View source for Mikhael Gromov (mathematician)

{{Short description|Russian-French mathematician}}
{{Use dmy dates|date=September 2022}}
{{Infobox scientist
| name              = Mikhael Gromov
| native_name       = Михаил Громов
| native_name_lang  = ru
| image             = Mikhael Gromov.jpg
| image_size        = 
| caption           = Gromov in 2014
| birth_date        = {{birth date and age|1943|12|23|df=y}}
| birth_place       = [[Boksitogorsk]], [[Russian SFSR]], [[Soviet Union]]
| death_date        = 
| death_place       = 
| nationality       = Russian and French
| alma_mater        = [[Saint Petersburg State University|Leningrad State University]] (PhD)
| doctoral_advisor  = [[Vladimir Rokhlin (Soviet mathematician)|Vladimir Rokhlin]]
| doctoral_students = [[Denis Auroux]]<br />[[François Labourie]]<br />[[Pierre Pansu]]<br />[[Mikhail Katz]]
| known_for         = [[Geometric group theory]]<br />[[Symplectic geometry]]<br />[[Systolic geometry]]<br />[[Gromov boundary]]<br />[[Gromov's compactness theorem (geometry)]]<br />[[Gromov's compactness theorem (topology)]]<br />[[Gromov's theorem on groups of polynomial growth]]<br />[[Gromov–Hausdorff convergence]]<br />[[Almost flat manifold|Gromov–Ruh theorem]]<br />[[Gromov–Witten invariant]]<br />[[Hyperbolic group|Gromov hyperbolic group]]<br />[[δ-hyperbolic space|Gromov δ-hyperbolic space]]<br />[[Simplicial volume|Gromov norm]]<br />[[Gromov product]]<br />[[Outer space (mathematics)#Gromov topology|Gromov topology]]<br />[[Gromov's inequality for complex projective space]]<br />[[Gromov's systolic inequality for essential manifolds|Gromov's systolic inequality]]<br />[[Bishop–Gromov inequality]]<br />[[Asymptotic dimension]]<br />[[Essential manifold]]<br />[[Filling area conjecture]]<br />[[Filling radius]]<br />[[Mean dimension]]<br />[[Minimal volume]]<br />[[Non-squeezing theorem]]<br />[[Pseudoholomorphic curve]]<br />[[Random group]]<br />[[Sofic group]]<br />[[Systolic freedom]]<br />[[2π theorem]]
| field             = Mathematics
| work_institutions = [[Institut des Hautes Études Scientifiques]]<br />[[New York University]]
| prizes            = {{no wrap|[[Oswald Veblen Prize in Geometry]] (1981)<br />[[Wolf Prize in Mathematics|Wolf Prize]] (1993)<br />[[Balzan Prize]] (1999)<br /> [[Kyoto Prize]] (2002)<br />[[Nemmers Prize in Mathematics]] (2004)<br />[[Bolyai Prize]] (2005)<br />[[Abel Prize]] (2009)}}
}}
'''Mikhael Leonidovich Gromov''' (also '''Mikhail Gromov''', '''Michael Gromov''' or '''Misha Gromov'''; {{langx|ru|link=no|Михаи́л Леони́дович Гро́мов}}; born 23 December 1943) is a Russian-French mathematician known for his work in [[geometry]], [[Mathematical analysis|analysis]] and [[group theory]]. He is a permanent member of [[Institut des Hautes Études Scientifiques]] 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 was Russian-Slavic and his mother Lea was of Jewish heritage. Both were [[pathologist]]s.<ref>Gromov, Mikhail. "A Few Recollections", in {{cite book|author1=Helge Holden|author2=Ragni Piene|title=The Abel Prize 2008–2012|url=https://books.google.com/books?id=tEprnQEACAAJ|date=3 February 2014|publisher=Springer Berlin Heidelberg|isbn=978-3-642-39448-5|pages=129–137}} (also available on Gromov's homepage: [https://www.ihes.fr/~gromov/PDF/autobiography-dec20-2010.pdf link])</ref> His mother was the cousin of World Chess Champion [[Mikhail Botvinnik]], as well as of the mathematician Isaak Moiseevich Rabinovich.<ref>[https://archive.today/20120710192722/http://centropaquarterly.org/index.php?nID=54&bioID=262 Воспоминания Владимира Рабиновича (генеалогия семьи М. Громова по материнской линии]. Лия Александровна Рабинович также приходится двоюродной сестрой известному [[Рига|рижскому]] математику, историку математики и популяризатору науки Исааку Моисеевичу Рабиновичу (род. 1911), автору книг «Математик Пирс Боль из Риги» (совместно с [[Мышкис, Анатолий Дмитриевич|А. Д. Мышкисом]] и с приложением комментария [[Ботвинник, Михаил Моисеевич|М. М. Ботвинника]] «О шахматной игре П. Г. Боля», 1965), «Строптивая производная» (1968) и др. Троюродный брат М. Громова – известный [[Латвия|латвийский]] адвокат и общественный деятель Александр Жанович Бергман ([[:pl:Aleksandr Bergman|польск.]], род. 1925).</ref> 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.<ref name="EMS">[http://www.ems-ph.org/journals/newsletter/pdf/2009-09-73.pdf Newsletter of the European Mathematical Society, No. 73, September 2009, p. 19]</ref> When Gromov was nine years old,<ref name="lemonde" /> 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.<ref name="EMS" />

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 (Soviet mathematician)|Vladimir Rokhlin]].<ref>{{cite magazine|url=http://cims.nyu.edu/newsletters/Spring2009.pdf|date=Spring 2009|title=Mikhael Gromov Receives the 2009 Abel Prize|magazine=CIMS Newsletter|publisher=Courant Institute of Mathematical Sciences|page=1}}</ref>

