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# Mikhael Gromov (mathematician)

Mikhael Leonidovich Gromov (also Mikhail Gromov, Michael Gromov or Misha 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.

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

## 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] His mother was the cousin of World Chess Champion 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, he was invited to give a presentation at the International Congress of Mathematicians in Nice, France. However, 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]

## Work

Gromov's style of geometry often features a "coarse" or "soft" viewpoint, analyzing asymptotic or large-scale properties.[G00] He is also interested in mathematical biology,[12] the structure of the brain and the thinking process, and the way scientific ideas evolve.[9]

Motivated by Nash and Kuiper's C1 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 1978, Gromov introduced the notion of almost flat manifolds.[G78] The famous quarter-pinched sphere theorem in Riemannian geometry says that if a complete Riemannian manifold has sectional curvatures which are all sufficiently close to a given positive constant, then M must be finitely covered by a sphere. In contrast, it can be seen by scaling that every closed Riemannian manifold has Riemannian metrics whose sectional curvatures are arbitrarily close to zero. Gromov showed that if the scaling possibility is broken by only considering Riemannian manifolds of a fixed diameter, then a closed manifold admitting such a Riemannian metric with sectional curvatures sufficiently close to zero must be finitely covered by a nilmanifold. The proof works by replaying the proofs of the Bieberbach theorem and Margulis lemma. Gromov's proof was given a careful exposition by Peter Buser and Hermann Karcher.[14][15][16]

In 1979, Richard Schoen and Shing-Tung Yau showed that the class of smooth manifolds which admit Riemannian metrics of positive scalar curvature is topologically rich. In particular, they showed that this class is closed under the operation of connected sum and of surgery in codimension at least three.[17] Their proof used elementary methods of partial differential equations, in particular to do with the Green's function. Gromov and Blaine Lawson gave another proof of Schoen and Yau's results, making use of elementary geometric constructions.[GL80] They also showed how purely topological results such as Stephen Smale's h-cobordism theorem could then be applied to draw conclusions such as the fact that every closed and simply-connected smooth manifold of dimension 5, 6, or 7 has a Riemannian metric of positive scalar curvature.

In 1981, Gromov formally introduced the Gromov–Hausdorff metric, which endows the set of all metric spaces with the structure of a metric space.[G81b] More generally, one can define the Gromov-Hausdorff distance between two metric spaces, relative to the choice of a point in each space. Although this does not give a metric on the space of all metric spaces, it is sufficient in order to define "Gromov-Hausdorff convergence" of a sequence of pointed metric spaces to a limit. Gromov formulated an important compactness theorem in this setting, giving a condition under which a sequence of pointed and "proper" metric spaces must have a subsequence which converges. This was later reformulated by Gromov and others into the more flexible notion of an ultralimit.[G93]

Gromov's compactness theorem had a deep impact on the field of geometric group theory. He applied it to understand the asymptotic geometry of the word metric of a group of polynomial growth, by taking the limit of well-chosen rescalings of the metric. By tracking the limits of isometries of the word metric, he was able to show that the limiting metric space has unexpected continuities, and in particular that its isometry group is a Lie group.[G81b] As a consequence he was able to settle the Milnor-Wolf conjecture as posed in the 1960s, which asserts that any such group is virtually nilpotent. Using ultralimits, similar asymptotic structures can be studied for more general metric spaces.[G93] Important developments on this topic were given by Bruce Kleiner, Bernhard Leeb, and Pierre Pansu, among others.[18][19]

Another consequence is Gromov's compactness theorem, stating that the set of compact Riemannian manifolds with Ricci curvaturec and diameterD is relatively compact in the Gromov–Hausdorff metric.[G81b] 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.[BGP92]

Along with Eliyahu Rips, Gromov introduced the notion of hyperbolic groups.[G87]

Gromov made foundational contributions to systolic geometry. Systolic geometry studies the relationship between size invariants (such as volume or diameter) of a manifold M and its topologically non-trivial submanifolds (such as non-contractible curves). In his 1983 paper "Filling Riemannian manifolds"[G83] Gromov proved that every essential manifold M with a Riemannian metric contains a closed non-contractible geodesic of length at most ${\displaystyle C(n)Vol(M)^{1/n}}$.[20]

Gromov founded the field of symplectic topology by introducing the theory of pseudoholomorphic curves.[G85] This led to Gromov–Witten invariants, which are used in string theory, and to his non-squeezing theorem.

