**Vladimir Voevodsky** was a Russian-born American mathematician who was awarded a Fields Medal in 2002 for his work in developing a homotopy theory for algebraic varieties.

- For example, Voevodsky's paper with Mikhail Kapranov on infinity-groupoids realised Grothendieck's idea, presented in his unpublished but widely circulated "letter to Quillen" (Pursuing stacks), of generalising the way that certain CW-complexes can, taking the viewpoint of homotopy, be described by groupoids.
- Voevodsky's paper on étale topologies again arose from a question that Grothendieck posed, this time in his 'Esquisse d'un programme'.
- Voevodsky moved to Harvard University in the United States where he completed his doctorate supervised by David Kazhdan.
- After the award of his doctorate, Voevodsky was a Member of the Institute for Advanced Study at Princeton from September 1992 to May 1993.
- In 1996 Voevodsky, in collaboration with Andrei Suslin, published Singular homology of abstract algebraic varieties.
- Also in 1996, Voevodsky published Homology of schemes which Claudio Pedrini describes as "an important step toward the construction of a category of the so-called mixed motives." In 1998 Voevodsky lectured at the International Congress of Mathematicians in Berlin on A'-homotopy theory.
- In a beautiful tour de force, Voevodsky has constructed all reasonable cohomology theories on schemes simultaneously by constructing a stable homotopy category of schemes.
- This is a triangulated category analogous to the stable homotopy category of spaces studied in algebraic topology; in particular, the Brown representability theorem holds, so that every cohomology theory on schemes is an object of the Voevodsky category.
- This work is, of course, the foundation of Voevodsky's proof of the Milnor conjecture.
- The exposition makes Voevodsky's ideas seem obvious; after the fact, of course.
- One of the most powerful advantages of the Voevodsky category is that one can construct cohomology theories by constructing their representing objects, rather than by describing the groups themselves.
- The author constructs singular homology (following the ideas of A Suslin and Voevodsky (1996), algebraic K-theory, and algebraic cobordism in this way.
- Throughout the paper, there are very clear indications of where Voevodsky thinks the theory needs further work, and the paper concludes with a discussion of possible future directions.
- As this review indicated, Voevodsky had proved the Milnor Conjecture.
- A major advance came when Voevodsky, building on a little-understood idea proposed by Andrei Suslin, created a theory of "motivic cohomology".
- In addition, Voevodsky provided a framework for describing many new cohomology theories for algebraic varieties.
- One consequence of Voevodsky's work, and one of his most celebrated achievements, is the solution of the Milnor Conjecture, which for three decades was the main outstanding problem in algebraic K-theory.
- From September 1998, Voevodsky was a Member of the Institute for Advanced Study at Princeton.
- Recent work by Voevodsky has shown that he has become interested in mathematical biology.
- Finally we mention some books which Voevodsky has written.
- More recently, Motivic homotopy theory (2007) is a book by several authors based on lectures they gave at the Summer School held in Nordfjordeid in August 2002 and contains Voevodsky's lectures on Motivic homotopy theory.
- Finally we mention Lecture notes on motivic cohomology (2006) which is written by several authors based on lectures given by Voevodsky.

Born 4 June 1966, Moscow, Russia. Died 30 September 2017, Princceton, New Jersey, USA.

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Prize Fields Medal, Origin Russia, Topology

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive