#### Luis A Caffarelli

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.

#### Contribution

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.

#### An Essay on the Prize

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.