## Publications

Books

• Werner Ballmann, Mikhael Gromov, and Viktor Schroeder. Manifolds of nonpositive curvature. Progress in Mathematics, 61. Birkhäuser Boston, Inc., Boston, MA, 1985. vi+263 pp. ISBN 0-8176-3181-X;[26] doi:10.1007/978-1-4684-9159-3
• Misha Gromov. 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;[27] doi:10.1007/978-0-8176-4583-0
• Mikhael Gromov. Partial differential relations. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 9. Springer-Verlag, Berlin, 1986. x+363 pp. ISBN 0-387-12177-3;[28] doi:10.1007/978-3-662-02267-2
• Misha Gromov. Great circle of mysteries. Mathematics, the world, the mind. Birkhäuser/Springer, Cham, 2018. vii+202 pp. ISBN 978-3-319-53048-2, 978-3-319-53049-9; doi:10.1007/978-3-319-53049-9

Major articles

 G78. M. Gromov. Almost flat manifolds. J. Differential Geom. 13 (1978), no. 2, 231–241. doi:10.4310/jdg/1214434488
 GL80. Mikhael Gromov and H. Blaine Lawson Jr. The classification of simply connected manifolds of positive scalar curvature. Ann. of Math. (2) 111 (1980), no. 3, 423–434. doi:10.2307/1971103
 G81a. Michael Gromov. Curvature, diameter and Betti numbers. Comment. Math. Helv. 56 (1981), no. 2, 179–195. doi:10.1007/BF02566208
 G81b. Mikhael Gromov. Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. doi:10.1007/BF02698687
 G81c. M. Gromov. 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. doi:10.1515/9781400881550-016
 CGT82. Jeff Cheeger, Mikhail Gromov, and Michael Taylor. Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differential Geom. 17 (1982), no. 1, 15–53. doi:10.4310/jdg/1214436699
 G82. Michael Gromov. Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99.
 G83. Mikhael Gromov. Filling Riemannian manifolds. J. Differential Geom. 18 (1983), no. 1, 1-147. doi:10.4310/jdg/1214509283
 GL83. Mikhael Gromov and H. Blaine Lawson Jr. Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Inst. Hautes Études Sci. Publ. Math. 58 (1983), 83–196. doi:10.1007/BF02953774
 GM83. M. Gromov and V.D. Milman. A topological application of the isoperimetric inequality. Amer. J. Math. 105 (1983), no. 4, 843–854. doi:10.2307/2374298
 G85. M. Gromov. Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82 (1985), no. 2, 307–347. doi:10.1007/BF01388806
 CG86a. Jeff Cheeger and Mikhael Gromov. Collapsing Riemannian manifolds while keeping their curvature bounded. I. J. Differential Geom. 23 (1986), no. 3, 309–346. doi:10.4310/jdg/1214440117
 CG86b. Jeff Cheeger and Mikhael Gromov. L2-cohomology and group cohomology. Topology 25 (1986), no. 2, 189–215. doi:10.1016/0040-9383(86)90039-X
 G87. M. Gromov. Hyperbolic groups. Essays in group theory, 75–263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987. doi:10.1007/978-1-4613-9586-7
 EG91. Yakov Eliashberg and Mikhael Gromov. Convex symplectic manifolds. Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), 135–162, Proc. Sympos. Pure Math., 52, Part 2, Amer. Math. Soc., Providence, RI, 1991. doi:10.1090/pspum/052.2
 G91. M. Gromov. Kähler hyperbolicity and L2-Hodge theory. J. Differential Geom. 33 (1991), no. 1, 263–292. doi:10.4310/jdg/1214446039
 BGP92. Yu. Burago, M. Gromov, and G. Perelʹman. A.D. Aleksandrov spaces with curvatures bounded below. Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3–51, 222; English translation in Russian Math. Surveys 47 (1992), no. 2, 1–58. doi:10.1070/rm1992v047n02abeh000877
 GS92. Mikhail Gromov and Richard Schoen. Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one. Inst. Hautes Études Sci. Publ. Math. 76 (1992), 165–246. doi:10.1007/bf02699433
 G93. M. Gromov. 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.[29]
 G96. Mikhael Gromov. Carnot–Carathéodory spaces seen from within. Sub-Riemannian geometry, 79–323, Progr. Math., 144, Birkhäuser, Basel, 1996. doi:10.1007/978-3-0348-9210-0_2
 G99. M. Gromov. Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. 1 (1999), no. 2, 109–197. doi:10.1007/pl00011162
 G00. Misha Gromov. Spaces and questions. GAFA 2000 (Tel Aviv, 1999). Geom. Funct. Anal. 2000, Special Volume, Part I, 118–161. doi:10.1007/978-3-0346-0422-2_5
 G03a. M. Gromov. Isoperimetry of waists and concentration of maps. Geom. Funct. Anal. 13 (2003), no. 1, 178–215. doi:10.1007/s000390300004 Mikhail Gromov. Erratum to: "Isoperimetry of waists and concentration of maps [GAFA 13:1 (2003), pp. 178–215]." Geom. Funct. Anal. 18 (2009), no. 5, 1786. doi:10.1007/s00039-009-0703-1
 G03b. Mikhaïl Gromov. On the entropy of holomorphic maps. Enseign. Math. (2) 49 (2003), no. 3-4, 217–235.
 G03c. M. Gromov. Random walk in random groups. Geom. Funct. Anal. 13 (2003), no. 1, 73–146. doi:10.1007/s000390300002

