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Overview
My research is in mathematical analysis, with a few points of contact with probability theory, and mathematical physics. I am interested in functional inequalities, optimal transport, and in the analysis of partial differential equations.
Projects and collaborations
Kinetic Optimal Transport (OTIKIN): the core of my current research is the development of an optimal transport framework adapted to kinetic equations. The challenge lies in the fact that kinetic equations combine conservative and dissipative effects. In the first part of the project (OTIKIN 1), with Jan Maas and Filippo Quattrocchi we propose a discrepancy between probability distributions in position and velocity, based on minimal acceleration. Currently, in collaboration with Guillaume Carlier and Jean Dolbeault, we are applying the new kinetic discrepancy to the analysis of Vlasov-Fokker-Planck equations. One advantage of our approach is that it treats nonlinear kinetic equations directly. In other words, our techniques will not involve linearisation nor perturbative arguments.
Kinetic equations: the theoretical framework of Kinetic Optimal Transport should be compared with constructive results, available for a few physically relevant equations. I have been working on constructive estimates for the long-time convergence to equilibrium for kinetic Fokker-Planck equations since 2020. I started a long-term collaboration with Gabriel Stoltz, and, then, with Lihan Wang, Andi Q. Wang, and Francis Lörler (for the interplay between hypocoercivity and the very recent theory of non-reversible lifts of diffusion processes). The outcomes may find application in molecular sampling and in the multi-scale analysis of thermonuclear fusion plasma. Finally, I have been involved in the redaction of the lecture notes of Clément Mouhot's course on de Giorgi's methods.
Diffusion limits, diffusion equations, and stability of functional inequalities: in a limit regime, in suitable cases, it is possible to reduce the analysis of a kinetic equation to the time-evolution of the macroscopic spatial density, which satisfies a diffusion equation. Such a diffusion limit procedure connects the mesoscopic and the macroscopic scales of description for a particle systems, in the spirit of Hilbert’s sixth problem (1900). Diffusion equations are linked (see this book and references quoted therein) with functional inequalities. The two sides of a functional inequality play the role of entropy and entropy production along the diffusion flow. The optimal functions and the optimal constant in the inequality are related with the large-time asymptotics of the flow. Finally, the stability estimates (which amount to establishing whether the difference between the two sides of the inequality controls some notion of distance from the set of optimal functions) can be translated into improved convergence rates along the flow, when far away from the asymptotic regime. All these ideas are protagonists in a series of work I wrote in collaboration with Jean Dolbeault and Nikita Simonov, about constructive stability of interpolation inequalities on the sphere and in the Gaussian space. More precise information about the geometry of solutions is available for the classical heat equation in the Euclidean space. With Francesco Pedrotti, I investigated the issue of log-concavity, with applications to Lipschitz transport maps and score-based diffusion models. I am currently interested in extending the analysis to inequalities arising in information theory and quantum mechanics.
Non-bilinear Dirichlet Forms: nonlinear semigroup in Hilbert spaces, induced by the sub-differential of convex energies, are an abstraction of diffusion equations. A special subclass is that of nonlinear Markov semigroups, which enjoy futher contraction and order-preserving topics. In this case, the energy associated with the semigroup is a (non-)bilinear Dirichlet form. I was introduced to this subject by Giuseppe Savaré, in occasion of my MSc thesis. With Ivailo Hartarsky, we proved a general normal contraction property for non-bilinear forms. In 2025, with Lorenzo Dello Schiavo, we worked at a systematic, complete, and general tretament of nonlinear Markov semigroups, and non-bilienar forms, adressing equivalence of various definitions, locality, and invariant subsets. In the long term, I aim at establishing connections with non-linear Markov processes, and with the structural properties of metric measure spaces.
Applications: of Optimal transport to statistics, and, consequently, to multi-dimensional inequality indices, also in view of the EU Multidimensional Inequality Monitoring Framework. This is a joint project with Gennaro Auricchio, Paolo Giudici, and Giuseppe Toscani.
Grants
Postdoc (2023-2025) : IST-Bridge International Postdoctoral Program (approx 175.000€) . This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 101034413.
PhD (2019-2023) : Cofund : MathInParis Fellowship (approx 175.000€). This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 754362. My PhD project was also partially funded by the EFI project of the French National Research Agency (ANR).
Prizes
PhD Thesis prize: Prix solennel de thèse de la Chancellerie des Universités de Paris 2024 en "sciences toutes spécialités" (10.000€)
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