The Work of Vladimir Voevodsky Christophe Soulé

Vladimir Voevodsky was born in 1966. He studied at Moscow State University and Harvard university. He is now Professor at the Institute for Advanced Study in Princeton. Among his main achievements are the following: he defined and developed motivic cohomology and the A-homotopy theory of algebraic varieties; he proved the Milnor conjectures on the K-theory of fields. Let us state the first Milnor conjecture. Let F be a field and n a positive integer. The Milnor K-group of F is the abelian group K n (F ) defined by the following generators and relations. The generators are sequences {a1, . . . , an} of n units ai ∈ F . The relations are {a1, . . . , ak−1, xy, ak+1, . . . , an} = {a1, . . . , ak−1, x, ak+1, . . . , an}+ {a1, . . . , ak−1, y, ak+1, . . . , an} for all ai, x, y ∈ F , 1 ≤ k ≤ n, and the Steinberg relation {a1, . . . , x, . . . , 1− x, . . . , an} = 0 for all ai ∈ F ∗ and x ∈ F − {0, 1}. On the other hand, let F be an algebraic closure of F and G = Gal(F/F ) the absolute Galois group of F , with its profinite topology. The Galois cohomology of F with Z/2 coefficients is, by definition,