Mikhail Leonidovich Gromov




For other people with the same name, see Gromov.














































Mikhail Leonidovich Gromov

Gromov Mikhail Leonidovich.jpg
Mikhail Gromov in 2009

Born
(1943-12-23) 23 December 1943 (age 75)

Boksitogorsk, Russian SFSR, Soviet Union
Residence France
Nationality Russian and French and American
Alma mater
Leningrad State University (PhD)
Known for Geometry
Awards Oswald Veblen Prize in Geometry (1981)
Wolf Prize (1993)
Kyoto Prize (2002)
Nemmers Prize in Mathematics (2004)
Bolyai Prize (2005)
Abel Prize (2009)
Scientific career
Fields Mathematics
Institutions
Institut des Hautes Études Scientifiques
New York University
Doctoral advisor Vladimir Rokhlin
Doctoral students
François Labourie
Pierre Pansu
Mikhail Katz


Mikhail Leonidovich Gromov (also Mikhael Gromov, Michael Gromov or Mischa Gromov; Russian: Михаи́л Леони́дович Гро́мов; born 23 December 1943), is an American-French-Russian mathematician known for work in geometry, analysis and group theory. He is a permanent member of IHÉS in France and a Professor of Mathematics at New York University.


Gromov has won several prizes, including the Abel Prize in 2009 "for his revolutionary contributions to geometry".




Contents






  • 1 Biography


  • 2 Work


  • 3 Prizes and honors


    • 3.1 Prizes


    • 3.2 Honors




  • 4 See also


  • 5 Books and other publications


  • 6 Notes


  • 7 References


  • 8 External links





Biography


Mikhail Gromov was born on 23 December 1943 in Boksitogorsk, Soviet Union. His father Leonid Gromov and his Jewish[1] mother Lea Rabinovitz[2][3] were pathologists.[4] Gromov was born during World War II, and his mother, who worked as a medical doctor in the Soviet Army, had to leave the front line in order to give birth to him.[5] When Gromov was nine years old,[6] his mother gave him the book The Enjoyment of Mathematics by Hans Rademacher and Otto Toeplitz, a book that piqued his curiosity and had a great influence on him.[5]


Gromov studied mathematics at Leningrad State University where he obtained a master's degree in 1965, a Doctorate in 1969 and defended his Postdoctoral Thesis in 1973. His thesis advisor was Vladimir Rokhlin.[7]


Gromov married in 1967. In 1970, invited to give a presentation at the International Congress of Mathematicians in France, he was not allowed to leave the USSR. Still, his lecture was published in the conference proceedings.[8]


Disagreeing with the Soviet system, he had been thinking of emigrating since the age of 14. In the early 1970s he ceased publication, hoping that this would help his application to move to Israel.[6][9] He changed his last name to that of his mother.[6] When the request was granted in 1974, he moved directly to New York where a position had been arranged for him at Stony Brook.[8]


In 1981 he left Stony Brook to join the faculty of University of Paris VI and in 1982 he became a permanent professor at the Institut des Hautes Études Scientifiques (IHES) where he remains today. At the same time, he has held professorships at the University of Maryland, College Park from 1991 to 1996, and at the Courant Institute of Mathematical Sciences since 1996.[3] He adopted French citizenship in 1992.[10]



Work


Gromov's style of geometry often features a "coarse" or "soft" viewpoint, analyzing asymptotic or large-scale properties.


Motivated by Nash and Kuiper's C1 embedding theorem and Stephen Smale's early results,[11] Gromov introduced in 1973 the notion of convex integration and the h-principle, a very general way to solve underdetermined partial differential equations and the basis for a geometric theory of these equations.


In the 1980s, Gromov introduced the Gromov–Hausdorff metric, a measure of the difference between two compact metric spaces. In this context he proved Gromov's compactness theorem, stating that the set of compact Riemannian manifolds with Ricci curvature ≥ c and diameter ≤ D is relatively compact in the Gromov–Hausdorff metric. The possible limit points of sequences of such manifolds are Alexandrov spaces of curvature ≥ c, a class of metric spaces studied in detail by Burago, Gromov and Perelman in 1992. Gromov was also the first to study the space of all possible Riemannian structures on a given manifold.