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.<ref name=simons/>

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 [[Aliyah|move to Israel]].<ref name=lemonde>{{Cite news|url=http://www.lemonde.fr/planete/article/2009/03/26/mikhail-gromov-le-genie-qui-venait-du-froid_1172835_3244.html|title=Mikhaïl Gromov, le génie qui venait du froid|last=Foucart|first=Stéphane|date=26 March 2009|newspaper=Le Monde.fr|language=fr|issn=1950-6244}}</ref><ref>{{Cite book|url=https://books.google.com/books?id=BoUFAQAAIAAJ&q=Vivre+savant+sous+le+communisme|title=Vivre savant sous le communisme|last=Ripka|first=Georges|date=1 January 2002|publisher=Belin|isbn=9782701130538|language=fr}}</ref> He changed his last name to that of his mother.<ref name=lemonde/> He received a coded letter saying that, if he could get out of the Soviet Union, he could go to [[Stony Brook University|Stony 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.<ref name="simons">{{cite web|url=https://www.simonsfoundation.org/science_lives_video/science-lives-mikhail-gromov/|title=Science Lives: Mikhail Gromov|last=Roberts|first=Siobhan|author-link=Siobhan Roberts|date=22 December 2014|publisher=Simons Foundation}}</ref>

In 1981 he left [[Stony Brook University]] to join the faculty of [[Pierre and Marie Curie University|University of Paris VI]] and in 1982 he became a permanent professor at the [[Institut des Hautes Études Scientifiques]] 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.<ref name=mactutor>{{MacTutor Biography|id=Gromov}}</ref> He adopted French citizenship in 1992.<ref>{{cite web|title=Mikhail Leonidovich Gromov|work=abelprize.no|url=http://www.abelprize.no/c53859/binfil/download.php?tid=53784
}}</ref>

==Work==
Gromov's style of geometry often features a "coarse" or "soft" viewpoint, analyzing asymptotic or large-scale properties.{{ran|G00}} He is also interested in [[mathematical biology]],<ref name=":0">{{citation|url=https://www.ams.org/notices/201003/rtx100300391p.pdf|title=Interview with Mikhail Gromov|journal=[[Notices of the AMS]]|date=March 2010|volume=57|issue=3|pages=391–403}}.</ref> the structure of the brain and the thinking process, and the way scientific ideas evolve.<ref name="simons" />

Motivated by [[Nash embedding theorem|Nash and Kuiper's isometric embedding theorems]] and the results on [[Immersion (mathematics)|immersion]]s by [[Morris Hirsch]] and [[Stephen Smale]],<ref name=":0" /> Gromov introduced the [[h-principle]] in various formulations. Modeled upon the special case of the Hirsch–Smale theory, he introduced and developed the general theory of ''microflexible sheaves'', proving that they satisfy an h-principle on [[open manifold]]s.{{ran|G69}} As a consequence (among other results) he was able to establish the existence of positively curved and negatively curved [[Riemannian metric]]s on any [[open manifold]] whatsoever. His result is in counterpoint to the well-known topological restrictions (such as the [[soul theorem|Cheeger–Gromoll soul theorem]] or [[Cartan–Hadamard theorem]]) on ''[[geodesically complete]]'' Riemannian manifolds of positive or negative curvature. After this initial work, he developed further h-principles partly in collaboration with [[Yakov Eliashberg]], including work building upon Nash and Kuiper's theorem and the [[Nash-Moser theorem|Nash–Moser implicit function theorem]]. There are many applications of his results, including topological conditions for the existence of [[Symplectic manifold#Lagrangian mapping|exact Lagrangian immersion]]s and similar objects in [[symplectic geometry|symplectic]] and [[contact geometry]].<ref>{{cite book|url=https://books.google.com/books?id=Mz_1CAAAQBAJ&q=Gromov-Lees+theorem&pg=PA215|last1=Arnold|first1=V. I.|last2=Goryunov|first2=V. V.|last3=Lyashko|first3=O. V.|last4=Vasilʹev|first4=V. A.|title=Singularity theory. I|translator-first1=A.|translator-last1=Iacob|series=Encyclopaedia of Mathematical Sciences|volume=6|publisher=[[Springer Publishing|Springer]]|location=Berlin|year=1993|mr=1660090|isbn=3-540-63711-7|edition=Translation of 1988 Russian original|doi=10.1007/978-3-642-58009-3|author-link1=Vladimir Arnold|author-link2=Victor Goryunov|author-link4=Victor Anatolyevich Vassiliev}}</ref><ref>{{cite book|last1=Eliashberg|first1=Y.|last2=Mishachev|first2=N.|title=Introduction to the h-principle|series=[[Graduate Studies in Mathematics]]|volume=48|publisher=[[American Mathematical Society]]|location=Providence, RI|year=2002|isbn=0-8218-3227-1|mr=1909245|author-link1=Yakov Eliashberg|doi=10.1090/gsm/048}}</ref> His well-known book ''Partial Differential Relations'' collects most of his work on these problems.{{ran|G86}} Later, he applied his methods to [[complex geometry]], proving certain instances of the ''Oka principle'' on deformation of [[continuous map]]s to [[holomorphic map]]s.{{ran|G89}} His work initiated a renewed study of the Oka–Grauert theory, which had been introduced in the 1950s.<ref>{{cite book|mr=3012475|last1=Cieliebak|first1=Kai|last2=Eliashberg|first2=Yakov|author-link2=Yakov Eliashberg|title=From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds|series=American Mathematical Society Colloquium Publications|volume=59|publisher=[[American Mathematical Society]]|location=Providence, RI|year=2012|isbn=978-0-8218-8533-8|doi=10.1090/coll/059|s2cid=118671586 }}</ref><ref>{{cite book|mr=3700709|last1=Forstnerič|first1=Franc|title=Stein manifolds and holomorphic mappings. The homotopy principle in complex analysis|edition=Second edition of 2011 original|series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3)|volume=56|publisher=[[Springer, Cham]]|year=2017|isbn=978-3-319-61057-3|doi=10.1007/978-3-319-61058-0}}</ref>

