Reaching the end of the project, I succeed to constitute a team by hiring postdocs and we have made progress on the two main aspects of the project. As explained, the project relies on recruiting young analysts as postdoc and so the progress and the organization of the whole project also depends of the candidates, in the job market for a postdoc position.
The main activity in pure mathematics relies on academic research which gives publications and the organization/participation of events, which is what has been done during and through project FAnFArE.
In terms of mathematical results, we have tried to follow the tasks explained and planned in the DoA. Answers to the initial questions have been obtained. Moreover, allowing to bring a better understanding, some investigation in new and very promising directions, in connexion with the topic of the project have also been conducted, At the end, the project has given rise to different kind of collaborations, invitations, visits of experts, achieved through research papers published in international journals. The results cover all the different six tasks. So mainly, project FAnFArE has succeed to contribute in the study of multilinear Fourier multipliers in the Euclidean space, by describing boundedness of some certain class of them. It has been related to the notion of sparse domination, vector-valued estimates, ... The team has also started the study of several objects in the multi-parameter setting and in a more geometrical setting involving some curvature. A deep understanding of analytical objects allow us to have sharp estimates and to prove sharp inequalities whose one of the consequence is the study of several kind of PDE (dispersive, fluids mechanic, ...). The time-frequency analysis associated to a semigroup of operators, which allows to tackle more abstract situations (as on a Riemannian manifold), has also been developed. With such a setting, we have focused on the study of several kinds of paraproducts according to the problem under consideration. That allows us to study the algebra property of Sobolev spaces (for a positive regularity) and to study singular (stochastic) PDE (for a negative regularity) for example.
So as expected and as planned initially, together the team FAnFArE has made important contributions in the time-frequency analysis both in the case of the Euclidean case (with a deep analysis for specific singular operators) and in the more abstract setting given by a semigroup of operators.
In parrallel of these mathematical achievements, several workshops / conferences have been organized, as well as various participations to international conferences in order to promote the project and the obtained results. The list of these events is updated on the website of the project :
https://www.math.sciences.univ-nantes.fr/fanfare/node/1(s’ouvre dans une nouvelle fenêtre).
Each year of the project, I have organized (or co-organized) with other members of the team, at least one main event focusing on inviting young promising analysts and more experienced experts of our field. By these events, the project is promoted and it allows to make connections and develop collaborations together, taking advantage of the skills of every members of the team.
