Professor Dr Alessio Figalli

Research Priorities: Calculus of variations, partial differential equations, optimal transportation, functional and geometric inequalities, free boundary problems

Alessio Figalli is an Italian mathematician. He investigates calculus of variations and partial differential equations, especially in optimal transport, geometric and functional inequalities, and free boundary problems. He examines how mass, energy, and shapes organise in the most efficient way, and his work has influenced both fundamental mathematics and applied areas such as meteorology, materials science, and models of collective behaviour.

Alessio Figalli’s research centres on optimal transport and its links with nonlinear partial differential equations. One of his most influential achievements, in collaboration with Guido De Philippis, established borderline second-order Sobolev regularity for solutions of the Monge-Ampère equation under very weak assumptions on the right-hand side. This result has important consequences for the regularity of optimal transport maps and for semigeostrophic equations used in meteorology.

A second major line of work concerns singularities in optimal transport. Figalli initiated the systematic study of the singular set of optimal transport maps and proved that, in full generality, this set has Lebesgue measure zero. Additionally, his work helped clarify how regularity can persist even when global smoothness fails, with further impact on metric geometry and probability.

A third priority in Figalli’s research is the use of transport methods to obtain sharp quantitative stability results for fundamental inequalities. Together with colleagues, he has proved influential stability estimates for the isoperimetric, Brunn-Minkowski, and Wulff inequalities, including results relevant to the description of low-energy crystal shapes.

In addition, Figalli has made major contributions to free boundary problems, developing new methods to analyse the regularity and structure of interfaces in classical and fractional obstacle problems, and more recently contributing to the classification of global solutions and the study of singular sets.