My research is about the theory of Partial Differential Equations, and especially about kinetic equations.
During the 1900 International Congress of Mathematicians in Paris, Hilbert announced his famous list of problems, which will play a decisive role in the research of the incoming decades. In particular, he proposed to axiomatize physics, this problem being known as the Hilbert's sixth problem. As Hilbert said himself : "Boltzmann's work on the principles of mechanics suggests the problem of developing mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua".
Indeed, irreversible processes appeared naturally along the study of the Boltzmann equation, such as an entropy (see the famous result known as the H-theorem), which is a decreasing in time function. Yet, the Boltzmann equation is obtained from a microscopic, and especially reversible description of the fluid studied. This is an apparent paradox, as it was noticed for example by Loschmidt and Zermelo.
In 1973, Lanford proved his famous theorem (see this page, giving, besides the statement of the result, a quite detailed sketch of the proof), which shows rigorously that irreversible phenomena can appear from reversible description of the matter.
Along my PhD, I studied the rigorous derivation of the Boltzmann equation from the BBGKY and Boltzmann hierarchies, and thier use in the proof of the Lanford's theorem. In particular, on the one hand I investigated the functional setting in which the objects used in the proof make sense, and on the other hand, I studied the problem of the derivation when particles evolve in a domain with boundary conditions, instead of focusing on the cases of the torus or the whole euclidian space, as in the article from Isabelle Gallagher, Laure Saint-Raymond and Benjamin Texier.
For more details, see this research statement.