Mikhael Leonidovich Gromov (alsoMikhail Gromov,Michael Gromov orMisha Gromov; Russian:Михаи́л Леони́дович Гро́мов; born 23 December 1943) is a Russian-French mathematician known for his work ingeometry,analysis andgroup theory. He is a permanent member ofInstitut des Hautes Études Scientifiques in France and a professor of mathematics atNew York University.
Gromov has won several prizes, including theAbel Prize in 2009 "for his revolutionary contributions to geometry".
Mikhail Gromov was born on 23 December 1943 inBoksitogorsk,Soviet Union. His father Leonid Gromov wasRussian-Slavic and his mother Lea was ofJewish heritage. Both werepathologists.[1] His mother was the cousin of World Chess ChampionMikhail Botvinnik, as well as of the mathematician Isaak Moiseevich Rabinovich.[2] Gromov was born duringWorld 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.[3] When Gromov was nine years old,[4] his mother gave him the bookThe Enjoyment of Mathematics byHans Rademacher andOtto Toeplitz, a book that piqued his curiosity and had a great influence on him.[3]
Gromov studied mathematics atLeningrad 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 wasVladimir Rokhlin.[5]
Gromov married in 1967. In 1970, he was invited to give a presentation at theInternational Congress of Mathematicians inNice, France. However, he was not allowed to leave the USSR. Still, his lecture was published in the conference proceedings.[6]
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 tomove to Israel.[4][7] He changed his last name to that of his mother.[4] He received a coded letter saying that, if he could get out of the Soviet Union, he could go toStony Brook, where a position had been arranged for him. When the request was granted in 1974, he moved directly to New York and worked at Stony Brook.[6]
In 1981 he leftStony Brook University to join the faculty ofUniversity of Paris VI and in 1982 he became a permanent professor at theInstitut des Hautes Études Scientifiques where he remains today. At the same time, he has held professorships at theUniversity of Maryland, College Park from 1991 to 1996, and at theCourant Institute of Mathematical Sciences in New York since 1996.[8] He adopted French citizenship in 1992.[9]
Gromov's style of geometry often features a "coarse" or "soft" viewpoint, analyzing asymptotic or large-scale properties.[G00] He is also interested inmathematical biology,[10] the structure of the brain and the thinking process, and the way scientific ideas evolve.[6]
Motivated byNash and Kuiper's isometric embedding theorems and the results onimmersions byMorris Hirsch andStephen Smale,[10] Gromov introduced theh-principle in various formulations. Modeled upon the special case of the Hirsch–Smale theory, he introduced and developed the general theory ofmicroflexible sheaves, proving that they satisfy an h-principle onopen manifolds.[G69] As a consequence (among other results) he was able to establish the existence of positively curved and negatively curvedRiemannian metrics on anyopen manifold whatsoever. His result is in counterpoint to the well-known topological restrictions (such as theCheeger–Gromoll soul theorem orCartan–Hadamard theorem) ongeodesically complete Riemannian manifolds of positive or negative curvature. After this initial work, he developed further h-principles partly in collaboration withYakov Eliashberg, including work building upon Nash and Kuiper's theorem and theNash–Moser implicit function theorem. There are many applications of his results, including topological conditions for the existence ofexact Lagrangian immersions and similar objects insymplectic andcontact geometry.[11][12] His well-known bookPartial Differential Relations collects most of his work on these problems.[G86] Later, he applied his methods tocomplex geometry, proving certain instances of theOka principle on deformation ofcontinuous maps toholomorphic maps.[G89] His work initiated a renewed study of the Oka–Grauert theory, which had been introduced in the 1950s.[13][14]