Gromov and [[Vitali Milman]] gave a formulation of the [[concentration of measure]] phenomena.{{ran|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 the [[law 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 of [[Riemannian manifold]]s in which the lower bound of the [[Ricci curvature]] or the first eigenvalue of the [[Laplace–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 as [[Michel Talagrand]].<ref>Talagrand, Michel A new look at independence. Ann. Probab. 24 (1996), no. 1, 1–34.</ref>

Since the seminal 1964 publication of [[James Eells]] and [[Joseph H. Sampson|Joseph Sampson]] on [[harmonic map]]s, 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.<ref>Eells, James, Jr.; Sampson, J. H. Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86 (1964), 109–160.</ref><ref>Yum Tong Siu. ''The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds.'' Ann. of Math. (2) 112 (1980), no. 1, 73–111.</ref><ref>Kevin Corlette. ''Archimedean superrigidity and hyperbolic geometry.'' Ann. of Math. (2) 135 (1992), no. 1, 165–182.</ref> Gromov had the insight that the extension of this program to the setting of mappings into [[metric space]]s would imply new results on [[discrete group]]s, following [[superrigidity|Margulis 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 standard [[Sobolev space]] theory.<ref>Korevaar, Nicholas J.; Schoen, Richard M. Sobolev spaces and harmonic maps for metric space targets. Comm. Anal. Geom. 1 (1993), no. 3-4, 561–659.</ref> A sample application of Gromov and Schoen's methods is the fact that [[Lattice (discrete subgroup)|lattices]] in the isometry group of the [[Hyperbolic quaternion|quaternionic hyperbolic space]] are [[Arithmetic group|arithmetic]].{{ran|GS92}}

===Riemannian geometry===
In 1978, Gromov introduced the notion of [[almost flat manifold]]s.{{ran|G78}} The famous [[sphere theorem|quarter-pinched sphere theorem]] in [[Riemannian geometry]] says that if a complete Riemannian manifold has [[sectional curvature]]s which are all sufficiently close to a given positive constant, then {{mvar|M}} must be finitely covered by a sphere. In contrast, it can be seen by scaling that every [[closed manifold|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 [[Flat manifold|Bieberbach theorem]] and [[Margulis lemma]]. Gromov's proof was given a careful exposition by [[Jürg Peter Buser|Peter Buser]] and Hermann Karcher.<ref>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.</ref><ref>Peter Buser and Hermann Karcher. ''Gromov's almost flat manifolds.'' Astérisque, 81. Société Mathématique de France, Paris, 1981. 148 pp.</ref><ref>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.</ref>

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 theory|surgery]] in codimension at least three.<ref>{{cite journal|last1=Schoen|first1=R.|author-link1=Richard Schoen|last2=Yau|first2=S. T.|title=On the structure of manifolds with positive scalar curvature|journal=Manuscripta Mathematica|zbl=0423.53032|volume=28|year=1979|issue=1–3|pages=159–183|url=http://eudml.org/doc/154634|doi=10.1007/BF01647970|mr=0535700| s2cid=121008386 |author-link2=Shing-Tung Yau}}</ref> Their proof used elementary methods of [[partial differential equation]]s, 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.{{ran|GL80b}} They also showed how purely topological results such as [[Stephen Smale]]'s [[h-cobordism|h-cobordism theorem]] could then be applied to draw conclusions such as the fact that every [[closed manifold|closed]] and [[simply-connected]] smooth manifold of dimension 5, 6, or 7 has a Riemannian metric of positive scalar curvature. They further introduced the new class of ''enlargeable manifolds'', distinguished by a condition in [[homotopy theory]].{{ran|GL80a}} They showed that Riemannian metrics of positive scalar curvature ''cannot'' exist on such manifolds. A particular consequence is that the [[torus]] cannot support any Riemannian metric of positive scalar curvature, which had been a major conjecture previously resolved by Schoen and Yau in low dimensions.<ref>{{cite book|author-link1=H. Blaine Lawson|author-link2=Marie-Louise Michelsohn|zbl=0688.57001|last1=Lawson|last2=Michelsohn|title=Spin geometry|first1=H. Blaine Jr. |first2=Marie-Louise|series=Princeton Mathematical Series|volume=38|publisher=[[Princeton University Press]]|location=Princeton, NJ|year=1989|isbn=0-691-08542-0|mr=1031992}}</ref>

In 1981, Gromov identified topological restrictions, based upon [[Betti number]]s, on manifolds which admit Riemannian metrics of [[sectional curvature|nonnegative sectional curvature]].{{ran|G81a}} The principal idea of his work was to combine [[Karsten Grove]] and Katsuhiro Shiohama's Morse theory for the Riemannian distance function, with control of the distance function obtained from the [[Toponogov's theorem|Toponogov comparison theorem]], together with the [[Bishop–Gromov inequality]] on volume of geodesic balls.<ref>Grove, Karsten; Shiohama, Katsuhiro A generalized sphere theorem. Ann. of Math. (2) 106 (1977), no. 2, 201–211.</ref> This resulted in topologically controlled covers of the manifold by geodesic balls, to which [[spectral 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 of [[Hodge theory]], [[Peter Li (mathematician)|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.<ref name="LiYau">Li, Peter; Yau, Shing-Tung. On the parabolic kernel of the Schrödinger operator. Acta Math. 156 (1986), no. 3-4, 153–201.</ref>

