... for his role in the development of number theory

... for revolutionizing the field of operator algebras, for inventing modern non-commutative geometry, and for discovering that these ideas appear everywhere, including the foundations of theoretical physics;

... for his work on the Langlands program

... for a lifetime of achievement, especially for pointing the way to unify apparently disparate fields of mathematics and to discover their elegant simplicity through links with the physical world

... for establishing the existence of the scaling limit of two-dimensional percolation, and for verifying John Cardy's conjectured relation

... for his work in combining analytic power with geometric insight in the field of random walks, percolation, and probability theory in general, especially for formulating stochastic Loewner evolution. His work opens new doors and reinvigorates research in these fields

... for finding, jointly with two undergraduate students, an algorithm that solves a modern version of a problem going back to the ancient Chinese and Greeks about how one can determine whether a number is prime in a time that increases polynomially with the size of the number

... for his discovery of the Ricci Flow Equation and its development into one of the most powerful tools of geometric analysis. Hamilton conceived of his work as a way to approach both the Poincaré Conjecture and the Thurston Geometrization Conjecture

... for his ground-breaking work in analysis, notably his optimal restriction theorems in Fourier analysis, his work on the wave map equation (the hyperbolic analogue of the harmonic map equation), his global existence theorems for KdV type equations, as well as significant work in quite distant areas of mathematics.

... for his joint work with Terry Tao on arithmetic progressions of prime numbers. These are equally spaced sequences of primes such as 31, 37, 43 or 13, 43, 73, 103.

... for their proof of the Fundamental Lemma for unitary groups.

... for his discovery of new composition laws for quadratic forms, and for his work on the average size of ideal class groups.

... for his complete resolution of a conjecture made by F Treves and L Nirenberg in 1970.

... for his work on rational billiards and geometric group theory, in particular, his crucial contribution to joint work with David Fisher and Kevin Whyte establishing the quasi-isometric rigidity of sol.

... for their work in advancing our understanding of the birational geometry of algebraic varieties in dimension greater than three, in particular, for their inductive proof of the existence of flips.

... for their work on local and global Galois representations, partly in collaboration with Clozel and Shepherd-Barron, culminating in the solution of the Sato-Tate conjecture for elliptic curves with non-integral j-invariants.

... for his proof of the Weinstein conjecture in dimension three.

... for her disproof of the Kodaira conjecture.

... for his work in p-adic harmonic analysis, particularly his contributions to the transfer conjecture and the fundamental lemma. This work, combined with that of others, makes it possible to finally resolve important, long-standing parts of the Langlands program.

... for their solutions of the Marden Tameness Conjecture, and, by implication through the work of Thurston and Canary, of the Ahlfors Measure Conjecture.

... for their spectacular work on stationary measures and orbit closures for actions of non-abelian groups on homogeneous spaces

... for his resolution of the André-Oort Conjecture in the case of products of modular curves

... for their work in hyperbolic geometry

For his recent outstanding work in enumerative geometry, specifically for his proof in a large class of cases of the MNOP conjecture that he formulated with Maulik, Okounkov and Nekrasov

... for her many and significant contributions to geometry and ergodic theory, in particular to the proof of an analogue of Ratner's theorem on unipotent flows for moduli of flat surfaces

... for his many and significant contributions to arithmetic algebraic geometry, particularly in the development and applications of the theory of perfectoid spaces

... for their solution of the Erdős distance problem and for other joint and separate contributions to combinatorial incidence geometry

... in recognition of their groundbreaking contributions to the understanding of mirror symmetry, in joint work generally known as the "Gross-Siebert Program"

... in recognition of his groundbreaking work in representation theory and related fields

... in recognition of their introduction of a novel geometric combinatorial method to study doubling properties of solutions to elliptic eigenvalue problems.

... in recognition of their groundbreaking and conceptually novel work on the geometry of the Gaussian free field and its application to the solution of open problems in the theory of two-dimensional random structures.

... in recognition of her groundbreaking work on sphere-packing problems in eight and twenty-four dimensions.

... in recognition of the profound contributions that each of them has made to the analysis of partial differential equations, particularly the Navier-Stokes and Euler equations.

... in recognition of his ground-breaking work in arithmetic geometry and arithmetic aspects of automorphic forms.

... for his groundbreaking achievements in commutative algebra, arithmetic algebraic geometry, and topology in the p-adic setting."

... for their profound contributions to the understanding of high dimensional manifolds and their diffeomorphism groups; they have transformed and reinvigorated the subject.

... in recognition of his wide-ranging and transformative work in geometry and topology, particularly his groundbreaking achievements in symplectic topology.

... for their groundbreaking advances in the understanding of singular solutions to the fundamental equations of fluids dynamics, including their construction of smooth self-similar solutions for the compressible Euler equation, and families of finite-energy blow-up solutions for both the compressible Euler and Navier-Stokes equations.