Mikhael Gromov (mathematician)
Mikhael Leonidovich Gromov  

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 
Mikhael Leonidovich Gromov (also Mikhail Gromov, Michael Gromov or Misha Gromov; Russian: Михаи́л Леони́дович Гро́мов; born 23 December 1943) is a RussianFrench 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[edit]
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 chessplayer 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[edit]
Gromov's style of geometry often features a "coarse" or "soft" viewpoint, analyzing asymptotic or largescale 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 C^{1} embedding theorem and Stephen Smale's early results,^{[12]} Gromov introduced in 1973 the method of convex integration and the hprinciple, 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 onetoone correspondence between the connected components of spaces.^{[13]}
In 1978, Gromov introduced the notion of almost flat manifolds.^{[G78]} The famous quarterpinched 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 ShingTung 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 hcobordism theorem could then be applied to draw conclusions such as the fact that every closed and simplyconnected 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 GromovHausdorff 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 "GromovHausdorff 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 wellchosen 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 MilnorWolf 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 curvature ≥ c and diameter ≤ D 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 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 nonsqueezing theorem.
Prizes and honors[edit]
Prizes[edit]
 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)^{[20]}
 Bolyai Prize in 2005
 Abel Prize in 2009 “for his revolutionary contributions to geometry”^{[21]}
Honors[edit]
 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)^{[22]}
 Member of the French Academy of Sciences (1997)^{[23]}
 Delivered the 2007 Paul Turán Memorial Lectures.^{[24]}
See also[edit]
 Gromov's compactness theorem (topology)
 Gromov's inequality for complex projective space
 Gromov's systolic inequality for essential manifolds
 Bishop–Gromov inequality
 Lévy–Gromov inequality
 Taubes's Gromov invariant
 Minimal volume
 Gromov norm
 Hyperbolic group
 Random group
 Ramsey–Dvoretzky–Milman phenomenon
 Systolic geometry
 Filling radius
 Gromov product
 Gromov δhyperbolic space
 Filling area conjecture
 Mean dimension
Publications[edit]
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 081763181X^{[25]}; doi:10.1007/9781468491593
 Misha Gromov. Metric structures for Riemannian and nonRiemannian 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 0817638989^{[26]}; doi:10.1007/9780817645830
 Mikhael Gromov. Partial differential relations. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 9. SpringerVerlag, Berlin, 1986. x+363 pp. ISBN 0387121773^{[27]}; doi:10.1007/9783662022672
 Misha Gromov. Great circle of mysteries. Mathematics, the world, the mind. Birkhäuser/Springer, Cham, 2018. vii+202 pp. ISBN 9783319530482, 9783319530499; doi:10.1007/9783319530499
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:10.1515/9781400881550016 
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, 1147. 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. L^{2}cohomology and group cohomology. Topology 25 (1986), no. 2, 189–215. doi:10.1016/00409383(86)90039X 
G87.  M. Gromov. Hyperbolic groups. Essays in group theory, 75–263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987. doi:10.1007/9781461395867 
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 L^{2}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 padic 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.^{[28]} 
G96.  Mikhael Gromov. Carnot–Carathéodory spaces seen from within. SubRiemannian geometry, 79–323, Progr. Math., 144, Birkhäuser, Basel, 1996. doi:10.1007/9783034892100_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:10.1007/9783034604222_5 
G03a.  M. Gromov. Isoperimetry of waists and concentration of maps. Geom. Funct. Anal. 13 (2003), no. 1, 178–215. doi:10.1007/s000390300004

G03b.  Mikhaïl Gromov. On the entropy of holomorphic maps. Enseign. Math. (2) 49 (2003), no. 34, 217–235. 
G03c.  M. Gromov. Random walk in random groups. Geom. Funct. Anal. 13 (2003), no. 1, 73–146. doi:10.1007/s000390300002 
Notes[edit]
 ^ 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 9781857430226.
 ^ ^{a} ^{b} O'Connor, John J.; Robertson, Edmund F., "Mikhael Gromov (mathematician)", 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 9783642394485. (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 19506244.
 ^ 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.
 ^ 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.
 ^ Peter Buser and Hermann Karcher. Gromov's almost flat manifolds. Astérisque, 81. Société Mathématique de France, Paris, 1981. 148 pp.
 ^ 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, BerlinNew York, 1981.
 ^ R. Schoen and S.T. Yau. On the structure of manifolds with positive scalar curvature. Manuscripta Math. 28 (1979), no. 13, 159–183.
 ^ Pierre Pansu. Métriques de CarnotCarathéodory et quasiisométries des espaces symétriques de rang un. Ann. of Math. (2) 129 (1989), no. 1, 1–60.
 ^ Bruce Kleiner and Bernhard Leeb. Rigidity of quasiisometries for symmetric spaces and Euclidean buildings. Inst. Hautes Études Sci. Publ. Math. No. 86 (1997), 115–197.
 ^ 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".
 ^ 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/s027309791987156035.
 ^ Grove, Karsten (2001). "Review: Metric structures for Riemannian and nonRiemannian spaces, by M. Gromov" (PDF). Bull. Amer. Math. Soc. (N.S.). 38 (3): 353–363. doi:10.1090/s0273097901009041.
 ^ McDuff, Dusa (1988). "Review: Partial differential relations, by Mikhael Gromov" (PDF). Bull. Amer. Math. Soc. (N.S.). 18 (2): 214–220. doi:10.1090/s027309791988156546.
 ^ 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/s0273097996006696.
References[edit]
 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[edit]
 Personal page at IHÉS
 Personal page at NYU
 Mikhail Gromov at the Mathematics Genealogy Project
 Anatoly Vershik, "Gromov's Geometry"
 1943 births
 Living people
 Jewish French scientists
 People from Boksitogorsk
 Russian people of Jewish descent
 Russian emigrants to France
 Foreign associates of the National Academy of Sciences
 Foreign Members of the Russian Academy of Sciences
 Kyoto laureates in Basic Sciences
 Differential geometers
 Russian mathematicians
 20thcentury French mathematicians
 21stcentury French mathematicians
 French people of RussianJewish descent
 Group theorists
 New York University faculty
 Wolf Prize in Mathematics laureates
 Geometers
 Members of the French Academy of Sciences
 Members of the Norwegian Academy of Science and Letters
 Abel Prize laureates
 Foreign Members of the Royal Society
 Highly Cited Researchers
 Soviet mathematicians