​We introduce the concept of quantum entanglement, fundamental in quantum physics. We review some basic properties of entangled states (pure and mixed), and explain how entanglement can be used as a resource in quantum information theory. This leads to various measures of entanglement (e.g. entanglement cost, distillable entanglement) to quantify how much entanglement a given system contains.

In the 1980’s, D. Ruelle, D. Bowen and others have introduced probabilistic and spectral methods in order to study deterministic chaos (“Ruelle resonances”). Following this approach and use of microlocal analysis, we will consider the geodesic flow on a smooth strictly negative curvature Riemannian manifold M. 1) the flow is Anosov (i.e. uniformly hyperbolic, with sensitivity to initial conditions) 2) the generator of the geodesic flow, namely the vector field on S*M, has an intrinsic discrete spectrum called Ruelle resonances (for this we need to consider specific Anisotropic Sobolev spaces). 3) The Ruelle resonances are structured in vertical bands. 4) The first (right most) band that governs long time fluctuations of classical probabilities is the spectrum of an effective quantum wave equation (this may be surprising because there is no added quantization procedure). 5) We will discuss consequences for the zeros of dynamical zeta functions. Relations between the Ruelle spectrum and periodic orbits.
This shows that the problematic of classical chaos and quantum chaos are closely related. Part 2) is from a joint work with J. Sjöstrand. Parts 3) and 4) are from a joint work with Masato Tsujii.

Given a quantum Hamiltonian, I will explain how the dynamical properties of the underlying classical Hamiltonian affect the behaviour of quantum eigenstates in the semiclassical limit. I will mostly focus on two opposite dynamical paradigms: completely integrable systems and chaotic ones. I will introduce tools from microlocal analysis and show how to use them in order to illustrate the classical-quantum correspondance and to compare properties of completely integrable and chaotic systems.

Systems of interest in physics are typically formed by a huge number of particles, way too large to extract quantitative information from the fundamental evolution equation, the Schrödinger equation. However, in some scaling regimes, the many-body evolution of physical observables can be approximated by nonlinear effective equations, depending on much less degrees of freedom than the full Schrödinger dynamics. In this talk, I will consider the evolution of many-body fermionic systems, in the mean-field regime. I will discuss mathematical methods for the derivation of the time-dependent Hartree-Fock equation, a well-known example of nonlinear effective evolution equation for interacting fermions. The methods combine ideas from semiclassical analysis together with tools from second quantization. If time permits, I will report about the extension of these techniques to the derivation of the Vlasov equation, or to the computation of the ground state energy for trapped fermionic systems beyond the quasi-free approximation.

​The 1983 discovery of the fractional quantum Hall effect marks a milestone in condensed matter physics: systems of “ordinary particles at ordinary energies” displayed highly exotic effects, most notably fractional quantum numbers. It was later recognized that this was due to emergent quasi-particles carrying a fraction of the charge of an electron. It was also conjectured that these quasi-particles had fractional statistics, i.e. a behavior interpolating between that of bosons and fermions, the only two types of fundamental particles.
These lectures will be an introduction to the basic physics of the fractional quantum Hall effect, with an emphasis on the challenges to rigorous many-body quantum mechanics emerging thereof. Some progress has been made on some of these, but lots remains to be done, and open problems will be mentioned.