The purpose of the project is to study, from a mathematical and a numerical perspective, linear and nonlinear models arising in physics and chemistry and which serve as tools to describe matter at the microscopic and nanoscopic scales. The main originality of the projet is its emphasis on large or infinite quantum systems. Those systems are well known for being hard to describe both from a physical and mathematical point of view. Specific mathematical tools need to be employed or invented.

The mathematical fields of concern for this project are:

• nonlinear analysis and partial differential equations,
• spectral theory,
• numerical analysis.

The physical systems that will be considered come from

• atomic physics or quantum chemistry,
• condensed matter,
• stellar or nuclear physics.

The scientific project contains four parts.

The first part is devoted to the study of relativistic atoms and molecules, while taking into account quantum electrodynamics effects like the polarization of the vacuum. The models are all based on the Dirac operator.

The second part is focused on the study of quantum crystals. The goal is to develop new strategies for describing their behavior in the presence of defects and local deformations. Both insulators, semiconductors and metals are considered.

In the third part, attractive systems are considered (like stars or a few nucleons interacting via strong forces in a nucleus). The project aims at rigorously understanding some of their specific properties, like Cooper pairing or the possible dynamical collapse of massive gravitational objects.

Finally, the last part is devoted to general properties of infinite quantum systems, in particular the proof of the existence of the thermodynamic limit.