Mathematics is the basic language of all natural sciences and all modern technology. In the twentieth century mathematics made tremendous strides both in opening new frontiers and in solving important and difficult old problems. Its influence permeates every creative scientific and technological discipline, and extends into the social science. With the developments in computer science, information technology, and statistics in the twentieth century, the importance of mathematics to mankind will be further enhanced in the twenty-first century.

... for his initiation of the field of global differential geometry and his continued leadership of the field, resulting in beautiful developments that are at the centre of contemporary mathematics, with deep connections to topology, algebra and analysis, in short, to all major branches of mathematics of the last sixty years.

... for his proof of Fermat's Last Theorem.

... for his contributions to mathematics, and to the new interdisciplinary fields of pattern theory and vision research.

... for his contributions to the new interdisciplinary field of mathematics mechanization.

... for initiating and developing a grand unifying vision of mathematics that connects prime numbers with symmetry.

... for their widespread and influential contributions to Mathematical Physics.

... for their many brilliant contributions to geometry in 3 and 4 dimensions.

... for his profound work in mathematical analysis and its application to partial differential equations, mathematical physics, combinatorics, number theory, ergodic theory and theoretical computer science.

... for their highly innovative works on nonlinear partial differential equations in Lorentzian and Riemannian geometry and their applications to general relativity and topology.

... for his pioneering works in algebra, geometry and mathematical physics and in particular deformation quantization, motivic integration and mirror symmetry.

... for his profound contributions to modern mathematical statistics and in particular the development of optimal algorithms for statistical estimation in the presence of noise and of efficient techniques for sparse representation and recovery in large data-sets.

... for his fundamental contributions to algebra, algebraic geometry, and representation theory, and for weaving these subjects together to solve old problems and reveal beautiful new connections.

... for their introduction and development of fundamental tools in number theory, allowing them as well as others to resolve some longstanding classical problems.

... for his far-reaching contributions to geometry, representation theory and theoretical physics. The fundamental and elegant concepts and techniques that he has introduced have had wide impact and are of lasting importance.

... for their remarkable results in many central areas of algebraic geometry, which have transformed the field and led to the solution of long-standing problems that had appeared out of reach.

... 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.

... for his work on concentration inequalities, on suprema of stochastic processes and on rigorous results for spin glasses.

... for their huge influence on and profound contributions to representation theory, as well as many other areas of mathematics

... for their remarkable insights that have transformed, and continue to transform, modern geometry.

... for their remarkable contributions to discrete mathematics and model theory with interaction notably with algebraic geometry, topology and computer sciences.

... for their contributions related to mathematical physics, to arithmetic geometry, to differential geometry and to Kähler geometry.