The second theme will address some of the most challenging computational problems in physics and chemistry. We have selected the following two fields in which particular momentum has been already created by bridging new physical observations with mathematical analysis of physical models and the development of novel computational approaches.

The first one concerns the physics of superfluid quantum systems: the Bose-Einstein condensates (BEC) and superfluid Helium (He). Since the first experimental realization of a Bose-Einstein Condensate in dilute atomic vapors by Cornell, Ketterle and Wieman in 1995 (2001 Nobel Prize in Physics), the study of ultracold gases has become one of the most dynamic research areas in modern quantum physics. Advances in this field have the potential to trigger a new technological revolution, similar to the progress induced when the laser technology was transferred from laboratory experiments to every-day applications. In the wake of advances made by experimental and theoretical physicists, computational mathematics emerges as a key player in this field which will likely catalyze new developments.

However, while numerical simulations are now often used to explore physical theories, their degree of maturity is still far from that achieved by the computational techniques employed in fields with a longer history, such as, for example, computational fluid dynamics. Recently progress has been made with modeling and simulation of superfluid quantum systems described by the macroscopic model of Gross and Pitaevskii. In particular, considerable effort was devoted to the simulation of systems with topological defects (quantized vortices), superfluid turbulence in He and rotating Bose-Einstein condensates. Recent experimental findings in this area lead to new open questions for theoretical physics and computational mathematics. Some of these challenges raise the bar quite high for the scientific computation community and are therefore very timely topics for the proposed thematic program. They include:

• efficient numerical methods for systems of coupled Gross-Pitaevskii equations, with applications to multi-component or spinor Bose-Einstein condensates,
• high-order numerical schemes for long time integration of Schrodinger type equations, with applications to oscillating condensates or Abrikosov lattices of quantized vortices,
• mathematical and numerical analysis of Schrodinger type equations with non-local terms or singularly perturbed, with applications to nonlinear optics,
• efficient numerical methods for Gross-Pitaevskii equations with stochastic terms, with applications to condensate-thermal cloud interactions and, finally, to quantum turbulence.

This sub-theme will survey the state-of-the-art computational methods for superfluid systems and will generate a lively debate on new mathematical and numerical ideas in this field. A strong interaction with the first theme, concerning classical fluids, is expected since, due to similarities of the mathematical models used in these fields, many ideas and computational techniques can be borrowed from numerical fluid dynamics.

The second sub-theme concerns the related problems of molecular simulation with their obvious connections to the broad field of multiscale computation (as evidenced, for example, by the Nobel Prize awarded to Martin Karplus, Michael Levitt and Arieh Warshel for "the development of multiscale models for complex chemical systems".} which will be the third main theme of the thematic program. This is research area with manifold applications in chemistry, solid-state physics, materials science, molecular biology and nanosciences. The methods used in this field serve both as theoretical tools to understand subtle phenomena taking place at the atomic scale and as a guide for experimentalists to design new molecules and materials for targeted applications. We wish to emphasize that molecular simulation accounts for about 20% of the total workload in scientific computing centres around the world.

Finding the properties of complex evolutionary molecular systems in which molecules interact with environments (e.g., solvents) is analytically intractable. Molecular dynamic simulation circumvents this problem in particular by introducing potential functions and by using suitably designed numerical methods. It gives access to key dynamic properties of the system such as transport coefficients, time-dependent responses to perturbations, rheological properties and spectra. However, long-time molecular dynamics (MD) simulations are numerically ill-conditioned and generate cumulative errors in numerical integration that cannot be entirely eliminated. Hybrid or coupled approaches attempt to merge classical and quantum methods in order to resolve the molecular problem at all scales. Therefore, to some extent, computer simulations bridge the gap between microscopic length and time scales and the macroscopic world. To date, controlling the accuracy of such simulations at every scale is a formidable challenge. Although the question of controling the accuracy of numerical simulation is a ubiquitous in scientific computing, its penetration tends to be quite different in different applications fields. In the domain of molecular simulation these ideas are rather new and are not a simple extension of what already exists, for example, in fluid mechanics.

New advances in mathematical modeling, numerical methods, and hardware will certainly have tremendous implications for the field of molecular simulation. They will enable researchers to simulate real systems of practical interest such as drug-protein complexes or catalytic surfaces in the presence of electric fields. This subtheme will therefore address some of the most important theoretical and numerical questions in this highly-strategic field:

• How to couple classical chemistry models with quantum chemistry models using multi-scale methods and algorithms?
• How to combine deterministic and stochastic models from the mathematical and computational point of view?
• Is it possible to increase the efficiency of molecular simulations by one or more orders of magnitude as was recently achieved for solvation models?
• Is it possible to select a proper combination of model, discretization and algorithm to provide a consistent approximation of a phenomenon of interest?

Since many models in quantum chemistry are based on Schrodinger-type equations, there is an obvious connection with the first sub-theme concerning superfluid quantum systems. Numerical methods developed for quantum fluids (such as multigrid, multiscale spectral or finite elements methods) are beginning to be used for the solution of macroscopic models in quantum chemistry, while stochastic modelling, already well advanced in chemistry, is now also used to take into account new physical phenomena in quantum fluid systems. The workshop will therefore offer a good opportunity to exchange ideas on possible extensions of numerical methods or models developed in one field to another field.