The Fields Medal is awarded to recognize outstanding mathematical achievement for existing work and for the promise of future achievement.
The medals and cash prizes are funded by a trust established by J.C.Fields at the University of Toronto, which has been supplemented periodically, but is still significantly underfunded. The discrepancy in 2018 was made up by the University of Toronto and the Fields Institute.
For the proof of the boundedness of Fano varieties and for contributions to the minimal model program.
For contributions to the theory of optimal transport and its applications in partial differential equations, metric geometry and probability.
For transforming arithmetic algebraic geometry over p-adic fields through his introduction of perfectoid spaces, with application to Galois representations, and for the development of new cohomology theories.
For his synthesis of analytic number theory, homogeneous dynamics, topology, and representation theory, which has resolved long-standing problems in areas such as the equidistribution of arithmetic objects.