Vladimir Alexandrovich Voevodsky (, ; 4 June 1966 – 30 September 2017) was a

Russia Russia, or the Russian Federation, is a country spanning Eastern Europe and North Asia. It is the list of countries and dependencies by area, largest country in the world, and extends across Time in Russia, eleven time zones, sharing Borders ...

America The United States of America (USA), also known as the United States (U.S.) or America, is a country primarily located in North America. It is a federal republic of 50 U.S. state, states and a federal capital district, Washington, D.C. The 48 ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

. His work in developing a

homotopy theory In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipli ...

for

algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...

and formulating motivic cohomology led to the award of a

Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...

in 2002. He is also known for the proof of the Milnor conjecture and motivic Bloch–Kato conjectures and for the

univalent foundations Univalent foundations are an approach to the foundations of mathematics in which mathematical Structuralism (philosophy of mathematics), structures are built out of objects called ''types''. Types in univalent foundations do not correspond exactly ...

of mathematics and homotopy type theory.

Early life and education

Vladimir Voevodsky's father, Aleksander Voevodsky, was head of the Laboratory of High Energy Leptons in the Institute for Nuclear Research at the

Russian Academy of Sciences The Russian Academy of Sciences (RAS; ''Rossíyskaya akadémiya naúk'') consists of the national academy of Russia; a network of scientific research institutes from across the Russian Federation; and additional scientific and social units such ...

. His mother Tatyana was a chemist. Voevodsky attended

Moscow State University Moscow State University (MSU), officially M. V. Lomonosov Moscow State University,. is a public university, public research university in Moscow, Russia. The university includes 15 research institutes, 43 faculties, more than 300 departments, a ...

for a while, but was expelled without a diploma for refusing to attend classes and failing academically. He received his Ph.D. in mathematics from

Harvard University Harvard University is a Private university, private Ivy League research university in Cambridge, Massachusetts, United States. Founded in 1636 and named for its first benefactor, the History of the Puritans in North America, Puritan clergyma ...

in 1992 after being recommended without even applying and without a formal college degree, following several independent publications; he was advised there by

David Kazhdan David Kazhdan (), born Dmitry Aleksandrovich Kazhdan (), is a Soviet and Israeli mathematician known for work in representation theory. Kazhdan is a 1990 MacArthur Fellow. Biography Kazhdan was born on 20 June 1946 in Moscow, USSR. His father ...

. While he was a first year undergraduate, he was given a copy of " Esquisse d'un Programme" (submitted a few months earlier by

Alexander Grothendieck Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...

CNRS The French National Centre for Scientific Research (, , CNRS) is the French state research organisation and is the largest fundamental science agency in Europe. In 2016, it employed 31,637 staff, including 11,137 tenured researchers, 13,415 eng ...

) by his advisor George Shabat. He learned the

French language French ( or ) is a Romance languages, Romance language of the Indo-European languages, Indo-European family. Like all other Romance languages, it descended from the Vulgar Latin of the Roman Empire. French evolved from Northern Old Gallo-R ...

"with the sole purpose of being able to read this text" and started his research on some of the themes mentioned there.

Work

Voevodsky's work was in the intersection of

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

with

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

. Along with Fabien Morel, Voevodsky introduced a

for schemes. He also formulated what is now believed to be the correct form of motivic cohomology, and used this new tool to prove Milnor's conjecture relating the Milnor

K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...

of a field to its

étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectu ...

. For the above, he received the

at the 24th

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...

held in

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China China, officially the People's Republic of China (PRC), is a country in East Asia. With population of China, a population exceeding 1.4 billion, it is the list of countries by population (United Nations), second-most populous country after ...

. In 1998 he gave a plenary lecture (''A¹-Homotopy Theory'') at the International Congress of Mathematicians in

Berlin Berlin ( ; ) is the Capital of Germany, capital and largest city of Germany, by both area and List of cities in Germany by population, population. With 3.7 million inhabitants, it has the List of cities in the European Union by population withi ...

. He coauthored (with Andrei Suslin and Eric M. Friedlander) ''Cycles, Transfers and Motivic Homology Theories'', which develops the theory of motivic cohomology in some detail. From 2002, Voevodsky was a professor at the

Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...

Princeton, New Jersey The Municipality of Princeton is a Borough (New Jersey), borough in Mercer County, New Jersey, United States. It was established on January 1, 2013, through the consolidation of the Borough of Princeton, New Jersey, Borough of Princeton and Pri ...

. In January 2009, at an anniversary conference in honor of

, held at the

Institut des Hautes Études Scientifiques The Institut des hautes études scientifiques (IHÉS; English: Institute of Advanced Scientific Studies) is a French research institute supporting advanced research in mathematics and theoretical physics (also with a small theoretical biology g ...

, Voevodsky announced a proof of the full Bloch–Kato conjectures. In 2009, he constructed the univalent model of Martin-Löf type theory in

simplicial set In mathematics, a simplicial set is a sequence of sets with internal order structure ( abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs. Every simplicial set gives rise to a "n ...

s. This led to important advances in type theory and in the development of new

of mathematics that Voevodsky worked on in his final years. He worked on a Coq library ''UniMath'' using univalent ideas. In April 2016, the

University of Gothenburg The University of Gothenburg () is a List of universities in Sweden, university in Sweden's second largest city, Gothenburg. Founded in 1891, the university is the third-oldest of the current List of universities in Sweden#Public universities, S ...

awarded an honorary doctorate to Voevodsky.

Death and legacy

Voevodsky died on 30 September 2017 at his home in

, aged 51, from an

aneurysm An aneurysm is an outward :wikt:bulge, bulging, likened to a bubble or balloon, caused by a localized, abnormal, weak spot on a blood vessel wall. Aneurysms may be a result of a hereditary condition or an acquired disease. Aneurysms can also b ...

. He was survived by his daughters, Diana Yasmine Voevodsky and Natalia Dalia Shalaby.

Selected works

* *Review: ''Lecture Notes on Motivic Cohomology'', European Mathematical Society
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Notes

References

External links

Vladimir Voevodsky on GitHub
Contains the slides of many of his recent lectures.

Интервью с Владимиром Воеводским и Лораном Лаффоргом * Julie Rehmeyer

New York Times, 6 October 2017 * * {{DEFAULTSORT:Voevodsky, Vladimir 1966 births 2017 deaths 20th-century Russian mathematicians 21st-century Russian mathematicians Fields Medalists Algebraic geometers Topologists Harvard Graduate School of Arts and Sciences alumni Institute for Advanced Study faculty Soviet mathematicians Sloan Research Fellows Harvard Fellows Russian expatriates in the United States Scientists from Moscow Russian scientists