# Luigi Ambrosio

## Italy

### 2019 Balzan Prize for Theory of Partial Differential Equations

#### Luigi Ambrosio is a remarkable mathematician whose astonishing capacity for synthesis has made it possible to create hitherto unimaginable bridges between partial differential equations and the calculus of variation. His influence on the analysis of very general spaces is exceptional.

Luigi Ambrosio is a remarkable mathematician renowned for his important contributions in the field of partial differential equations and of the calculus of variation.

Born in Italy in 1963, he is Director of the Scuola Normale Superiore in Pisa (since 2019) after having served as Professor there since 1998. He has also held professorships at the universities of Roma Tor Vergata, Benevento and Pavia. He has received a great many prestigious awards, including the Prize of the Italian Mathematical Union and the Fermat Prize. He has been invited to give plenary lectures at the International Congress of Mathematicians, as well as the European Congress. He is a Member of the Accademia dei Lincei.

*Partial differential equations* concern the dynamics of quantities that depend on several parameters, which are most often time and position. There are numerous examples: a vibrating string, a moving electromagnetic wave, the motion of a fluid, or more generally, any moving object.

The *calculus of variation* tries to describe curves or surfaces minimising a certain quantity. For example, which surface limited by a closed curve has a minimal area? These “minimal surfaces” are analogous to soap bubbles. What shape should one give to an airplane wing so as to minimize air resistance? One may also try to direct a large number of cars in a city in order to minimize traffic jams.

Luigi Ambrosio’s astonishing capacity for synthesis has allowed him to create hitherto unimaginable bridges between these two important fields of mathematics. For example, his discoveries concern transport equations and conservation laws, optimal transport, evolution equations in geometry, and analysis in metric spaces. Historically, mathematicians looked for smooth solutions with no regularities, but unfortunately this does not correspond to physical reality. Over the past few decades, in particular thanks to Ambrosio, new methods have made it possible to find non-smooth solutions. This has led to the fundamental renewal of the venerable field of infinitesimal calculus.

Luigi Ambrosio has introduced functional mathematical spaces, enabling him to make analyses under very weak conditions of regularity. For example, he has extended the theory of currents (which goes back to the 1960s) to show the existence of minimal surfaces beyond Euclidean space to any geometry whatsoever, something that had not previously been shown. This result was quite a surprise to the community of scholars in the field of geometric measure theory.

Other important advances include his work on the mathematical curvature of metric measure spaces as well as on gradient flows in these spaces, which are a priori not very regular.

The problem of optimal transport, which goes back to Gaspard Monge in 1781, consists in moving one distribution of masses towards another in the most efficient way. This problem has had a long and complex history, but thanks to Luigi Ambrosio’s contribution, some of the bases for definitive proof of the existence of solutions have been established, precisely in the situation described by Monge.

Luigi Ambrosio is not only known for his mathematical discoveries. He is an exceptional teacher who has founded a school of thought and who has trained a great number of students who today make up a community of mathematicians working in prestigious universities throughout the world.

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