In [[Jeff Cheeger]]'s fundamental compactness theory for Riemannian manifolds, a key step in constructing coordinates on the limiting space is an [[injectivity radius]] estimate for [[closed manifold]]s.<ref>Cheeger, Jeff. Finiteness theorems for Riemannian manifolds. Amer. J. Math. 92 (1970), 61–74.</ref> Cheeger, Gromov, and [[Michael E. Taylor|Michael Taylor]] localized Cheeger's estimate, showing how to use [[Bishop–Gromov inequality|Bishop−Gromov volume comparison]] to control the injectivity radius in absolute terms by curvature bounds and volumes of geodesic balls.{{ran|CGT82}} Their estimate has been used in a number of places where the construction of coordinates is an important problem.<ref>Anderson, Michael T. Ricci curvature bounds and Einstein metrics on compact manifolds. J. Amer. Math. Soc. 2 (1989), no. 3, 455–490.</ref><ref>Bando, Shigetoshi; Kasue, Atsushi; Nakajima, Hiraku. On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth. Invent. Math. 97 (1989), no. 2, 313–349.</ref><ref>Tian, G. On Calabi's conjecture for complex surfaces with positive first Chern class. Invent. Math. 101 (1990), no. 1, 101–172.</ref> A particularly well-known instance of this is to show that [[Grigori Perelman]]'s "noncollapsing theorem" for [[Ricci flow]], which controls volume, is sufficient to allow applications of [[Richard S. Hamilton|Richard Hamilton]]'s compactness theory.<ref>Grisha Perelman. The entropy formula for the Ricci flow and its geometric applications.</ref><ref>Hamilton, Richard S. A compactness property for solutions of the Ricci flow. Amer. J. Math. 117 (1995), no. 3, 545–572.</ref><ref>Cao, Huai-Dong; Zhu, Xi-Ping. A complete proof of the Poincaré and geometrization conjectures—application of the Hamilton-Perelman theory of the Ricci flow. Asian J. Math. 10 (2006), no. 2, 165–492.</ref> Cheeger, Gromov, and Taylor applied their injectivity radius estimate to prove [[Gaussian function|Gaussian]] control of the [[heat kernel]], although these estimates were later improved by Li and Yau as an application of their gradient estimates.<ref name="LiYau" />

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"{{ran|G83}} Gromov [[Gromov's systolic inequality for essential manifolds|proved]] that every [[essential manifold]] <math>M</math> with a Riemannian metric contains a closed non-contractible [[geodesic]] of length at most <math>C(n)\operatorname{Vol}(M)^{1/n}</math>.<ref>Katz, M.  Systolic geometry and topology.  With an appendix by J. Solomon.   Mathematical Surveys and Monographs, volume 137.  [[American Mathematical Society]], 2007.</ref>

===Gromov−Hausdorff convergence and geometric group theory===
In 1981, Gromov introduced the [[Gromov–Hausdorff metric]], which endows the set of all [[metric space]]s with the structure of a metric space.{{ran|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]].{{ran|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 [[Growth rate (group theory)|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]].{{ran|G81b}} As a consequence he was able to settle the [[Gromov's theorem on groups of polynomial growth|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.{{ran|G93}} Important developments on this topic were given by [[Bruce Kleiner]], Bernhard Leeb, and [[Pierre Pansu]], among others.<ref>Pierre Pansu. ''Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un.'' Ann. of Math. (2) 129 (1989), no. 1, 1–60.</ref><ref>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.</ref>

Another consequence is [[Gromov's compactness theorem (geometry)|Gromov's compactness theorem]], stating that the set of compact [[Riemannian manifold]]s with [[Ricci curvature]] ≥ ''c'' and [[diameter]] ≤ ''D'' is [[relatively compact]] in the Gromov–Hausdorff metric.{{ran|G81b}} The possible limit points of sequences of such manifolds are [[Alexandrov space]]s of curvature ≥ ''c'', a class of [[metric space]]s studied in detail by [[Yuri Burago|Burago]], Gromov and [[Grigori Perelman|Perelman]] in 1992.{{ran|BGP92}}

Along with [[Eliyahu Rips]], Gromov introduced the notion of [[hyperbolic group]]s.{{ran|G87}}

===Symplectic geometry===
Gromov's theory of [[pseudoholomorphic curves]] is one of the foundations of the modern study of [[symplectic geometry]].{{ran|G85}} Although he was not the first to consider pseudo-holomorphic curves, he uncovered a "bubbling" phenomena paralleling [[Karen Uhlenbeck]]'s earlier work on [[Yang–Mills connection]]s, and Uhlenbeck and Jonathan Sack's work on [[harmonic map]]s.<ref>Uhlenbeck, Karen K. Connections with Lp bounds on curvature. Comm. Math. Phys. 83 (1982), no. 1, 31–42.</ref><ref>Sacks, J.; Uhlenbeck, K. The existence of minimal immersions of 2-spheres. Ann. of Math. (2) 113 (1981), no. 1, 1–24.</ref> In the time since Sacks, Uhlenbeck, and Gromov's work, such bubbling phenomena has been found in a number of other geometric contexts. The corresponding [[Gromov's compactness theorem (topology)|compactness 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 for [[minimal surface]]s, is the "[[non-squeezing theorem]]," which provided a striking qualitative feature of symplectic geometry. Following ideas of [[Edward Witten]], Gromov's work is also fundamental for [[Gromov-Witten theory]], which is a widely studied topic reaching into [[string theory]], [[algebraic geometry]], and [[symplectic geometry]].<ref>Witten, Edward Two-dimensional gravity and intersection theory on moduli space. Surveys in differential geometry (Cambridge, MA, 1990), 243–310, Lehigh Univ., Bethlehem, PA, 1991.</ref><ref>Eliashberg, Y.; Givental, A.; Hofer, H. Introduction to symplectic field theory. GAFA 2000 (Tel Aviv, 1999). Geom. Funct. Anal. 2000, Special Volume, Part II, 560–673.</ref><ref>Bourgeois, F.; Eliashberg, Y.; Hofer, H.; Wysocki, K.; Zehnder, E. Compactness results in symplectic field theory. Geom. Topol. 7 (2003), 799–888.</ref> From a different perspective, Gromov's work was also inspirational for much of [[Andreas Floer]]'s work.<ref>Floer, Andreas. Morse theory for Lagrangian intersections. J. Differential Geom. 28 (1988), no. 3, 513–547.</ref>

[[Yakov Eliashberg]] and Gromov developed some of the basic theory for symplectic notions of convexity.{{ran|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 an [[h-principle]] to hold for the problem of constructing certain [[symplectomorphism]]s. They also introduced analogous notions in [[contact geometry]]; the existence of convex contact structures was later studied by [[Emmanuel Giroux]].<ref>Giroux, Emmanuel. Convexité en topologie de contact. [[Comment. Math. Helv.]] 66 (1991), no. 4, 637–677.</ref>

