Periodic systems, i.e., systems that are translationally invariant at least in one dimension, have been of keen interest not only in physics and chemistry, but also mathematics. Scientists in these disciplines have been trying to solve important problems related to a large variety of periodic systems, with important theoretical and technological impact. We mention here as an exemple, among many others, some active research fields in Mathematics, Physics and Chemistry, related to the treatment of periodic systems.

Mathematics:

1) lattice sums, such as the Madelung constants in which one determines the electrostatic potential experienced by an ion in a periodic crystal. Because of the long range nature of the Coulomb potential, these sums are conditionally convergent and, therefore, the result depends on the order in which the terms are summed up [1].
2) the large N limit of the Thomson problem, in which a fixed number of electrons is constrained to the surface of (usually a sphere) and one seeks the electronic configuration giving the lowest energy. The problem has been solved for various finite values of N [2], but the general case is not trivial, and the large N limit is still problematic.
3) With some important exceptions [3], the topology of the supercell (a finite portion of the system that represents the crystal bulk) has not received a particular attention. However, in the presence of long-range interactions, this is a crucial aspect.

Physics:

1) The description of properties of crystalline solids [4,5], in particular using density-functional theory or many-body perturbation theory [6]. These methods are very powerful but are limited to weakly correlated materials since the electron-electron interaction is hidden in an effective potential.
2) The study of the uniform electron gas and its various phases (Fermi liquid, Wigner crystal, etc.) [7,8,9]. The uniform electron gas is a model to study electron correlation in many-electron systems.  Although many of its properties are known, its features at low density where it is predicted to form a Wigner crystal is still largely unknown.

Chemistry:

1) the ab initio treatment of crystals [10,11,12]. Ab initio chemistry approaches such as configuration interaction and coupled cluster theory are known for their great accuracy but their explicit treatment of the Coulomb interaction between electrons makes their application to periodic solids non-trivial.
2)  the study of polymers and nanotubes [13]. Since in this case the system is periodic in only one direction, one often extrapolates results of finite systems to estimate the infinite-size limit. However, results may depend on the choice of the extrapolation [14].

The aim of this workshop is to bring together some of the best experts in the various disciplines, in order to tackle a given problem from different points of view. For example: lattice sums can be related to the electrostatic interaction in crystalline solids and large molecular systems such as those treated in molecular dynamics simulations; the large N limit of the Thomson problem is linked to the structure of a Wigner crystal; the treatment of correlation using effective potentials can be linked to the explicit treatment of the Coulomb interaction in ab initio approaches.