# Mikhail Leonidovich Gromov

**Mikhail Leonidovich Gromov** (also **Mikhael Gromov**, **Michael Gromov** or **Mischa Gromov**; Russian: Михаи́л Леони́дович Гро́мов; born 23 December 1943) is a Russian-French mathematician known for his 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.

Mikhail Leonidovich Gromov | |
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Mikhail Gromov in 2009 | |

Born | |

Nationality | Russian and French |

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 |

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

## BiographyEdit

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]} His mother was the cousin of chess-player Mikhail Botvinnik, as well as of the mathematician Isaak Moiseevich Rabinovich. ^{[5]} 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.^{[6]} When Gromov was nine years old,^{[7]} 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.^{[6]}

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.^{[8]}

Gromov married in 1967. In 1970, invited to give a presentation at the International Congress of Mathematicians in Nice, France, he was not allowed to leave the USSR. Still, his lecture was published in the conference proceedings.^{[9]}

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.^{[7]}^{[10]} He changed his last name to that of his mother.^{[7]} When the request was granted in 1974, he moved directly to New York where a position had been arranged for him at Stony Brook.^{[9]}

In 1981 he left Stony Brook University 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 in New York since 1996.^{[3]} He adopted French citizenship in 1992.^{[11]}

## WorkEdit

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

Motivated by Nash and Kuiper's C^{1} embedding theorem and Stephen Smale's early results,^{[12]} Gromov introduced in 1973 the method 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. One application is the Gromov–Lees Theorem, named for him and Jack Alexander Lees, concerning Lagrangian immersions and a one-to-one correspondence between the connected components of spaces.^{[13]}

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,^{[12]} the structure of the brain and the thinking process, and the way scientific ideas evolve.^{[9]}

## Prizes and honorsEdit

### PrizesEdit

- 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)
^{[14]} - Bolyai Prize in 2005
- Abel Prize in 2009 “for his revolutionary contributions to geometry”
^{[15]}

### HonorsEdit

- 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)
^{[16]} - Member of the French Academy of Sciences (1997)
^{[17]} - Delivered the 2007 Paul Turán Memorial Lectures.
^{[18]}

## See alsoEdit

- 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 publicationsEdit

- 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.
^{[19]} - 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. ISBN 0-8176-3898-9
^{[20]} - 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
^{[21]} - 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
^{[22]} - 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).

## NotesEdit

**^**Masha Gessen (2011).*Perfect Rigour: A Genius and the Mathematical Breakthrough of a Lifetime*. Icon Books Ltd.**^***The International Who's Who, 1997–98*. Europa Publications. 1997. p. 591. ISBN 978-1-85743-022-6.- ^
^{a}^{b}O'Connor, John J.; Robertson, Edmund F., "Mikhail Leonidovich Gromov",*MacTutor History of Mathematics archive*, University of St Andrews. **^**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)**^**Воспоминания Владимира Рабиновича (генеалогия семьи М. Громова по материнской линии. Лия Александровна Рабинович также приходится двоюродной сестрой известному рижскому математику, историку математики и популяризатору науки Исааку Моисеевичу Рабиновичу (род. 1911), автору книг «Математик Пирс Боль из Риги» (совместно с А. Д. Мышкисом и с приложением комментария М. М. Ботвинника «О шахматной игре П. Г. Боля», 1965), «Строптивая производная» (1968) и др. Троюродный брат М. Громова — известный латвийский адвокат и общественный деятель Александр Жанович Бергман (польск., род. 1925).- ^
^{a}^{b}Newsletter of the European Mathematical Society, No. 73, September 2009, p. 19 - ^
^{a}^{b}^{c}Foucart, Stéphane (26 March 2009). "Mikhaïl Gromov, le génie qui venait du froid".*Le Monde.fr*(in French). ISSN 1950-6244. **^**http://cims.nyu.edu/newsletters/Spring2009.pdf- ^
^{a}^{b}^{c}Roberts, Siobhan (22 December 2014). "Science Lives: Mikhail Gromov". Simons Foundation. **^**Ripka, Georges (1 January 2002).*Vivre savant sous le communisme*(in French). Belin. ISBN 9782701130538.**^**"Mikhail Leonidovich Gromov".*abelprize.no*.- ^
^{a}^{b}"Interview with Mikhail Gromov" (PDF),*Notices of the AMS*,**57**(3): 391–403, March 2010. **^**Arnold, V. I.; Goryunov, V. V.; Lyashko, O. V.; Vasil'Ev, V. A. (6 December 2012).*Singularity Theory I*. ISBN 9783642580093.**^**Gromov Receives Nemmers Prize**^**Abel Prize for 2009, Laureates 2009**^**Professor Mikhail Gromov ForMemRS | Royal Society**^**Mikhaël Gromov — Membre de l’Académie des sciences**^**"Turán Memorial Lectures".**^**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.**^**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.**^**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.**^**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.

## ReferencesEdit

- 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 linksEdit

Wikimedia Commons has media related to .Mikhail Leonidovich Gromov |

- Personal page at IHÉS
- Personal page at NYU
- Mikhail Gromov at the Mathematics Genealogy Project
- Anatoly Vershik, "Gromov's Geometry"