# Research

My research focuses on the study of problems arising from the broad context of **statistical mechanics**. More precisely, I am interested in the analysis of partial differential equations in **kinetic theory**.

Kinetic equations describe the evolution of a N-particle system at a mesoscopic level, which is an intermediate viewpoint between the Newtonian dynamic of the microscopic particles and a macroscopic, hydrodynamical model.

Part of my work concerns the analysis of singular limits for **Vlasov-type equations**. In particular, I’m working on validating the quasineutral limit for the Vlasov-Poisson equation and the Vlasov Poisson equation for massless electrons. I’m also interested in the relationship between the **quasineutral limit** and the **mean field limit**, i.e. the derivation of mesoscopic equations of Vlasov type from the N particle system described by the Newton dynamic.

More recently, I started studying the geometric structure of stationary solutions for the 3-dimensional **gravitational Vlasov-Poisson system** in the spirit of the geometric interpretation of 2D Euler.

I am also interested in **quantization of measures**, which concerns the approximation in some optimal way of a diffuse measure with discrete ones. The problem of quantization of probability measures arises in a great variety of contexts, such as signal processing, pattern or speech recognition, economics, and much more. In particular, I worked on a dynamical approach to the quantization problem in space dimensions one and two, using gradient flows to recast the problem through (possibly degenerate) **parabolic equations**. This study also led me to the necessity of understanding parabolic equations of ultrafast diffusion type. The natural questions are related to the well-posedness and asymptotic behaviour to such degenerate evolution equations.

The common root of these topics is the general task of finding a rigorous justification of the description of a large number of identical objects -typically physical particles as gas moleculs or ions and electrons in a plasma- via approximated models that describe the behavior of the “generic object” of a physical system. In terms of mathematical tools, I make extensive use of PDEs techniques, optimal transport, probability, calculus of variations, and Riemannian geometry.

For more information about some of my research interests, here are some expository notes:

**A gradient flow perspective on the quantization problem**.*PDE Models for Multi-Agent Phenomena*, Springer INdAM Ser, 28 (2018), pages 145–165.**Recent developments on the well-posedness theory for Vlasov-type equations**(with Megan Griffin-Pickering). To be included in the proceedings of the conference "Particle Systems and Partial Differential Equations”, editions VI, VII and VIII.**Recent developments on quasineutral limits for Vlasov-type equations**(with Megan Griffin-Pickering).*Recent advances in kinetic equations and applications*, Springer INdAM Series. To appear.