Gromov andVitali Milman gave a formulation of theconcentration of measure phenomena.[GM83] They defined a "Lévy family" as a sequence of normalized metric measure spaces in which any asymptotically nonvanishing sequence of sets can be metrically thickened to include almost every point. This closely mimics the phenomena of thelaw of large numbers, and in fact the law of large numbers can be put into the framework of Lévy families. Gromov and Milman developed the basic theory of Lévy families and identified a number of examples, most importantly coming from sequences ofRiemannian manifolds in which the lower bound of theRicci curvature or the first eigenvalue of theLaplace–Beltrami operator diverge to infinity. They also highlighted a feature of Lévy families in which any sequence of continuous functions must be asymptotically almost constant. These considerations have been taken further by other authors, such asMichel Talagrand.[15]
Since the seminal 1964 publication ofJames Eells andJoseph Sampson onharmonic maps, various rigidity phenomena had been deduced from the combination of an existence theorem for harmonic mappings together with a vanishing theorem asserting that (certain) harmonic mappings must be totally geodesic or holomorphic.[16][17][18] Gromov had the insight that the extension of this program to the setting of mappings intometric spaces would imply new results ondiscrete groups, followingMargulis superrigidity.Richard Schoen carried out the analytical work to extend the harmonic map theory to the metric space setting; this was subsequently done more systematically by Nicholas Korevaar and Schoen, establishing extensions of most of the standardSobolev space theory.[19] A sample application of Gromov and Schoen's methods is the fact thatlattices in the isometry group of thequaternionic hyperbolic space arearithmetic.[GS92]
In 1978, Gromov introduced the notion ofalmost flat manifolds.[G78] The famousquarter-pinched sphere theorem inRiemannian geometry says that if a complete Riemannian manifold hassectional curvatures which are all sufficiently close to a given positive constant, thenM must be finitely covered by a sphere. In contrast, it can be seen by scaling that everyclosed 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 anilmanifold. The proof works by replaying the proofs of theBieberbach theorem andMargulis lemma. Gromov's proof was given a careful exposition byPeter Buser and Hermann Karcher.[20][21][22]
In 1979,Richard Schoen andShing-Tung Yau showed that the class ofsmooth manifolds which admit Riemannian metrics of positivescalar curvature is topologically rich. In particular, they showed that this class is closed under the operation ofconnected sum and ofsurgery in codimension at least three.[23] Their proof used elementary methods ofpartial differential equations, in particular to do with theGreen's function. Gromov andBlaine Lawson gave another proof of Schoen and Yau's results, making use of elementary geometric constructions.[GL80b] They also showed how purely topological results such asStephen Smale'sh-cobordism theorem could then be applied to draw conclusions such as the fact that everyclosed andsimply-connected smooth manifold of dimension 5, 6, or 7 has a Riemannian metric of positive scalar curvature. They further introduced the new class ofenlargeable manifolds, distinguished by a condition inhomotopy theory.[GL80a] They showed that Riemannian metrics of positive scalar curvaturecannot exist on such manifolds. A particular consequence is that thetorus cannot support any Riemannian metric of positive scalar curvature, which had been a major conjecture previously resolved by Schoen and Yau in low dimensions.[24]
