Mikhail Leonidovich Gromov

Mikhail Leonidovich Gromov
Mikhail Gromov
Born	23 December 1943 Boksitogorsk, Russian SFSR, Soviet Union
Residence	France
Nationality	Russian and French
Fields	Mathematics
Institutions	Institut des Hautes Études Scientifiques New York University
Alma mater	Leningrad State University (PhD)
Doctoral advisor	Vladimir Rokhlin
Doctoral students	Denis Auroux Christophe Bavard François Labourie Yashar Memarian Pierre Pansu Abdelghani Zeghib
Known for	Geometry
Notable awards	Oswald Veblen Prize in Geometry (1981) Wolf Prize (1993) Kyoto Prize (2002) Nemmers Prize in Mathematics (2004) Bolyai Prize (2005) Abel Prize (2009)

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.^[1][2] Both his parents were pathologists.^[3] 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.^[4] When Gromov was nine years old,^[5] 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.^[4][6]

Gromov studied for a doctorate (1973) in Leningrad, where he was a student of Vladimir Rokhlin.^[7] 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 differential relations is the basis for a geometric theory of partial differential equations.

Gromov is also interested in mathematical biology.^[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)^[9]
• Bolyai Prize in 2005
• Abel Prize in 2009 “for his revolutionary contributions to geometry”^[10]

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).^[11]
• Member of the French Academy of 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
• 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.^[12]
• 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^[13]
• 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^[14]
• 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^[15]
• 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. The International Who's Who, 1997–98. Europa Publications. 1997. p. 591. ISBN 978-1-85743-022-6.
2. O'Connor, John J.; Robertson, Edmund F., "Mikhail Leonidovich Gromov", MacTutor History of Mathematics archive, University of St Andrews.
3. 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)
4. Newsletter of the European Mathematical Society, No. 73, September 2009, p. 19
5. Le Monde – Mikhaïl Gromov, le génie qui venait du froid (in French)
6. Note: This book was translated to English as The Enjoyment of Mathematics, the original title in German is Von Zahlen und Figuren.
7. http://cims.nyu.edu/newsletters/Spring2009.pdf
8. "Interview with Mikhail Gromov" (PDF), Notices of the AMS 57 (3), March 2010: 391–403.
9. • Gromov Receives Nemmers Prize
10. Abel Prize for 2009, Laureates 2009
11. • Professor Mikhail Gromov ForMemRS | Royal Society
12. 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.
13. 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.
14. 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.
15. 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

• 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"