Gromov introduced geometric group theory, the study of infinite groups via the geometry of their Cayley graphs and their word metric. In 1981 he proved Gromov's theorem on groups of polynomial growth: a finitely generated group has polynomial growth (a geometric property) if and only if it is virtually nilpotent (an algebraic property). The proof uses the Gromov–Hausdorff metric mentioned above. Along with Eliyahu Rips he introduced the notion of hyperbolic groups.


Gromov founded the field of symplectic topology by introducing the theory of pseudoholomorphic curves. This led to Gromov–Witten invariants which are used in string theory and to his non-squeezing theorem.


Gromov is also interested in mathematical biology,[11] the structure of the brain and the thinking process, and the way scientific ideas evolve.[8]



Prizes and honors



Prizes




  • Prize of the Mathematical Society of Moscow (1971)


  • Oswald Veblen Prize in Geometry (AMS) (1981)


  • Prix Elie Cartan de l'Academie des Sciences de Paris (1984)


  • Prix de l'Union des Assurances de Paris (1989)


  • Wolf Prize in Mathematics (1993)


  • Leroy P. Steele Prize for Seminal Contribution to Research (AMS) (1997)


  • Lobachevsky Medal (1997)


  • Balzan Prize for Mathematics (1999)


  • Kyoto Prize in Mathematical Sciences (2002)


  • Nemmers Prize in Mathematics (2004)[12]


  • Bolyai Prize in 2005


  • Abel Prize in 2009 “for his revolutionary contributions to geometry”[13]



Honors



  • Invited speaker to International Congress of Mathematicians: 1970 (Nice), 1978 (Helsinki), 1982 (Warsaw), 1986 (Berkeley)

  • Foreign member of the National Academy of Sciences ((1989), the American Academy of Arts and Sciences ((1989), the Norwegian Academy of Science and Letters, and the Royal Society (2011)[14]

  • Member of the French Academy of Sciences (1997)[15]



See also




  • Gromov's theorem on groups of polynomial growth

  • Gromov's theorem on almost flat manifolds

  • Gromov's compactness theorem (geometry)

  • Gromov's compactness theorem (topology)

  • Gromov's inequality for complex projective space

  • Gromov's systolic inequality for essential manifolds

  • Gromov–Hausdorff convergence

  • Bishop–Gromov inequality

  • Lévy–Gromov inequality

  • Gromov–Witten invariants

  • Taubes's Gromov invariant

  • Minimal volume

  • Localisation on the sphere

  • Gromov norm

  • Hyperbolic group

  • Random group

  • Ramsey–Dvoretzky–Milman phenomenon

  • Systolic geometry

  • Filling radius

  • Gromov product

  • Gromov δ-hyperbolic space

  • Filling area conjecture

  • Metric Structures for Riemannian and Non-Riemannian Spaces

  • Mean dimension




Books and other publications



  • Gromov, M. Hyperbolic manifolds, groups and actions. Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), pp. 183–213, Ann. of Math. Stud., 97, Princeton Univ. Press, Princeton, N.J., 1981.

  • Gromov, M. Hyperbolic groups. Essays in group theory, 75–263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987.

  • Gromov, M. Asymptotic invariants of infinite groups. Geometric group theory, Vol. 2 (Sussex, 1991), 1–295, London Math. Soc. Lecture Note Ser., 182, Cambridge Univ. Press, Cambridge, 1993.[16]

  • Gromov, Misha: Metric structures for Riemannian and non-Riemannian spaces. Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, 152. Birkhäuser Boston, Inc., Boston, MA, 1999. xx+585 pp. .mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
    ISBN 0-8176-3898-9[17]

  • Gromov, M. Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82 (1985), no. 2, 307–347.

  • Gromov, Mikhael Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. No. 53 (1981), 53–73.

