As with many important concepts it is not easy to give a unique short definition of complex systems. They comprise in any case systems which, like biological systems, have many interrelated components which interact in a nonlinear, not easily identifiable, often chaotic way. It has become increasingly clear in recent years that for the modeling of such systems traditional tools of mathematics and physics must be supplemented by entirely new methods from mathematical physics, stochastic analysis and the theory of dynamical systems, working in cooperation, to construct, in particular, a conceptual and technical framework permitting to understand better the relations between finite dimensional and infinite dimensional systems. The aim of the conference was to bring together scientists of various areas, and in particular of the above mentioned disciplines, to present their new results and interact constructively in searching for better tools to handle complex systems. The conference successfully achieved these goals. It was an exciting experience to see mathematicians from very different areas discussing with physicists and biologists about their models and methods, and receiving from them new stimulations for their own research, particularly on the challenging problems posed by complex systems.
Dynamical and chaotic formations of structures are typical examples of phenomena connected with complex systems. They are observed e.g. in turbulent fluids or disordered systems of solid state physics (spin glasses), as well as in biological pattern formation, in the functioning of the immune systems or in the neuronal dynamics. As another example of interactions achieved at the conference, we mention that ideas from Mathematical Physics, in particular Statistical Mechanics and Quantum Physics, such as the renormalization group, which have been employed intensively in the analysis of phenomena involving a multiplicity of scales (as e.g. in the chaotic behavior of strongly nonlinear systems) have been brought together with methods belonging to the area of asymptotic analysis of scaled stochastic dynamical systems. Powerful tools in this mathematical area turned out to be functional integration, to handle an infinite number of degrees of freedom, and the theory of dynamical systems, concerning non-linear phenomena.
Among the areas and topics discussed at the conference were:
Dynamical systems: block spin renormalization group, ergodic behavior, chaotic systems and time correlations, periodic orbits, integrable systems, billiards (classical and relativistic). Stochastic systems: structural complexity, stochastic (partial) differential equations, infinite dimensional Laplacians, image data compression, random complex networks, information structures, random graphs, classical statistics, random walks.
Quantum systems: anharmonic lattice systems, relations between quantum and classical properties, decoherence, nonlinear phenomena, gauge transformations, scattering theory, quantum systems on graphs, decoupling, quantum fields, bound states, many degrees of freedom, quantum information. Statistical mechanics: percolation, self-organized criticality, self-similar patterns, molecular networks, phase transitions, spin models, spin glass models.
Analytic methods: spectral theory, Kortweg-de Vries equation, nonlinear evolution, pseudoholomorphic curves, non-convex minimization, infinite dimensional entire functions.
Biological systems: neuroscience, discrete optimization and DNA, sequencing, immunization, data compression, information differential geometry.