==Prizes and honors==

===Prizes===
* [[Prize of the Mathematical Society of Moscow]] (1971)
* [[Oswald Veblen Prize in Geometry]] ([[American Mathematical Society|AMS]]) (1981)
* [[Elie Cartan Prize|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 ([[American Mathematical Society|AMS]]) (1997)
* [[Lobachevsky Medal]] (1997)
* [[Balzan Prize]] for Mathematics (1999)
* [[Kyoto Prize]] in Mathematical Sciences (2002)
* [[Nemmers Prize in Mathematics]] (2004)<ref>[https://www.ams.org/notices/200407/comm-nemmers.pdf Gromov Receives Nemmers Prize]
</ref>
* [[Bolyai Prize]] in 2005
* [[Abel Prize]] in 2009 "for his revolutionary contributions to geometry"<ref>{{Cite web |title=2009: Mikhail Leonidovich Gromov |url=https://abelprize.no/abel-prize-laureates/2009 |website=www.abelprize.no}}</ref>

===Honors===
* Invited speaker to [[International Congress of Mathematicians]]: 1970 (Nice), 1978 (Helsinki), 1983 (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]], the [[Royal Society]] (2011), <ref>[http://royalsociety.org/people/mikhail-gromov/ Professor Mikhail Gromov ForMemRS | Royal Society]</ref> and the [[National Academy of Sciences of Ukraine]] (2023).<ref>[https://www.nas.gov.ua/UA/PersonalSite/Pages/default.aspx?PersonID=0000031337 | National Academy of sciences of Ukraine, communication]</ref>
* Member of the [[French Academy of Sciences]] (1997)<ref>[http://www.academie-sciences.fr/fr/Liste-des-membres-de-l-Academie-des-sciences-/-G/mikhail-gromov.html Mikhaël Gromov — Membre de l’Académie des sciences]</ref>
* Delivered the 2007 [[Paul Turán]] Memorial Lectures.<ref>{{cite web|url=https://old.renyi.hu/turanlectures_vk.html|title=Turán Memorial Lectures}}</ref>

==See also==
{{div col|colwidth=25em}}
* [[Cartan–Hadamard conjecture]]
* [[Cartan–Hadamard theorem]]
* [[Collapsing manifold]]
* [[Lévy–Gromov inequality]]
* [[Taubes's Gromov invariant]]
* [[Mostow rigidity theorem]]
* [[Ramsey–Dvoretzky–Milman phenomenon]]
* [[Systoles of surfaces]]
{{div col end}}

