The classical fields of mathematics harmonic analysis and theory of partial differential equation lead to many applications in increasingly varied fields. Most of the proposed internships are essentially of a theoretical nature () but the model phenomena cover a very wide scientific field. It is an illustration and not a limitative list. In most of the Analysis sector, other topics can be proposed case-by-case, in France or abroad.
Here are some example of themes that can be covered:
- Equations of Quantum mechanics:
the aim is to study the Schrödinger equation in different settings. For example, the study of resonances () for the linear Schrödinger equation is related to the metastable state research. Non-linear Schrödinger equations intervene in molecular chemistry. A large use of functional analysis joins asymptotic methods with a nature more geometric.
- Wave equations and general relativity:
solving equations as a system of evolutionary partial differential equations raise problems. On simplified models such as "equations of waves", disperive phenomena appear. One of the key tools for understanding these phenomena is what we call microlocale analysis, a sector from the Fourier analysis in the 1970's.
- Kinetic models:
Kinetic models describe different physic systems (gas, ionized gas, plasmas...) through a statistical approach at the microscopic level (molecual or atomic). These are generally partial differential equations with non-local terms, that raise a wide range of major problems of mathematics physics (such as, for example: questions related to convergence speed towards equilibrium states, or some dispertive effects). The study of these models traditional tools of analysis, in some intances with interesting interpretations from the probabilistic perspective.
- Stroke modelisation
The simplified models of geophysical fluids take into account the effects of the Earth's rotation. The understanding of the damping phenomenon and side effects improved recently. Mathematical tools used are the study of parabolical systems and Fourier analysis.
- Geophysics fluid mechanics:
- Solitons and qualitative study of solutions:
The description of the real-time behavior of solutions of Korteweg de Vries (KdV) and non-linear Schrödinger is a mathematically active research topic and very important from a physical pespective. KdV and Schrödinger equations are considered as universal models of Hamiltonian systems in finite dimension and appearing in a very large number of physical phenomena. Solitons are specific solutions of these equations, travelling or periodical waves. The aim of the internship is to understand recent results on the study of solutions that are in a neighbourhood of solitons, and in a second phase continue the study of solutions that have a large-time behavior similar to that of solitons.
Example of internships of the earlier years:
- Wavelets and chirps characterization (ENS Cachan).
- Variational problems with 2 phases and their free boundaries (MIT, Cambridge, USA).
- Topological methods in fluid mechanics (Cérémade, University of Paris IX Dauphine).
- Wavelets and compression (ENS Cachan).
- Automatic analysis of sleep–wake condition (ENS Cachan).
- Asymptotic development at high orders of the field diffracted by a semi-infinite cone with mixed surface conditions (CEA, Bruyères-le-Châtel).
- Interfaces in phase transition problems (Pierre et Marie Curie University).
- Carleman's inequality and applications (Centre de Mathématiques de l’Ecole polytechnique).
- Wavelets and non-linear applications (Princeton University, USA).
Course language: French
- Teaching coordinator: Dubach Guillaume