Alessio Figalli

fl. 1984, Rome, Italy

AnalysisDifferential equations and dynamical systemsFields Medal 2018

Italian mathematician at ETH Zurich working on optimal transport, partial differential equations and the calculus of variations; Fields Medalist 2018.

Alessio Figalli was born in Rome and, encouraged by a mathematician uncle, entered the Scuola Normale Superiore in Pisa after excelling in the Italian mathematical olympiads. He completed his PhD in 2007 under Luigi Ambrosio at the Scuola Normale and Cédric Villani at the École normale supérieure de Lyon, already working at the interface of optimal transport and partial differential equations.

After positions in Paris and a professorship at the University of Nice, he spent a year at the University of Texas at Austin in 2009 and returned there as professor in 2011, holding the R. L. Moore chair. In 2016 he moved to ETH Zurich as full professor, where he now also directs the Mathematics Research Institute (FIM).

Figalli's research centers on optimal transport — the efficient rearrangement of mass described by the Monge–Ampère equation — and on its consequences for PDEs, geometry and probability. With Guido De Philippis he developed a sharp regularity theory for transport maps; he has also made major advances on the obstacle problem, the parabolic Monge–Ampère equation, isoperimetric and functional inequalities, rough transport flows, and the dynamics of crystal surfaces. He received the Fields Medal in 2018.

Workplaces: ETH Zurich, University of Texas at Austin

Italy

Award-winning work

Fields Medal 2018Figalli was awarded the medal for contributions to the theory of optimal transport and its applications in partial differential equations, metric geometry and probability. He developed a powerful regularity theory for optimal transport maps, proving with De Philippis and others that, outside a small singular set, the Monge–Ampère equation governing the transport map is smooth — a question going back to Caffarelli. He applied optimal transport to the semigeostrophic equations of atmospheric dynamics, proving existence of distributional solutions in three dimensions along the lines of Brenier's program, and resolved long-standing problems on the obstacle problem and the regularity of free boundaries.

Contributions, linked to the library

Library topicOptimal transportDeveloped optimal transport into a sharp analytic tool: regularity theory for transport maps and the Monge–Ampère equation, with applications to the semigeostrophic equations and free-boundary problems.