In 1981, Gromov identified topological restrictions, based uponBetti numbers, on manifolds which admit Riemannian metrics ofnonnegative sectional curvature.[G81a] The principal idea of his work was to combineKarsten Grove and Katsuhiro Shiohama's Morse theory for the Riemannian distance function, with control of the distance function obtained from theToponogov comparison theorem, together with theBishop–Gromov inequality on volume of geodesic balls.[25] This resulted in topologically controlled covers of the manifold by geodesic balls, to whichspectral sequence arguments could be applied to control the topology of the underlying manifold. The topology of lower bounds on sectional curvature is still not fully understood, and Gromov's work remains as a primary result. As an application ofHodge theory,Peter Li and Yau were able to apply their gradient estimates to find similar Betti number estimates which are weaker than Gromov's but allow the manifold to have convex boundary.[26]
InJeff Cheeger's fundamental compactness theory for Riemannian manifolds, a key step in constructing coordinates on the limiting space is aninjectivity radius estimate forclosed manifolds.[27] Cheeger, Gromov, andMichael Taylor localized Cheeger's estimate, showing how to useBishop−Gromov volume comparison to control the injectivity radius in absolute terms by curvature bounds and volumes of geodesic balls.[CGT82] Their estimate has been used in a number of places where the construction of coordinates is an important problem.[28][29][30] A particularly well-known instance of this is to show thatGrigori Perelman's "noncollapsing theorem" forRicci flow, which controls volume, is sufficient to allow applications ofRichard Hamilton's compactness theory.[31][32][33] Cheeger, Gromov, and Taylor applied their injectivity radius estimate to proveGaussian control of theheat kernel, although these estimates were later improved by Li and Yau as an application of their gradient estimates.[26]
Gromov made foundational contributions tosystolic 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] Gromovproved that everyessential manifold with a Riemannian metric contains a closed non-contractiblegeodesic of length at most.[34]
In 1981, Gromov introduced theGromov–Hausdorff metric, which endows the set of allmetric 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 anultralimit.[G93]
Gromov's compactness theorem had a deep impact on the field ofgeometric group theory. He applied it to understand the asymptotic geometry of theword metric of agroup 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 aLie group.[G81b] As a consequence he was able to settle theMilnor-Wolf conjecture as posed in the 1960s, which asserts that any such group isvirtually nilpotent. Using ultralimits, similar asymptotic structures can be studied for more general metric spaces.[G93] Important developments on this topic were given byBruce Kleiner, Bernhard Leeb, andPierre Pansu, among others.[35][36]
Another consequence isGromov's compactness theorem, stating that the set of compactRiemannian manifolds withRicci curvature ≥c anddiameter ≤D isrelatively compact in the Gromov–Hausdorff metric.[G81b] The possible limit points of sequences of such manifolds areAlexandrov spaces of curvature ≥c, a class ofmetric spaces studied in detail byBurago, Gromov andPerelman in 1992.[BGP92]
Along withEliyahu Rips, Gromov introduced the notion ofhyperbolic groups.[G87]
Gromov's theory ofpseudoholomorphic curves is one of the foundations of the modern study ofsymplectic geometry.[G85] Although he was not the first to consider pseudo-holomorphic curves, he uncovered a "bubbling" phenomena parallelingKaren Uhlenbeck's earlier work onYang–Mills connections, and Uhlenbeck and Jonathan Sack's work onharmonic maps.[37][38] In the time since Sacks, Uhlenbeck, and Gromov's work, such bubbling phenomena has been found in a number of other geometric contexts. The correspondingcompactness theorem encoding the bubbling allowed Gromov to arrive at a number of analytically deep conclusions on existence of pseudo-holomorphic curves. A particularly famous result of Gromov's, arrived at as a consequence of the existence theory and the monotonicity formula forminimal surfaces, is the "non-squeezing theorem," which provided a striking qualitative feature of symplectic geometry. Following ideas ofEdward Witten, Gromov's work is also fundamental forGromov-Witten theory, which is a widely studied topic reaching intostring theory,algebraic geometry, andsymplectic geometry.[39][40][41] From a different perspective, Gromov's work was also inspirational for much ofAndreas Floer's work.[42]