==Publications==
'''Books'''
{{longitem|{{rma|BGS85|tw=4em|{{cite book|author-link1=Hans Werner Ballmann|first1=Werner|last1=Ballmann|first2=Mikhael|last2=Gromov|first3=Viktor|last3=Schroeder|title=Manifolds of nonpositive curvature|series=Progress in Mathematics|volume=61|publisher=[[Birkhäuser|Birkhäuser Boston, Inc.]]|location=Boston, MA|year=1985|isbn=0-8176-3181-X|doi=10.1007/978-1-4684-9159-3|mr=0823981|zbl=0591.53001}}<ref>{{cite journal|author=Heintze, Ernst|title=Review: ''Manifolds of nonpositive curvature'', by W. Ballmann, M. Gromov & V. Schroeder|journal=Bull. Amer. Math. Soc. (N.S.)|year=1987|volume=17|issue=2|pages=376–380|doi=10.1090/s0273-0979-1987-15603-5|doi-access=free}}</ref>}}}}
{{longitem|{{rma|G86|tw=4em|{{cite book|last1=Gromov|first1=Mikhael|title=Partial differential relations|series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3)|volume=9|publisher=[[Springer-Verlag]]|location=Berlin|year=1986|isbn=3-540-12177-3|mr=0864505|doi=10.1007/978-3-662-02267-2|zbl=0651.53001}}<ref>{{cite journal|author=McDuff, Dusa|author-link=Dusa McDuff|title=Review: ''Partial differential relations'', by Mikhael Gromov|journal=Bull. Amer. Math. Soc. (N.S.)|year=1988|volume=18|issue=2|pages=214–220|doi=10.1090/s0273-0979-1988-15654-6|doi-access=free}}</ref>}}}}
{{longitem|{{rma|G99a|tw=4em|{{cite book|last1=Gromov|first1=Misha|title-link=Metric Structures for Riemannian and Non-Riemannian Spaces|title=Metric structures for Riemannian and non-Riemannian spaces|edition=Based on the 1981 French original|series=Progress in Mathematics|volume=152|publisher=[[Birkhäuser|Birkhäuser Boston, Inc.]]|location=Boston, MA|year=1999|isbn=0-8176-3898-9|mr=1699320|translator-last1=Bates|translator-first1=Sean Michael|others=With appendices by [[Mikhail Katz|M. Katz]], [[Pierre Pansu|P. Pansu]], and [[Stephen Semmes|S. Semmes]].|doi=10.1007/978-0-8176-4583-0|zbl=0953.53002}}<ref>{{cite journal|author=Grove, Karsten|title=Review: ''Metric structures for Riemannian and non-Riemannian spaces'', by M. Gromov|journal=Bull. Amer. Math. Soc. (N.S.)|year=2001|volume=38|issue=3|pages=353–363|doi=10.1090/s0273-0979-01-00904-1|doi-access=free}}</ref>}}}}
{{longitem|{{rma|G18|tw=4em|{{cite book|first1=Misha|last1=Gromov|title=Great circle of mysteries. Mathematics, the world, the mind|publisher=[[Springer, Cham]]|year=2018|mr=3837512|isbn=978-3-319-53048-2|doi=10.1007/978-3-319-53049-9|zbl=1433.00004}}}}}}
'''Major articles'''
{{refbegin|30em}}
{{longitem|{{rma|G69|tw=4em|{{cite journal|mr=0263103|last1=Gromov|first1=M. L.|title=Stable mappings of foliations into manifolds|journal=[[Mathematics of the USSR-Izvestiya]]|volume=33|year=1969|issue=4|pages=671–694|zbl=0205.53502|doi=10.1070/im1969v003n04abeh000796| bibcode=1969IzMat...3..671G }}}}}}
{{longitem|{{rma|G78|tw=4em|{{cite journal|first1=M.|last1=Gromov|title=Almost flat manifolds|journal=[[Journal of Differential Geometry]]|volume=13|year=1978|issue=2|pages=231–241|doi=10.4310/jdg/1214434488|doi-access=free|mr=0540942|zbl=0432.53020}}}}}}
{{longitem|{{rma|GL80a|tw=4em|{{cite journal|last1=Gromov|first1=Mikhael|last2=Lawson|first2=H. Blaine Jr. |mr=0569070|title=Spin and scalar curvature in the presence of a fundamental group. I|journal=[[Annals of Mathematics]]|series=Second Series|volume=111|year=1980|issue=2|pages=209–230|author-link2=Blaine Lawson|doi=10.2307/1971198| jstor=1971198 |zbl=0445.53025|s2cid=14149468}}}}}}
{{longitem|{{rma|GL80b|tw=4em|{{cite journal|last1=Gromov|first1=Mikhael|last2=Lawson|first2=H. Blaine Jr. |doi=10.2307/1971103|mr=0577131|zbl=0463.53025|title=The classification of simply connected manifolds of positive scalar curvature|journal=[[Annals of Mathematics]]|series=Second Series|volume=111|year=1980|issue=3|pages=423–434| jstor=1971103 |author-link2=Blaine Lawson|url=https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/827.pdf}}}}}}
{{longitem|{{rma|G81a|tw=4em|{{cite journal|first1=Michael|last1=Gromov|url=http://www.digizeitschriften.de/dms/resolveppn/?PID%3DGDZPPN002064243|title=Curvature, diameter and Betti numbers|journal=[[Commentarii Mathematici Helvetici]]|volume=56|year=1981|issue=2|pages=179–195|doi=10.1007/BF02566208|mr=0630949|zbl=0467.53021| s2cid=120818147 }}}}}}
{{longitem|{{rma|G81b|tw=4em|{{cite journal|last1=Gromov|first1=Mikhael|title=Groups of polynomial growth and expanding maps|journal=[[Publications Mathématiques de l'IHÉS|Publications Mathématiques de l'Institut des Hautes Études Scientifiques]]|volume=53|year=1981|pages=53–73|zbl=0474.20018|doi=10.1007/BF02698687|url=http://www.numdam.org/item/?id=PMIHES_1981__53__53_0|mr=0623534| s2cid=121512559 }}}}}}
{{longitem|{{rma|G81c|tw=4em|{{cite conference|first1=M.|last1=Gromov| title=Hyperbolic manifolds, groups and actions |url=https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/hyperbolic_manifold.pdf|book-title=Riemann surfaces and related topics|conference=Proceedings of the 1978 Stony Brook Conference (State University of New York, Stony Brook, NY, 5–9 June 1978)|pages=183–213|series=Annals of Mathematics Studies|volume=97|publisher=[[Princeton University Press]]|location=Princeton, NJ|year=1981|doi=10.1515/9781400881550-016|mr=0624814|isbn=0-691-08264-2|editor1-first=Irwin|editor1-last=Kra|editor2-first=Bernard|editor2-last=Maskit|editor-link2=Bernard Maskit|editor-link1=Irwin Kra|zbl=0467.53035}}}}}}
{{longitem|{{rma|CGT82|tw=4em|{{cite journal|first1=Jeff|last1=Cheeger|first2=Mikhail|last2=Gromov|first3=Michael|last3=Taylor|title=Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds|journal=[[Journal of Differential Geometry]]|volume=17|year=1982|issue=1|pages=15–53|doi=10.4310/jdg/1214436699|doi-access=free|mr=0658471|author-link1=Jeff Cheeger|author-link3=Michael E. Taylor|zbl=0493.53035}}}}}}
{{longitem|{{rma|G82|tw=4em|{{cite journal|first1=Michael|last1=Gromov|url=http://www.numdam.org/item/PMIHES_1982__56__5_0/|title=Volume and bounded cohomology|journal=[[Publications Mathématiques de l'IHÉS|Publications Mathématiques de l'Institut des Hautes Études Scientifiques]]|volume=56|year=1982|pages=5–99|mr=0686042|zbl=0515.53037}}}}}}
{{longitem|{{rma|G83|tw=4em|{{cite journal|first1=Mikhael|last1=Gromov|title=Filling Riemannian manifolds|journal=[[Journal of Differential Geometry]]|volume=18|year=1983|issue=1|pages=1–147|doi=10.4310/jdg/1214509283|doi-access=free|mr=0697984|zbl=0515.53037}}}}}}
{{longitem|{{rma|GL83|tw=4em|{{cite journal|last1=Gromov|first1=Mikhael|last2=Lawson|first2=H. Blaine Jr. |mr=0720933|title=Positive scalar curvature and the Dirac operator on complete Riemannian manifolds|zbl=0538.53047|journal=[[Publications Mathématiques de l'IHÉS|Publications Mathématiques de l'Institut des Hautes Études Scientifiques]]|volume=58|year=1983|pages=83–196|author-link2=Blaine Lawson|doi=10.1007/BF02953774| s2cid=123212001 |url=http://www.numdam.org/item/PMIHES_1983__58__83_0/}}}}}}
{{longitem|{{rma|GM83|tw=4em|{{cite journal|first1=M.|last1=Gromov|first2=V. D.|last2=Milman|url=https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/1037.pdf|title=A topological application of the isoperimetric inequality|journal=[[American Journal of Mathematics]]|volume=105|year=1983|issue=4|pages=843–854|doi=10.2307/2374298| jstor=2374298 |mr=0708367|author-link2=Vitali Milman|zbl=0522.53039}}}}}}
{{longitem|{{rma|G85|tw=4em|{{cite journal|first1=M.|last1=Gromov|url=https://eudml.org/doc/143289|title=Pseudo holomorphic curves in symplectic manifolds|journal=[[Inventiones Mathematicae]]|volume=82|year=1985|issue=2|pages=307–347|doi=10.1007/BF01388806| bibcode=1985InMat..82..307G |mr=0809718|zbl=0592.53025| s2cid=4983969 }}}}}}
