Xavier is an ICREA Research Professor since 2020. He is a mathematician who works on PDE. He has been the PI of an ERC Starting Grant (2019-2024) and an ERC Consolidator Grant (2024-2029), and has received several awards for young mathematicians in Spain, including the 'Premio Nacional de Investigación' for researchers under 40 in Mathematics and Computer Science. He also received the Scientific Research Award from the Fundación Princesa de Girona in 2019, as well as the Stampacchia Gold Medal in 2021, an international prize awarded every three years in recognition of outstanding contributions to the Calculus of Variations. In 2022, he was elected member of the Spanish Royal Academy of Sciences. In 2025, he received the 'Premi Nacional de Recerca al Talent Jove' from the Generalitat de Catalunya.

My research interests are on Partial Differential Equations (PDE), a vast and very active field of research in both pure and applied mathematics. PDE are used in essentially all sciences and engineering, and have important connections with several branches of pure mathematics. I work mainly on topics related to the regularity of solutions to nonlinear PDE. This is one of the most basic and important question in PDE theory: to understand whether all solutions to a given PDE are smooth or if, instead, they may have singularities. Some of my main contributions have been in the context of free boundary problems. These are PDE problems that involve unknown/moving interfaces, such as ice melting to water (phase transitions). From the mathematical point of view, they give rise to extremely challenging questions, and their study is closely connected to geometric measure theory. In particular, the study of free boundary problems has a strong geometrical flavor.