Vladimir Voevodsky

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Vladimir Voivodsky (2011)

Vladimir Alexandrovich Wojewodski ( Russian Владимир Александрович Воеводский , scientific. Transliteration Vladimir Aleksandrovich Voevodskij , mostly under the English name as Vladimir Voevodsky quotes; * 4. June 1966 in Moscow ; † the thirtieth September 2017 in Princeton , New Jersey ) was an American Mathematician of Russian origin and Fields Medal winner . He worked in the areas of homotopy theory of algebraic varieties and motivic cohomology .

Life

His father Alexander Wojewodski was an experimental physicist with a laboratory at an institute of the Soviet Academy of Sciences, his mother Tatiana a chemistry professor at Lomonossow University. Voevodsky flew from school in Moscow several times, once because he contradicted his teacher's view that Dostoyevsky was a communist. He attended the Lomonossow University in Moscow with the intermediate diploma in 1989. Since he did not attend all courses out of boredom, he flew from the Lomonossow University and continued to study mathematics privately. Due to publications with Mikhail Mikhailovich Kapranov , he was admitted to Harvard University at the suggestion of Kapranov despite the lack of formal academic requirements and although he had not applied . Kapranow worked on higher category theory and both proved a connection between -gruppoids and homotopy types , which Alexander Grothendieck had suspected in his manuscript Esquisse d'un program from 1984 (Voevodsky only learned a little French for the purpose of understanding the text). Even at Harvard, he did not attend the prescribed courses, but this did not bother anyone because of his research performance and he received his doctorate in 1992 with the dissertation Homology of schemes and covariant motives , supervised by David Kazhdan . In 1992/1993 he was at the Institute for Advanced Study (IAS) in Princeton, New Jersey . From 1993 to 1996 he was a Junior Fellow and 1996/1997 visiting scholar at Harvard. He has been a member of the IAS since 1998, where he has been a professor since 2002. In 1996/97 he was a visiting scientist at the Max Planck Institute for Mathematics in Bonn and at the same time an associate professor at Northwestern University from 1996 to 1999 . From 2006 to 2008 he was a visiting scholar at Harvard University. He died of an aneurysm in his Princeton home .

He was a Sloan Research Fellow from 1996 to 1998 and a Clay Prize Fellow from 1999 to 2001. He has been an honorary professor at Wuhan University since 2004 and a member of the European Academy of Sciences since 2003 .

He was married to Natalia Shalaby and had two daughters. In addition to mathematics, he was also interested in biology, nature photography and politics.

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Wojewodski dealt with the interfaces between algebraic geometry and topology , initially dealing with the assumptions and ideas of Alexander Grothendieck from the 1980s. Together with Fabien Morel , he founded the homotopy theory of schemes . He is the author of the modern formulation of motivic cohomology and used this to prove the Milnor conjecture . For this work he and Laurent Lafforgue were awarded the Fields Medal in 2002 at the 24th International Congress of Mathematicians in Beijing . In 1998 he gave a plenary lecture at the International Congress of Mathematicians in Berlin (A 1 -Homotopy Theory) . In continuation of his proof of the Milnor conjecture, he also proved the Bloch-Kato conjecture with Markus Rost (via the Galoisohomological description of Milnor K groups, the Milnor conjecture is part of it).

The existence of motivic cohomology was suggested in a work by Alexander Beilinson , Robert MacPherson and Vadim Schechtman in 1987. Wojewodski worked on it with Kapranow and when he went to Cornell University in the early 1990s alone. In this area there were several attempts at proof that later proved to be flawed, including by Spencer Bloch in 1986. The area was therefore considered speculative and uncertain. The many errors and the complexity of the proofs later led Voevodsky to develop his own (topological) theory of automatic mathematical proofs. The path taken by Wojewodski with Eric Friedlander and Andrei Suslin bypassed Spencer Bloch's flawed lemma and was instead based on a work by Wojewodski Cohomological Theory of Presheaves with Transfers from 1992/93. The work also turned out to be flawed, as Pierre Deligne and Wojewodski discovered when Wojewodski gave lectures on it at the IAS in 1999/2000. The mistake was corrected by Wojewodski and a correct proof was published in 2006. The fact that Vojewodski's fundamental work had been available since 1993, but the error was not noticed until 2000, was a main motivation for Vojewodski to deal with computer-assisted evidence soon after. In his 1989 work on groupoids with Kapranow, a mistake was later found, which reinforced Wojewodski in his distrust of the system of mutual trust in published evidence from reputable mathematicians. In addition, he dealt with higher-dimensional category theory in the 2000s, which led to very technical and extensive evidence with a corresponding susceptibility to hard-to-detect errors. In his own words, he stopped his research, which was mainly driven by curiosity about structures that had not yet been discovered, and turned to the question of how the security of evidence could be improved with computers. According to Voevodsky, the area was frowned upon by pure mathematicians back then (early 2000s), few worked on it (like Thomas Hales and Carlos Simpson) and the evidence assistants existing at the time were unsuitable for the type of mathematical research Voivodsky had in mind. Wojewodski recognized that this required a new foundation of mathematics, which until then had been based on predicate logic and Zermelo-Fraenkel set theory on the one hand and category theory on the other. The realization that the category theory was also inadequate cost Wojewodski, in his own words, the greatest overcoming. The analogues of sets in higher dimensions were not categories, as he had previously thought, but groupoids.