{{longitem|{{rma|CG86a|tw=4em|{{cite journal|first1=Jeff|last1=Cheeger|first2=Mikhael|last2=Gromov|title=Collapsing Riemannian manifolds while keeping their curvature bounded. I|journal=[[Journal of Differential Geometry]]|volume=23|year=1986|issue=3|pages=309–346|doi=10.4310/jdg/1214440117|doi-access=free|mr=0852159|author-link1=Jeff Cheeger|zbl=0606.53028}}}}}}
{{longitem|{{rma|CG86b|tw=4em|{{cite journal|first1=Jeff|last1=Cheeger|first2=Mikhael|last2=Gromov|title={{math|L<sup>2</sup>}}-cohomology and group cohomology|journal=[[Topology (journal)|Topology]]|volume=25|year=1986|issue=2|pages=189–215|doi=10.1016/0040-9383(86)90039-X|doi-access=free|mr=0837621|author-link1=Jeff Cheeger|zbl=0597.57020}}}}}}
{{longitem|{{rma|G87|tw=4em|{{cite encyclopedia|first1=M.|last1=Gromov|url=https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/657.pdf|title=Hyperbolic groups|encyclopedia=Essays in group theory|pages=75–263|series=Mathematical Sciences Research Institute Publications|volume=8|publisher=[[Springer-Verlag]]|location=New York|year=1987|doi=10.1007/978-1-4613-9586-7|editor-last1=Gersten|editor-first1=S. M.|isbn=0-387-96618-8|mr=0919829|editor-link1=Stephen M. Gersten|zbl=0634.20015}}}}}}
{{longitem|{{rma|G89|tw=4em|{{cite journal|mr=1001851|last1=Gromov|first1=M.|title=Oka's principle for holomorphic sections of elliptic bundles|journal=[[Journal of the American Mathematical Society]]|volume=2|year=1989|issue=4|pages=851–897|doi=10.1090/S0894-0347-1989-1001851-9|doi-access=free|zbl=0686.32012}}}}}}
{{longitem|{{rma|EG91|tw=4em|{{cite conference|first1=Yakov|last1=Eliashberg|first2=Mikhael|last2=Gromov|url=https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/976.pdf|title=Convex symplectic manifolds|book-title=Several complex variables and complex geometry. Part 2|conference=Proceedings of the Thirty-Seventh Annual Summer Research Institute held at the University of California (Santa Cruz, CA, 10–30 July 1989)|editor1-first=Eric|editor1-last=Bedford|editor2-first=John P.|editor2-last=D'Angelo|editor3-first=Robert E.|editor3-last=Greene|editor4-first=Steven G.|editor4-last=Krantz|pages=135–162|series=Proceedings of Symposia in Pure Mathematics|volume=52|publisher=[[American Mathematical Society]]|location=Providence, RI|year=1991| issue=2 |doi=10.1090/pspum/052.2|mr=1128541|isbn=0-8218-1490-7|editor-link4=Steven G. Krantz|editor-link3=Robert Everist Greene|author-link1=Yakov Eliashberg|zbl=0742.53010}}}}}}
{{longitem|{{rma|G91|tw=4em|{{cite journal|first1=M.|last1=Gromov|title=Kähler hyperbolicity and {{math|L<sup>2</sup>}}-Hodge theory|journal=[[Journal of Differential Geometry]]|volume=33|year=1991|issue=1|pages=263–292|doi=10.4310/jdg/1214446039|doi-access=free|mr=1085144|zbl=0719.53042}}}}}}
{{longitem|{{rma|BGP92|tw=4em|{{cite journal|last1=Burago|author-link1=Yuri Burago|author-link3=Grigori Perelman|first1=Yu.|last2=Gromov|first2=M.|last3=Perelʹman|first3=G.|title=A. D. Aleksandrov spaces with curvatures bounded below|journal=[[Russian Mathematical Surveys]]|year=1992|volume=47|issue=2|pages=1–58|doi=10.1070/RM1992v047n02ABEH000877|mr=1185284|zbl=0802.53018|s2cid=10675933}}}}}}
{{longitem|{{rma|GS92|tw=4em|{{cite journal|mr=1215595|last1=Gromov|first1=Mikhail|last2=Schoen|first2=Richard|title=Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one|zbl=0896.58024|journal=[[Publications Mathématiques de l'IHÉS|Publications Mathématiques de l'Institut des Hautes Études Scientifiques]]|volume=76|year=1992|pages=165–246|doi=10.1007/bf02699433|s2cid=118023776|author-link2=Richard Schoen|url=http://www.numdam.org/item/PMIHES_1992__76__165_0/}}}}}}
{{longitem|{{rma|G93|tw=4em|{{cite conference|last1=Gromov|first1=M.|author-link1=Mikhael Gromov (mathematician)|title=Asymptotic invariants of infinite groups|zbl=0841.20039|book-title=Geometric group theory. Vol. 2|conference=Symposium held at Sussex University (Sussex, July 1991)|pages=1–295|series=London Mathematical Society Lecture Note Series|issue=182|publisher=[[Cambridge University Press]]|location=Cambridge|year=1993|mr=1253544|isbn=0-521-44680-5|editor1-last=Niblo|editor1-first=Graham A.|editor2-last=Roller|editor2-first=Martin A.|url=https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/492.pdf}}<ref>{{cite journal|author=Toledo, Domingo|title=Review: ''Geometric group theory, Vol. 2: Asymptotic invariants of infinite groups'', by M. Gromov|journal=Bull. Amer. Math. Soc. (N.S.)|year=1996|volume=33|issue=3|pages=395–398|doi=10.1090/s0273-0979-96-00669-6|doi-access=free}}</ref>}}}}
{{longitem|{{rma|G96|tw=4em|{{cite encyclopedia|first1=Mikhael|last1=Gromov|title=Carnot-Carathéodory spaces seen from within |url=https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/carnot_caratheodory.pdf|encyclopedia=Sub-Riemannian geometry|pages=79–323|series=Progress in Mathematics|volume=144|publisher=[[Birkhäuser]]|location=Basel|year=1996|doi=10.1007/978-3-0348-9210-0_2|mr=1421823|isbn=3-7643-5476-3|zbl=0864.53025|editor-last1=Bellaïche|editor-first1=André|editor-last2=Risler|editor-first2=Jean-Jacques}}}}}}
{{longitem|{{rma|G99b|tw=4em|{{cite journal|first1=M.|last1=Gromov|title=Endomorphisms of symbolic algebraic varieties|journal=[[Journal of the European Mathematical Society]]|volume=1|year=1999|issue=2|pages=109–197|doi=10.1007/PL00011162 |doi-access=free|mr=1694588|zbl=0998.14001}}}}}}
{{longitem|{{rma|G00|tw=4em|{{cite conference|first1=Misha|last1=Gromov| title=Spaces and questions |url=https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/SpacesandQuestions.pdf|book-title=Visions in mathematics: GAFA 2000 Special Volume, Part I|conference=Proceedings of the meeting held at Tel Aviv University, Tel Aviv, 25 August – 3 September 1999|series=[[Geometric and Functional Analysis]]|pages=118–161|doi=10.1007/978-3-0346-0422-2_5|mr=1826251|editor1-first=N.|editor1-last=Alon|editor2-first=J.|editor2-last=Bourgain|editor3-first=A.|editor3-last=Connes|editor4-first=M.|editor4-last=Gromov|editor5-first=V.|editor5-last=Milman|editor-link1=Noga Alon|editor-link2=Jean Bourgain|editor-link3=Alain Connes|editor-link5=Vitali Milman|publisher=[[Birkhäuser]]|location=Basel|year=2000|isbn=978-3-0346-0421-5|zbl=1006.53035}}}}}}
{{longitem|{{rma|G03a|tw=4em|{{cite journal|first1=M.|last1=Gromov|title=Isoperimetry of waists and concentration of maps|journal=[[Geometric and Functional Analysis]]|volume=13|year=2003|issue=1|pages=178–215|doi=10.1007/s000390300004|doi-access=free|mr=1978494|zbl=1044.46057}} {{erratum|doi=10.1007/s00039-009-0703-1|checked=yes}}
* See also: {{cite journal|mr=2784762|last1=Memarian|first1=Yashar|title=On Gromov's waist of the sphere theorem|journal=Journal of Topology and Analysis|volume=3|year=2011|issue=1|pages=7–36|doi=10.1142/S1793525311000507|arxiv=0911.3972|zbl=1225.46055| s2cid=115178123 }}}}}}
{{longitem|{{rma|G03b|tw=4em|{{cite journal|first1=Mikhaïl|last1=Gromov|url=https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/1024.pdf|title=On the entropy of holomorphic maps|journal=[[L'Enseignement mathématique|L'Enseignement Mathématique. Revue Internationale]]|series=2e Série|volume=49|year=2003|issue=3–4|pages=217–235|mr=2026895|zbl=1080.37051}}}}}}
{{longitem|{{rma|G03c|tw=4em|{{cite journal|first1=M.|last1=Gromov|title=Random walk in random groups|journal=[[Geometric and Functional Analysis]]|volume=13|year=2003|issue=1|pages=73–146|doi=10.1007/s000390300002|doi-access=free|mr=1978492|zbl=1122.20021}}
* See also: {{cite journal|mr=1978493|last1=Silberman|first1=L.|title=Addendum to: ''Random walk in random groups'' by M. Gromov|journal=[[Geometric and Functional Analysis]]|volume=13|year=2003|issue=1|pages=147–177|doi=10.1007/s000390300003|citeseerx=10.1.1.124.6500|zbl=1124.20027| s2cid=120354073 }}}}}}
{{refend}}

