The transport properties of quantum systems of interacting particles are of central importance in Mathematical Physics, both from a fundamental theoretical point of view and for applications to close-to-equilibrium physical systems, such as metals, quantum liquids, nano-materials, cold atoms, crystalline solids.
While we are still far from a complete, fundamental, theory of transport based on a microscopic dynamical law, the last few years witnessed important progresses in the understanding of several specific questions arising in this context, such as the quantization properties of transverse conductivity in interacting electron systems (as in the Quantum Hall effect and in the recently synthesized Chern insulators), the bulk-edge correspondence in topological insulators, the role of disorder and localization in interacting many-body systems as well as the crucial role of the unitary gas in the transition between condensation and superfluidity/superconductivity.
The mathematical techniques underlying these developments are diverse, as they include the spectral theory of operators, analysis techniques for non-linear PDEs as well as techniques from Differential Geometry.
At present these approaches are somewhat disjoint and their specific applicability, which is still the subject of intense research, is restricted to distinct problems and different models. However, these methods share common underlying features, such as the reduction to simpler effective theories, and the identification of geometrical and dynamical obstructions preventing the system to freely transfer energy or momentum from one part to the other.
In this project, we propose to analyze a few model cases by a combination of several techniques, with the two-sided purpose of pushing forward our mathematical knowledge on the transport properties of these specific systems, on the one side, and of developing new techniques based on a combination of different techniques and ideas, on the other side.
