Hyperplane arrangements (A. Dimca and A. Parusinski - 6 ECTS) : descriptionFinal exam : 27 February, 9h-11h, salle 1
Hyperbolic surfaces (F. Labourie - 6 ECTS) Final exam : 16 February, 9h-12h, salle 1
The purpose of this course is to give an introduction to Riemann surfaces/ hyperbolic surfaces. First, we give examples of constructions of Riemann surfaces. First by gluing pants, which will lead us to the description of the Teichmüller space and its Fenchel-Nielsen coordinates. Then the construction of arithmetic subgroups via the quaternion algebra. Once we have finished these constructions, we will explain the basic concepts related to the spectrum of the Laplacian and give the proof in the compact case of the trace formula. If time permits we will go to the proof of the existence of the spectral gap for modular surfaces and will approach the unique quantum ergodicity. The course will be based on the book of Bergeron.
Bloc Analysis
Non-linear dynamics and formation of singularities (P. Raphael - 6 ECTS) Final exam: 26 February, 9h-12h, salle 1
This course is an introduction to the problem of the concentration of energy and singularities training for PDEs. The underlying phenomena emerge in various classical areas of mathematical physics (especially optics, gravity, fluid mechanics and turbulence) and are at the heart of an international research activity. After a general presentation of the problem in different contexts such as the nonlinear Schrodinger equation or the equations of fluid mechanics, we introduce the basic objects at the center of singular mechanics: the solitary wave. We will then present in different contexts the dynamic renormalization approach that in the last ten years has allowed spectacular advances on the construction and classification of concentration bubbles. The course will be self-contained modulo knowledge of basic tools for EDPs, including Sobolev spaces and injection properties.
Numerical methods in fluid mechanics (B. Nkonga - 6 ECTS) Final exam: 28 February, 9h-12h, salle 2
The simulation of fluid flows is essential in various sectors, including engineering and environment. The problems are of various nature: flow of compressible or incompressible fluids, viscous or non-viscous, Newtonian and non-Newtonian, laminar or turbulent... The numerical methods to address these problems are also very diverse. The course aims to give the basic elements to understand them. Some of the topics will be reminders of problems of evolution (explicit / implicit schemes, consistency, stability and convergence), spatial approximation methods (finite difference, finite volume, finite elements), stabilization of an approximation, Stokes problem, Uzawa methods, turbulences, etc.
Bloc Probability and Statistics (joint lectures with the Master MathMods)
Stochastic control for mathematical finance (F. Delarue - 6 ECTS)
This course will be an introduction to the theory of optimal stochastic control, which is widely used in mathematical finance for portfolio optimization and option pricing. Lectures will consist of a description of mathematical tools for characterizing the optimal strategies: dynamic programming principle and Hamilton-Jacobi-Bellman equations, stochastic Pontryagin principle and backward stochastic differential equations.
Numerical methods for PDEs (R. Masson - 6 ECTS)
This course will focus on the use of parabolic PDEs in mathematical finance. A first part of the lectures will consist of a reformulation of the Black Scholes model and of the option pricing problem in terms of PDE methods. Then, advanced numerical methods, based on finite volume methods, will be discussed.