Yakov Eliashberg and Gromov developed some of the basic theory for symplectic notions of convexity.[EG91] They introduce various specific notions of convexity, all of which are concerned with the existence of one-parameter families of diffeomorphisms which contract the symplectic form. They show that convexity is an appropriate context for anh-principle to hold for the problem of constructing certainsymplectomorphisms. They also introduced analogous notions incontact geometry; the existence of convex contact structures was later studied byEmmanuel Giroux.[43]
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Books
| BGS85. | Ballmann, Werner; Gromov, Mikhael; Schroeder, Viktor (1985).Manifolds of nonpositive curvature. Progress in Mathematics. Vol. 61. Boston, MA:Birkhäuser Boston, Inc.doi:10.1007/978-1-4684-9159-3.ISBN 0-8176-3181-X.MR 0823981.Zbl 0591.53001.[50] |
| G86. | Gromov, Mikhael (1986).Partial differential relations. Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Vol. 9. Berlin:Springer-Verlag.doi:10.1007/978-3-662-02267-2.ISBN 3-540-12177-3.MR 0864505.Zbl 0651.53001.[51] |
| G99a. | Gromov, Misha (1999).Metric structures for Riemannian and non-Riemannian spaces. Progress in Mathematics. Vol. 152. Translated by Bates, Sean Michael. With appendices byM. Katz,P. Pansu, andS. Semmes. (Based on the 1981 French original ed.). Boston, MA:Birkhäuser Boston, Inc.doi:10.1007/978-0-8176-4583-0.ISBN 0-8176-3898-9.MR 1699320.Zbl 0953.53002.[52] |
| G18. | Gromov, Misha (2018).Great circle of mysteries. Mathematics, the world, the mind.Springer, Cham.doi:10.1007/978-3-319-53049-9.ISBN 978-3-319-53048-2.MR 3837512.Zbl 1433.00004. |
Major articles
| G69. | Gromov, M. L. (1969). "Stable mappings of foliations into manifolds".Mathematics of the USSR-Izvestiya.33 (4):671–694.Bibcode:1969IzMat...3..671G.doi:10.1070/im1969v003n04abeh000796.MR 0263103.Zbl 0205.53502. |
| G78. | Gromov, M. (1978)."Almost flat manifolds".Journal of Differential Geometry.13 (2):231–241.doi:10.4310/jdg/1214434488.MR 0540942.Zbl 0432.53020. |
| GL80a. | Gromov, Mikhael;Lawson, H. Blaine Jr. (1980). "Spin and scalar curvature in the presence of a fundamental group. I".Annals of Mathematics. Second Series.111 (2):209–230.doi:10.2307/1971198.JSTOR 1971198.MR 0569070.S2CID 14149468.Zbl 0445.53025. |
| GL80b. | Gromov, Mikhael;Lawson, H. Blaine Jr. (1980)."The classification of simply connected manifolds of positive scalar curvature"(PDF).Annals of Mathematics. Second Series.111 (3):423–434.doi:10.2307/1971103.JSTOR 1971103.MR 0577131.Zbl 0463.53025. |
| G81a. | Gromov, Michael (1981)."Curvature, diameter and Betti numbers".Commentarii Mathematici Helvetici.56 (2):179–195.doi:10.1007/BF02566208.MR 0630949.S2CID 120818147.Zbl 0467.53021. |
| G81b. |
| G81c. | Gromov, M. (1981)."Hyperbolic manifolds, groups and actions"(PDF). InKra, Irwin;Maskit, Bernard (eds.).Riemann surfaces and related topics. Proceedings of the 1978 Stony Brook Conference (State University of New York, Stony Brook, NY, 5–9 June 1978). Annals of Mathematics Studies. Vol. 97. Princeton, NJ:Princeton University Press. pp. 183–213.doi:10.1515/9781400881550-016.ISBN 0-691-08264-2.MR 0624814.Zbl 0467.53035. |
| CGT82. |
| G82. | Gromov, Michael (1982)."Volume and bounded cohomology".Publications Mathématiques de l'Institut des Hautes Études Scientifiques.56:5–99.MR 0686042.Zbl 0515.53037. |
| G83. | Gromov, Mikhael (1983)."Filling Riemannian manifolds".Journal of Differential Geometry.18 (1):1–147.doi:10.4310/jdg/1214509283.MR 0697984.Zbl 0515.53037. |
