### Vladimir Voevodsky †

Born 4 June 1966, Moscow, Russia

Died 30 September 2017

Fields Medal (2002) “especially for his work on motivic cohomology and homotopy theory for algebraic varieties and his proof of the Milnor conjecture.”

Vladimir Voevodsky comes from a scientific household: his mother was a chemist, his father an experimental physicist. He began to study Mathematics at the Lomonossov Moscow State University, but, although his academic achievements were later accredited with a Bachelor of Science (1989) he actually gave up formal study. Instead, Voevodsky worked in algebraic geometry on his own. To learn, he sought direct contact with scientists, first of all with Yuri Shabat and later with Mikhail Kapranov (now at Yale University). Kapranov helped Voevodsky later to become a Ph.D. student at Harvard University in the USA where he wrote a doctoral thesis on “Homology of schemes and covariant motif” (Ph.D. 1992). His thesis advisor was David Kazhdan.

Voevodsky did research at the Institute for Advanced Study in Princeton (1992-93). He was a junior fellow of the Harvard Society of Fellows at Harvard University (1993- 96) and then associate professor at Northwestern University (1996-99). At the same time he was guest at Harvard (1996-97 and again 2006-08) and at the Max Planck Institute for Mathematics in Bonn (1996-97). He returned to the Institute for Advanced Study (1998-2001), where he has worked as a professor since 2002. Voevodsky has been a member of the European Academy of Sciences since 2003, and he holds an honorary degree from the Wuhan University, China (2004).

Vladimir Voevodsky has worked in many areas of mathematics. His earlier work was related to the ideas introduced by Alexander Grothendieck in his famous manuscript “Esquisse d’un Programme”, in particular to “Dessin d’enfant”, anabelian algebraic geometry and $\infty$-groupoids as models for homotopy types. From 1990 to 2009 most of Voevodsky’s work was related to the development of motivic homotopy theory. This development was in part guided and motivated by the conjectures of Beilinson and Lichtenbaum addressing the properties of hypothetical (at that time) “motivic cohomology”. In 1995 Voevodsky found a proof of Milnor’s Conjecture – a particular case of Beilinson-Lichtenbaum Conjectures on motivic cohomology with finite coefficients which earned him a Fields Medal in 2002. It took him about 12 years, from 1997 to 2009, to work out the details of the proof of the general Beilinson-Lichtenbaum Conjectures for finite coefficients which was published in Annals of Mathematics in 2011.

Since about 2002 Voevodsky has also been actively thinking about the problem of computer proof verification in pure mathematics. The first results in the form of the theory of univalent fibrations and its connection with type theory appeared in 2005 but it was only in the fall of 2009 that it became clear that the existing system of Coq is largely adequate for formalization of mathematics based on the univalent ideas. The Univalent Foundations program was formally announced by Voevodsky in the spring of 2010. Since then it has become the main focus of his mathematical work.