01 · A life in mathematics
The person behind the work
Vladimir Voevodsky was born in Moscow in the Soviet Union and received the 2002 Fields Medal shortly after becoming a professor at the Institute for Advanced Study. He brought methods inspired by topology into algebraic geometry, helping create motivic cohomology and an algebraic version of homotopy theory. These tools yielded his proof of the Milnor conjecture, relating algebraic K-theory and Galois cohomology. His later research turned toward foundations and computer-checked proofs, including univalent foundations. That later programme grew from his concern for reliable mathematical reasoning, but the Fields Medal primarily recognised his earlier transformation of algebraic geometry and its arithmetic connections.
The contribution, explained
Topology studies features that survive continuous deformation. Algebraic geometry studies spaces constrained by polynomial equations, where ordinary deformation methods do not transfer directly. Voevodsky developed a framework that gives these rigid spaces analogues of topological tools. Motivic cohomology organises algebraic cycles into groups and connects them with other invariants. In the Milnor conjecture, symbols built from nonzero field elements, taken modulo two, match appropriate Galois cohomology classes. Proving this required a new conceptual setting in which relations invisible in a single calculation could become structural facts. The machinery reaches far beyond this one conjecture.
Why it matters
Motivic methods changed how arithmetic and geometric invariants could be compared, proving the Milnor conjecture and supporting later work on the Bloch–Kato conjecture and modern algebraic homotopy theory.