Vladimir Voevodsky

Native spelling: Владимир Воеводский

1966–2017, Moscow, Soviet Union, Princeton, New Jersey, United States

AlgebraTopologyFoundations of mathematicsFields Medal 2002

Russian-American mathematician who created motivic homotopy theory, proved the Milnor conjecture, and later initiated univalent foundations for mathematics.

Vladimir Aleksandrovich Voevodsky was born in Moscow and studied at Moscow State University, where he first worked on categorical algebra, before moving to Harvard University and completing his PhD in 1992 under David Kazhdan. After positions at Harvard and Northwestern University, he became a permanent professor at the Institute for Advanced Study in Princeton in 2002.

In the 1990s Voevodsky, partly in collaboration with Andrei Suslin, Eric Friedlander and Fabien Morel, built motivic cohomology and A^1-homotopy theory — an algebraic version of homotopy theory in which spaces are replaced by schemes and the affine line replaces the interval. With this framework he proved the Milnor conjecture on the relation between Milnor K-theory and Galois cohomology, which won him the Fields Medal in 2002. The same ideas later yielded, jointly with Charles Weibel, Markus Rost and others, the full norm residue theorem (the Bloch–Kato conjecture), relating algebraic K-theory to étale cohomology for all primes.

From around 2006 Voevodsky turned to the foundations of mathematics. Concerned that mathematics as practiced contained uncorrectable errors that refereed journals could not catch, he developed univalent foundations: a version of Martin-Löf type theory in which equality is interpreted as homotopy equivalence, suitable for machine verification. His UniMath library in the Coq proof assistant implemented this vision, and his 2014 special year at the IAS with the book 'Homotopy Type Theory' launched a new community. He died suddenly in Princeton in 2017, aged 51.

Workplaces: Institute for Advanced Study, Princeton, Northwestern University

Russia, United States

Award-winning work

Fields Medal 2002Voevodsky received the medal for his construction of motivic cohomology and his proof of the Milnor conjecture. Motivated by Grothendieck's vision of a 'motivic' cohomology theory for algebraic varieties, he defined the motivic cohomology groups through the triangulated category of mixed motives, using algebraic analogues of homotopy theory in which the affine line plays the role of the unit interval — the A^1-homotopy theory he developed with Fabien Morel. Applying this machinery, he proved the Milnor conjecture identifying the mod-2 Milnor K-theory of a field with its Galois cohomology, a result at the heart of algebraic K-theory and quadratic forms, later extended with collaborators to the full norm residue theorem.

Contributions, linked to the library

Library topicHomological algebraMotivic cohomology and the triangulated category of motives are homological-algebraic constructions at the heart of his Fields Medal work.Library topicFormal proof with proof assistants (Lean)He initiated univalent foundations and built the UniMath library of computer-verified mathematics in Coq.Library topicAlgebraic geometryHis motivic homotopy theory applies homotopy-theoretic methods to algebraic varieties and schemes.