... for excellence of research in the mathematical sciences published within the past ten years.

... for his extraordinary vision, which has brought the theory of group representations into a revolutionary new relationship with the theory of automorphic forms and number theory.

... for his role in the introduction and application of radically new approaches to the topology of singular spaces, including characteristics classes, intersection homology, perverse sheaves, and stratified Morse theory.

... for his proof of Fermat's Last Theorem by discovering a beautiful strategy to establish a major portion of the Shimura-Taniyama conjecture, and for his courage and technical power in bringing his idea to completion.

... for fundamental discoveries on wavelets and wavelet expansions and for her role in making wavelet methods a practical basic tool of applied mathematics.

... for the theory of free probability, in particular, using random matrices and a new concept of entropy to solve several hitherto intractable problems in von Neumann algebras.

... for groundbreaking work relating to Seiberg-Witten and Gromov-Witten invariants of symplectic 4-manifolds, and his proof of Weinstein conjecture for all contact 3-manifolds.

... for his leading role in the development of homotopy theory, which has both reinvigorated algebraic topology as a central field in mathematics and led to the resolution of the Kervaire invariant problem for framed manifolds.

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