| GL83. |
| GM83. | Gromov, M.;Milman, V. D. (1983)."A topological application of the isoperimetric inequality"(PDF).American Journal of Mathematics.105 (4):843–854.doi:10.2307/2374298.JSTOR 2374298.MR 0708367.Zbl 0522.53039. |
| G85. | Gromov, M. (1985)."Pseudo holomorphic curves in symplectic manifolds".Inventiones Mathematicae.82 (2):307–347.Bibcode:1985InMat..82..307G.doi:10.1007/BF01388806.MR 0809718.S2CID 4983969.Zbl 0592.53025. |
| CG86a. | Cheeger, Jeff; Gromov, Mikhael (1986)."Collapsing Riemannian manifolds while keeping their curvature bounded. I".Journal of Differential Geometry.23 (3):309–346.doi:10.4310/jdg/1214440117.MR 0852159.Zbl 0606.53028. |
| CG86b. | Cheeger, Jeff; Gromov, Mikhael (1986)."L2-cohomology and group cohomology".Topology.25 (2):189–215.doi:10.1016/0040-9383(86)90039-X.MR 0837621.Zbl 0597.57020. |
| G87. | Gromov, M. (1987)."Hyperbolic groups"(PDF). InGersten, S. M. (ed.).Essays in group theory. Mathematical Sciences Research Institute Publications. Vol. 8. New York:Springer-Verlag. pp. 75–263.doi:10.1007/978-1-4613-9586-7.ISBN 0-387-96618-8.MR 0919829.Zbl 0634.20015. |
| G89. |
| EG91. | Eliashberg, Yakov; Gromov, Mikhael (1991)."Convex symplectic manifolds"(PDF). In Bedford, Eric; D'Angelo, John P.;Greene, Robert E.;Krantz, Steven G. (eds.).Several complex variables and complex geometry. Part 2. Proceedings of the Thirty-Seventh Annual Summer Research Institute held at the University of California (Santa Cruz, CA, 10–30 July 1989). Proceedings of Symposia in Pure Mathematics. Vol. 52. Providence, RI:American Mathematical Society. pp. 135–162.doi:10.1090/pspum/052.2.ISBN 0-8218-1490-7.MR 1128541.Zbl 0742.53010. |
| G91. | Gromov, M. (1991)."Kähler hyperbolicity andL2-Hodge theory".Journal of Differential Geometry.33 (1):263–292.doi:10.4310/jdg/1214446039.MR 1085144.Zbl 0719.53042. |
| BGP92. | Burago, Yu.; Gromov, M.;Perelʹman, G. (1992). "A. D. Aleksandrov spaces with curvatures bounded below".Russian Mathematical Surveys.47 (2):1–58.doi:10.1070/RM1992v047n02ABEH000877.MR 1185284.S2CID 10675933.Zbl 0802.53018. |
| GS92. |
| G93. | Gromov, M. (1993)."Asymptotic invariants of infinite groups"(PDF). In Niblo, Graham A.; Roller, Martin A. (eds.).Geometric group theory. Vol. 2. Symposium held at Sussex University (Sussex, July 1991). London Mathematical Society Lecture Note Series. Cambridge:Cambridge University Press. pp. 1–295.ISBN 0-521-44680-5.MR 1253544.Zbl 0841.20039.[53] |
| G96. | Gromov, Mikhael (1996)."Carnot-Carathéodory spaces seen from within"(PDF). In Bellaïche, André; Risler, Jean-Jacques (eds.).Sub-Riemannian geometry. Progress in Mathematics. Vol. 144. Basel:Birkhäuser. pp. 79–323.doi:10.1007/978-3-0348-9210-0_2.ISBN 3-7643-5476-3.MR 1421823.Zbl 0864.53025. |
| G99b. | Gromov, M. (1999)."Endomorphisms of symbolic algebraic varieties".Journal of the European Mathematical Society.1 (2):109–197.doi:10.1007/PL00011162.MR 1694588.Zbl 0998.14001. |
| G00. | Gromov, Misha (2000)."Spaces and questions"(PDF). InAlon, N.;Bourgain, J.;Connes, A.; Gromov, M.;Milman, V. (eds.).Visions in mathematics: GAFA 2000 Special Volume, Part I. Proceedings of the meeting held at Tel Aviv University, Tel Aviv, 25 August – 3 September 1999.Geometric and Functional Analysis. Basel:Birkhäuser. pp. 118–161.doi:10.1007/978-3-0346-0422-2_5.ISBN 978-3-0346-0421-5.MR 1826251.Zbl 1006.53035. |
| G03a. | Gromov, M. (2003)."Isoperimetry of waists and concentration of maps".Geometric and Functional Analysis.13 (1):178–215.doi:10.1007/s000390300004.MR 1978494.Zbl 1044.46057. (Erratum: doi:10.1007/s00039-009-0703-1)
|
| G03b. | Gromov, Mikhaïl (2003)."On the entropy of holomorphic maps"(PDF).L'Enseignement Mathématique. Revue Internationale. 2e Série.49 (3–4):217–235.MR 2026895.Zbl 1080.37051. |
| G03c. | Gromov, M. (2003)."Random walk in random groups".Geometric and Functional Analysis.13 (1):73–146.doi:10.1007/s000390300002.MR 1978492.Zbl 1122.20021.
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