to further sets of integers. The Green–Tao theorem then arises as a special case, although it is not trivial to show that the prime numbers satisfy the conditions of Green and Tao's extension of the Szemerédi theorem. In 2010, Green and Tao gave a multilinear extension of Dirichlet's celebrated theorem on arithmetic progressions. Given a matrix and a matrix whose components are all integers, Green and Tao give conditions on when there exist infinitely many matrices such that all components of are prime numbers. The proof of Green and Tao was incomplete, as it was conditioned upon unproven conjectures. Those conjectures were proved in later work of Green, Tao, and Tamar Ziegler. Notable awards 1999 – Packard Fellowship 2000 – Salem Prize for: "his work in harmonic analysis and on related questions in geometric measure theory and partial differential equations." 2002 – Bôcher Memorial Prize for: Global regularity of wave maps I. Small critical Sobolev norm in high dimensions. Internat. Math. Res. Notices (2001), no. 6, 299-328. Global regularity of wave maps II. Small energy in two dimensions. Comm. Math. Phys. 2244 (2001), no. 2, 443-544. in addition to "his remarkable series of papers, written in collaboration with J. Colliander, M. Keel, G. Staffilani, and H. Takaoka, on global regularity in optimal Sobolev spaces for KdV and other equations, as well as his many deep contributions to Strichartz and bilinear estimates." 2003 – Clay Research Award for: his restriction theorems in Fourier analysis, his work on wave maps, his global existence theorems for KdV-type equations, and for his solution with Allen Knutson of Horn's conjecture 2005 – Australian Mathematical Society Medal 2005 – Ostrowski Prize (with Ben Green) for: "their exceptional achievements in the area of analytic and combinatorial number theory" 2005 – Levi L.Conant Prize (with Allen Knutson) for: their expository article "Honeycombs and Sums of Hermitian Matrices" (Notices of the AMS. 48 (2001), 175–186.) 2006 – Fields Medal for: "his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory" 2006 – MacArthur Award 2006 – SASTRA Ramanujan Prize 2006 – Sloan Fellowship 2007 – Fellow of the Royal Society 2008 – Alan T. Waterman Award for: "his surprising and original contributions to many fields of mathematics, including number theory, differential equations, algebra, and harmonic analysis" 2008 – Onsager Medal for: "his combination of mathematical depth, width and volume in a manner unprecedented in contemporary mathematics".