAP2007 (mathematical proofs of algorithms in computer algebra). The goal was to bring together people from the communities of formal proofs, constructive mathematics and computer algebra (in a wide sense). One objective of the workshop is to bridge the gap between conceptual (abstract) and computational (constructive) mathematics, by providing a computational understanding of abstract mathematics. It is becoming clear that many parts of abstract mathematics can be made constructive and even computational and that abstract mathematical techniques contain an underlying constructive content. We are not only interested in algorithms however, but also in formal proofs of the correctness of these algorithms. We received many very positive reactions, both on the scientific quality and on the stimulating well-organised center. 3: Nonlinear Collective Behavior: Networks, Swarming and Reaction-Diffusion Dynamics (complex biological and biochemical systems from cells to the spread of diseases). The focus was on the confluence of interdisciplinary research involving nonlinear cooperative behaviour in complex systems. This is a broad research area with applications in many fields since cooperative behaviour underlies much of the observed phenomena in nature. Instead of surveying all aspects of the field of nonlinear cooperative behaviour, the workshop emphasized some of the most exciting and least well developed aspects of this topic: reaction-diffusion dynamics in heterogeneous media, the dynamics of networks and cooperative behaviour and swarming in systems with self-propelled elements. The workshop succeeded in identifying the common tools and phenomena in the broad field of collective phenomena', and in setting the perspective for future work in this field. 4: The Computational Complexity of Quantum Hamiltonian Problems. (Information technologies, Quantum computing). A diverse set of researchers who were interested in the rigorous mathematical analysis of techniques for simulating the properties of quantum systems described by many-body Hamiltonians were brought together and important results were extensively discussed at this workshop for the first time. An example is the result that determining the ground-state energy of a quantum Hamiltonian of a one-dimensional chain is a hard problem to solve for quantum computers (in technical terms, the problem is QMA-complete). The QMA-classification, and the various other quantum classes that now exist in computer science beyond the class NP, were thoroughly reviewed and discussed. Many participants felt that the workshop had truly identified an emerging field of interdisciplinary theoretical science.