We organize a masterclass in Angers, France, from December 16 to 18, 2025. The masterclass is mainly intended for second-year master students and PhD students. There will be two parallel lectures:

Introduction to shifted symplectic structures, by Tristan Bozec

The aim of this course is to provide some basic algebraic tools needed to set up derived symplectic geometry. The main idea is to use homological data to deal with the singular spaces that naturally arise in physics (classical mechanics, field theories etc). We will thus start with linear shifted symplectic structures, and then move on to lagrangian ones, illustrated by exercises and examples coming from standard differential geometry and Lie theory. If time permits, we will see a bit of actual derived geometry through the fundamental example of the critical locus.

Semiclassical analysis of the high-frequency Helmholtz equation and its numerical approximation, by Martin Averseng

These lectures will be an introduction to recently obtained theoretical results concerning the numerical resolution of the Helhmoltz equation at high-frequency. The main novelty of these results is that their proof uses tools from semiclassical analysis: the aim of the course will be to illustrate the interaction between these two areas of mathematics. In its simplest form, the Helmholtz equation reads Delta u + k^2 u = 0 where k>0 (that is, the wave equation in Fourier space). When k is large, this PDE describes the propagation of high-frequency waves; despite its linearity, it is notoriously difficult to design efficient computer methods for its numerical resolution. One famous manifestation of this difficulty is the so-called ''pollution effect'' plaguing one of the standard methods for the discretization of PDEs, the Finite-Element Method (FEM). The pollution effect is the fact FEM approximations are less and less optimal -- with respect to the best approximation available in the considered approximation space -- as k goes to infinity. We will show that this pollution effect is related to error propagation along billiard trajectories, or rays, and we will present results obtained with Galkowski and Spence on the design of non-uniform meshes that take these informations into account to reduce the pollution effect. 

The lectures will be in english.

The organisation board will pay for accommodation, including breakfast and lunches. Dinners are not included. Travel costs will be covered for students from the Centre Henri Lebesgue only.

The number of participants is limited to 40. Registration will open soon. When registering to the masterclass, don't forget to indicate which lecture you will attend.