AG Hofmanova
Higher Education Cooperation with the African Institute for Mathematical Sciences (AIMS) in Senegal 2024-2028
The German Academic Exchange Service (Deutscher Akademischer Austauschdienst – DAAD) finances the programme ‘Higher Education Cooperation with the African Institute for Mathematical Sciences (AIMS)’ with funding provided by the Federal Ministry of Education and Research (BMBF).
PDEs & Randomness School/Conference
The HEC-AIMS Newsletter 2024/2 reports about the Welcome Event in Bielefeld in June 2024.
Project Aims
This project aims to contribute to a sustainable empowerment of scientific research in Africa. In Senegal, as for the rest of the surrounding countries, despite the local efforts to establish good scientific training, there are still several lacks at the academic level in terms of research perspectives and possibility for early career researchers to pursue their careers. For instance, the absence of a postdoctoral programme is responsible for the severe phenomenon of drop-out for PhD earners who end up leaving at a very early stage of their research career and, sometimes, without a reasonable possibility of career reorientation. Employability and competitiveness of these early career scientists is one of the important challenges that one has to solve in order to build up a productive and dynamic research system. The AIMS experience is very positive in many respects; the German Research Chair (GRC) programme is particularly valuable to improve the conditions of mathematical development in the continent. Within this framework, our project will tackle the problematics stated above by taking a certain number of actions that are expected to have improving effects in short, mean and long terms.
We aim to establish a school of stochastic partial differential equations (PDEs) in the landscape of the Senegalese mathematical community, in collaboration with the GRC Prof. Mouhamadou Sy at AIMS Senegal.
The school will have the form of an umbrella under which several consistent activities will be coordinated. We aim to hire a team of researchers including two PhD students and three postdoctoral researchers. This team will be trained under the supervision of the GRC in order to conduct high quality research in stochastic PDEs and to build a strong international profile in close cooperation with Bielefeld University.
Mathematics
At the mathematical level, the project focuses on the analysis of the Navier-Stokes (NS) equations, which model the behavior of fluids, and the nonlinear Schroedinger (NLS) equations, which model the propagation of dispersive waves in nonlinear media. These equations have wide applications in aerodynamics and hydrodynamics engineering, as well as in optics and telecommunication. Despite their importance in practice, the rigorous mathematical theory remains underdeveloped. The global regularity problem (existence, uniqueness and continuity of solutions) possesses many intriguing open questions for the three-dimensional version of NS, and is an attractive challenge for supercritical regimes of NLS (i.e. in the case when nonlinearities of the medium of propagation are stronger than linear dispersive properties that are inherent to the wave itself). On the one hand, simplified models of NS have been studied, for instance the model of Katz-Pavlovic or the model of Tao, with the hope to obtain valuable insights into the original dynamics. In these simplified models, explosion in finite time was observed. On the other hand, an explosion has also been proved by Merle-Martel-Rodnianski-Szeftel for NLS in high dimension.
In the stochastic NS setting, Hofmanova and collaborators have proved nonuniqueness in law. Sy has developed the Inviscid-Infinite-Dimensional limit approach to establish probabilistic global regularity for supercritical NLS for smooth data.
The doctoral and postdoctoral scientific programmes aim to build upon these advances to address the following open questions:
- Regularity/explosion of approximate models for NS perturbed by a random noise,
- Probabilistic global regularity of NLS for singular data (towards the probabilistic scaling regularity),
- Properties of invariant measures constructed via the fluctuation-dissipation approach (with consideration of the application to turbulence),
- Studying the averages of carefully chosen observables under invariant measure of the NLS (with consideration of the application to weak turbulence theory).
Programme
To facilitate these scientific goals, several events will be organised at AIMS Senegal. These include an annual summer school (~10 days) which finishes with a three-day symposium (topics may include analysis of PDEs, stochastic analysis, geometry, applied mathematics). Workshops will be regularly organised. We will also, through the doctoral and postdoctoral team, be part of the current initiative at Bielefeld university, named SPDEvent, which is fully intended for doctoral and postdoctoral researchers. We aim to set up a short stay visitor programme intended for researchers from other Senegalese, African or oversea institutions. These guest researchers will stay typically for one week at the AIMS Centre to participate on ongoing activities. AIMS Senegal, by its geographical location and its academic opportunities, will play a critical role in the coordination of the Senegalese mathematical community.
Outreach events will be organised in Senegal, intended to attract people to mathematics in an inclusive manner and to contribute to disseminating math culture in the society. During these events, promotion of gender equality will be particularly emphasised. The GRC at AIMS Senegal has in recent years written three books in mathematics in African languages as part of his commitment to a deep mathematical development in Africa. We plan to organise, during outreach events, animations in this line.
