# Mikhail Leonidovich Gromov

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**Mikhail Leonidovich Gromov** (also spelled **Mikhael Gromov** or **Michael Gromov**; Russian: Михаи́л Леони́дович Гро́мов; born 23 December 1943), is a French–Russian mathematician known for important contributions in many different areas of mathematics. He is considered a geometer in a very broad sense of the word. In 2009 he was awarded the Abel Prize "for his revolutionary contributions to geometry".

## Biography

Mikhail Gromov was born on 23 December 1943 in Boksitogorsk, Soviet Union. His father was Leonid Gromov and his mother was Lea Rabinovitz.^{}^{} Both his parents were pathologists.^{} Gromov was born during World War II, and his mother, who worked as a medical doctor in the Soviet Army, had to distance herself from the front line in order to give birth to him.^{} When Gromov was nine years old,^{} his mother gave him the book *Numbers and Figures*, by Hans Rademacher and Otto Toeplitz, a book that piqued his curiosity and had a great influence on him.^{}^{}

Gromov studied for a doctorate (1973) in Leningrad, where he was a student of Vladimir Rokhlin.^{} He is now a permanent member of IHÉS, and a Professor of Mathematics at New York University.

## Work

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

His impact has been felt most heavily in geometric group theory, where he characterized groups of polynomial growth and created, along with Eliyahu Rips, the notion of hyperbolic group; symplectic topology, where he introduced pseudoholomorphic curves, and in Riemannian geometry. His work, however, has delved deeply into analysis and algebra, where he will often formulate a problem in "geometric" terms. For example, his homotopy principle (h-principle) on is the basis for a geometric theory of partial differential equations.

Gromov is also interested in mathematical biology.^{}

## Prizes and honors

### Prizes

- (1971)
- Oswald Veblen Prize in Geometry (AMS) (1981)
- (1984)
- (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)
^{} - Bolyai Prize in 2005
- Abel Prize in 2009 “for his revolutionary contributions to geometry”
^{}

### Honors

- Invited speaker to International Congress of Mathematicians: 1970 (Nice), 1978 (Helsinki), 1982 (Warsaw), 1986 (Berkeley)
- Foreign member of the National Academy of Sciences, the American Academy of Arts and Sciences, the Norwegian Academy of Science and Letters, and the Royal Society (2011).
^{} - Membre de l'Institut de France – Académie des Sciences

## 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
- Gromov–Witten invariants
- Taubes's Gromov invariant
- Minimal volume
- 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.
^{} - 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
^{} - 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
^{} - 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
^{} - 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

## References

- 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