for his groundbreaking work on partial differential equations, including creating a theory of regularity for nonlinear equations such as the Monge-Ampère equation, and free-boundary problems such as the obstacle problem, work that has influenced a whole generation of researchers in the field.

The Shaw Prize in Mathematical Sciences 2018 is awarded to **Luis A Caffarelli**, Professor of Mathematics at the University of Texas at Austin, USA for his groundbreaking work on partial differential equations, including creating a theory of regularity for nonlinear equations such as the Monge–Ampère equation, and free-boundary problems such as the obstacle problem, work that has influenced a whole generation of researchers in the field.

Partial differential equations are fundamental to large parts of mathematics, physics, and indeed all the sciences. They are used to model heat flow, fluid motion, electromagnetic waves, quantum mechanics, the shape of soap bubbles, and innumerable other physical phenomena.

Differential equations are fundamental to large parts of mathematics, physics, and indeed all the sciences. For example, if a physical system is in a certain state, and obeys certain laws, then a differential equation will tell you the state of the system infinitesimally later, and by putting together all these infinitesimal changes one can follow the evolution of the system in time. Or if a static system is held together by certain forces, a differential equation can often say how one part of the system depends on its immediate neighbours, and putting together this local information can give a global description of the system.

The simplest differential equations, known as ordinary differential equations, concern functions that depend on just one variable. For example, if a stone is thrown vertically in the air, then there is an ordinary differential equation that describes how its height varies as a function of time, given its initial height and velocity. However, the most useful equations, known as partial differential equations, concern functions of several variables (such as, for example, three spatial coordinates and one temporal coordinate) and are significantly more complicated as a result. Partial differential equations can be used to model heat flow, fluid motion, electromagnetic waves, quantum mechanics, the shape of soap bubbles, and innumerable other physical phenomena.

**Luis A Caffarelli** was born in 1948 in Buenos Aires, Argentina and is currently Professor of Mathematics at the University of Texas at Austin, USA. He obtained his Master of Science in 1969 and his PhD in Mathematics in 1972 from the University of Buenos Aires, Argentina. He joined the University of Minnesota, USA, where he was successively Postdoctoral Fellow (1973–1974), Assistant Professor (1975–1977), Associate Professor (1977–1979) and Professor (1979–1983). He was a Professor at the Courant Institute of Mathematical Sciences, New York University, USA (1980–1982), the University of Chicago, USA (1983–1986), the Institute for Advanced Study in Princeton, USA (1986–1996) and the Courant Institute, New York University, USA (1994–1997). He is a member of the US National Academy of Sciences and a Fellow of the American Academy of Arts and Sciences.