## 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. ^ a b
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. ^ 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).
6. ^ a b Newsletter of the European Mathematical Society, No. 73, September 2009, p. 19
7. ^ a b c Foucart, Stéphane (26 March 2009). "Mikhaïl Gromov, le génie qui venait du froid". Le Monde.fr (in French). ISSN 1950-6244.
9. ^ a b c Roberts, Siobhan (22 December 2014). "Science Lives: Mikhail Gromov". Simons Foundation.
10. ^ Ripka, Georges (1 January 2002). Vivre savant sous le communisme (in French). Belin. ISBN 9782701130538.
11. ^ "Mikhail Leonidovich Gromov". abelprize.no.
12. ^ a b "Interview with Mikhail Gromov" (PDF), Notices of the AMS, 57 (3): 391–403, March 2010.
13. ^ Arnold, V. I.; Goryunov, V. V.; Lyashko, O. V.; Vasil'Ev, V. A. (6 December 2012). Singularity Theory I. ISBN 9783642580093.
14. ^ Hermann Karcher. Report on M. Gromov's almost flat manifolds. Séminaire Bourbaki (1978/79), Exp. No. 526, pp. 21–35, Lecture Notes in Math., 770, Springer, Berlin, 1980.
15. ^ Peter Buser and Hermann Karcher. Gromov's almost flat manifolds. Astérisque, 81. Société Mathématique de France, Paris, 1981. 148 pp.
16. ^ Peter Buser and Hermann Karcher. The Bieberbach case in Gromov's almost flat manifold theorem. Global differential geometry and global analysis (Berlin, 1979), pp. 82–93, Lecture Notes in Math., 838, Springer, Berlin-New York, 1981.
17. ^ R. Schoen and S.T. Yau. On the structure of manifolds with positive scalar curvature. Manuscripta Math. 28 (1979), no. 1-3, 159–183.
18. ^ Pierre Pansu. Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. of Math. (2) 129 (1989), no. 1, 1–60.
19. ^ Bruce Kleiner and Bernhard Leeb. Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings. Inst. Hautes Études Sci. Publ. Math. No. 86 (1997), 115–197.
20. ^ Katz, M. Systolic geometry and topology. With an appendix by J. Solomon. Mathematical Surveys and Monographs, volume 137. American Mathematical Society, 2007.
21. ^ Gromov Receives Nemmers Prize
22. ^ Abel Prize for 2009, Laureates 2009
23. ^ Professor Mikhail Gromov ForMemRS | Royal Society
24. ^ Mikhaël Gromov — Membre de l’Académie des sciences
25. ^
26. ^ 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.
27. ^ 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.
28. ^ 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.
29. ^ 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.

## References

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