The new univalent foundation of mathematics was based on two foundations. On the one hand, a deductive formal proof system based on the calculus of inductive constructions (CIC) by Thierry Coquand , the developer of the proof assistant Coq (from the late 1980s, it was still based on conventional type theory), and on the other hand, an interpretation of the theorems of the formal system the homotopy type theory (HoTT), a combination of homotopy theory to type theory . A third component, which, according to Wojewodski, is the least understood and the deepest, is the coding of mathematical questions into these homotopy types , where he goes back to his work with Kapranow on groupoids. Wojewodski has been working on this theory since 2005 and presented it publicly for the first time in November 2009 at a lecture at the Ludwig Maximilians University in Munich. From 2012/13 he organized a program for this at the IAS.

The new theory, which combines distant subjects such as topology and theory of programming languages ​​and mathematical logic, received a great deal of attention and Voevodsky became the central figure of a school of mathematicians working on this new foundation of mathematics. Among other things, she used Wojewodski for the development of programmable proof assistants for mathematicians to develop abstract mathematical theories and also applied them in his own research.

Around 2005 he also dealt with population genetics and, as a result, with a new category-theoretical approach to probability theory.

Fonts

  • with Andrei Suslin, Eric M. Friedlander: Cycles, transfers, and motivic homology theories . Annals of Mathematics Studies Vol. 143. Princeton University Press (2000).
  • Motivic Homotopy Theory in: Björn Dundas, Marc Levine u. a. (Ed.) Motivic homotopy theory , (Summer School Nordfjordeid, Norway, 2002), Springer 2006
  • -homotopy theory . Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998). Doc. Math. 1998, Extra Vol. I, 579-604
  • with Carlo Mazza, Charles Weibel Lectures on Motivic Cohomology , 1999/2000
  • with Suslin Bloch-Kato conjecture and motivic cohomology with finite coefficients
  • with Suslin: Singular homology of abstract algebraic varieties. Invent. Math. 123 (1996), no. 1, 61-94.
  • with Fabien Morel: -homotopy theory of schemes. Inst. Hautes Études Sci. Publ. Math. No. 90: 45-143 (2001) (1999).
  • Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic. Int. Math. Res. Not. 2002, no. 7, 351-355.
  • Reduced power operations in motivic cohomology. Publ. Math. Inst. Hautes Études Sci. No. 98: 1-57 (2003).
  • Motivic cohomology with Z / 2 coefficients. Publ. Math. Inst. Hautes Études Sci. No. 98: 59-104 (2003).
  • with D. Orlov, A. Vishik: An exact sequence for K * M / 2 with applications to quadratic forms. Ann. of Math. (2) 165 (2007), no. 1, 1-13.
  • Motivic Eilenberg-Maclane spaces . Publ. Math. Inst. Hautes Études Sci. No. 112 (2010): 1-99.
  • On motivic cohomology with Z / l coefficients. Ann. of Math. (2) 174 (2011), no. 1, 401-438.
  • Lectures on Motivic Cohomology , 1999/2000, Clay Monographs in Mathematics, AMS, Volume 2, 2006

literature

Web links

Individual evidence

  1. ↑ Obituary at the IAS
  2. ↑ Vladimir Voevodsky, June 4, 1966 - September 30, 2017 at golem.ph.utexas.edu, accessed October 2, 2017
  3. ↑ Julie Rehmeyer, Vladimir Voevodsky, Revolutionary Mathematician, Dies at 51 , New York Times obituary, Oct. 6, 2017
  4. ↑ Referred to as a Bachelor's degree in his English curriculum vitae. In contrast, the New York Times obituary stated that he had no formal undergraduate degree.
  5. ^ Voevodsky, The origins and motivations of univalent foundations , IAS 2014
  6. ↑ Carlos Simpson proved in 1998 a sentence that implied that there was a mistake in the work of Wojewodski and Kapranow, but could not explicitly show the mistake in the work of Kapranow and Wojewodski. Kapranow and Wojewodski, on the other hand, were convinced of the correctness of their work for a long time (until 2013) because they believed they had passed an argument similar to Simpson back then.
  7. ^ Voevodsky, The Origins and Motivations of Univalent Foundations, IAS 2014
  8. ↑ In 2013, the Institute for Advanced Study published the book Homotopy Type Theory: Univalent Foundations of Mathematics , Online
  9. ^ Bernay's Lecture by Voevodsky at ETH Zurich 2014
personal data
SURNAME Voivodsky, Vladimir
ALTERNATIVE NAMES Voivodsky, Vladimir Alexandrovich; Voevodsky, Vladimir; Воеводский, Владимир Александрович (Russian)
BRIEF DESCRIPTION American-Russian mathematician
DATE OF BIRTH 4th June 1966
PLACE OF BIRTH Moscow
DATE OF DEATH September 30, 2017
Place of death Princeton , New Jersey