From tuesday 19 to thursday 21, there will be two sessions a day; one in the morning and one in the afternoon. It is not possible to attend to both lectures.

The two lectures are :

**Semiclassical analysis and Bloch-Floquet theory**, by Clotilde Fermanian Kammerer

The aim of these lectures is to discuss different PDEs technics related with a Schrödinger equation describing the dynamics of an electron in a crystal in presence of impurities. Because the size of the cells of the crystal are supposed to be very small comparatively with the macroscopic scale, it is a multi-scale problem with periodic aspects. We shall use semi-classical measures (also called Wigner measures) to take care of the multi-scale features, and Bloch theory to deal with the periodicity. These notions will be explained and used for calculating the density of probability of presence of the electron in the limit where the size of the cells is much smaller than the macroscopic scale.**Regulous functions**, by Jean-Philippe Monnier

When we develop real algebraic geometry in a classical way, many defects appear: no classical Nullstellensatz and Cartan's Theorems A and B no longer work. A recent theory consists of replacing the use of regular functions with continuous rational functions (otherwise called regulous). This course is an introduction to regulous algebraic geometry. We will show that this new geometry repairs many defects of real algebraic geometry.