  • Gromov, Mikhael Structures métriques pour les variétés riemanniennes. (French) [Metric structures for Riemann manifolds] Edited by J. Lafontaine and P. Pansu. Textes Mathématiques [Mathematical Texts], 1. CEDIC, Paris, 1981. iv+152 pp.
    ISBN 2-7124-0714-8

  • Gromov, Mikhael: Partial differential relations. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 9. Springer-Verlag, Berlin, 1986. x+363 pp.
    ISBN 0-387-12177-3[18]


  • Ballmann, Werner; Gromov, Mikhael; Schroeder, Viktor: Manifolds of nonpositive curvature. Progress in Mathematics, 61. Birkhäuser Boston, Inc., Boston, MA, 1985. vi+263 pp.
    ISBN 0-8176-3181-X[19]

  • Gromov, Mikhael: Carnot–Carathéodory spaces seen from within. Sub-Riemannian geometry, 79–323, Progr. Math., 144, Birkhäuser, Basel, 1996.

  • Gromov, Michael: Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. No. 56 (1982), 5–99 (1983).



Notes





  1. ^ Masha Gessen (2011). Perfect Rigour: A Genius and the Mathematical Breakthrough of a Lifetime. Icon Books Ltd.


  2. ^ The International Who's Who, 1997–98. Europa Publications. 1997. p. 591. ISBN 978-1-85743-022-6.


  3. ^ ab O'Connor, John J.; Robertson, Edmund F., "Mikhail Leonidovich Gromov", MacTutor History of Mathematics archive, University of St Andrews.


  4. ^ Gromov, Mikhail. "A Few Recollections", in Helge Holden; Ragni Piene (3 February 2014). The Abel Prize 2008–2012. Springer Berlin Heidelberg. pp. 129–137. ISBN 978-3-642-39448-5. (also available on Gromov's homepage: link)


  5. ^ ab Newsletter of the European Mathematical Society, No. 73, September 2009, p. 19


  6. ^ abc Foucart, Stéphane (2009-03-26). "Mikhaïl Gromov, le génie qui venait du froid". Le Monde.fr (in French). ISSN 1950-6244.


  7. ^ http://cims.nyu.edu/newsletters/Spring2009.pdf


  8. ^ abc Roberts, Siobhan (December 22, 2014). "Science Lives: Mikhail Gromov". Simons Foundation.


  9. ^ Ripka, Georges (2002-01-01). Vivre savant sous le communisme (in French). Belin. ISBN 9782701130538.


  10. ^ "Mikhail Leonidovich Gromov". abelprize.no.


  11. ^ ab "Interview with Mikhail Gromov" (PDF), Notices of the AMS, 57 (3): 391–403, March 2010.


  12. ^ Gromov Receives Nemmers Prize


  13. ^ Abel Prize for 2009, Laureates 2009


  14. ^ Professor Mikhail Gromov ForMemRS | Royal Society


  15. ^ Mikhaël Gromov — Membre de l’Académie des sciences


  16. ^ Toledo, Domingo (1996). "Review: Geometric group theory, Vol. 2: Asymptotic invariants of infinite groups, by M. Gromov" (PDF). Bull. Amer. Math. Soc. (N.S.). 33 (3): 395–398. doi:10.1090/s0273-0979-96-00669-6.


  17. ^ Grove, Karsten (2001). "Review: Metric structures for Riemannian and non-Riemannian spaces, by M. Gromov" (PDF). Bull. Amer. Math. Soc. (N.S.). 38 (3): 353–363. doi:10.1090/s0273-0979-01-00904-1.


  18. ^ McDuff, Dusa (1988). "Review: Partial differential relations, by Mikhael Gromov" (PDF). Bull. Amer. Math. Soc. (N.S.). 18 (2): 214–220. doi:10.1090/s0273-0979-1988-15654-6.


  19. ^ Heintze, Ernst (1987). "Review: Manifolds of nonpositive curvature, by W. Ballmann, M. Gromov & V. Schroeder" (PDF). Bull. Amer. Math. Soc. (N.S.). 17 (2): 376–380. doi:10.1090/s0273-0979-1987-15603-5.




References


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  • Marcel Berger, Encounter with a Geometer, Part I, AMS Notices, Volume 47, Number 2

  • Marcel Berger, Encounter with a Geometer, Part II, AMS Notices, Volume 47, Number 3




External links







  • Personal page at IHÉS

  • Personal page at NYU


  • Mikhail Gromov at the Mathematics Genealogy Project

  • Anatoly Vershik, "Gromov's Geometry"















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