==Notes==
{{Reflist}}

==References==
{{Refbegin}}
* [[Marcel Berger]], "[https://www.ams.org/notices/200002/fea-berger.pdf Encounter with a Geometer, Part I]", ''[[AMS Notices]]'', Volume 47, Number 2
* Marcel Berger, "[https://www.ams.org/notices/200003/fea-berger.pdf Encounter with a Geometer, Part II]"", ''AMS Notices'', Volume 47, Number 3
{{Refend}}

==External links==
{{Commons category-inline|Mikhail Leonidovich Gromov}}
* [https://www.ihes.fr/~gromov/ Personal page at Institut des Hautes Études Scientifiques]
* [https://web.archive.org/web/20090408203911/http://as.nyu.edu/object/IO_3199.html Personal page at NYU]
* {{MathGenealogy|id=14999|name=Mikhail Gromov}}
* [http://www.pdmi.ras.ru/~vershik/gromov-vershik.pdf Anatoly Vershik, "Gromov's Geometry"]

{{Wolf Prize in Mathematics}}
{{Abel Prize laureates}}
{{Veblen Prize recipients}}
{{FRS 2011}}
{{Authority control}}

{{DEFAULTSORT:Gromov, Mikhail}}
[[Category:1943 births]]
[[Category:Living people]]
[[Category:Jewish French scientists]]
[[Category:People from Boksitogorsk]]
[[Category:Russian people of Jewish descent]]
[[Category:Russian emigrants to France]]
[[Category:Foreign associates of the National Academy of Sciences]]
[[Category:Foreign members of the Russian Academy of Sciences]]
[[Category:Kyoto laureates in Basic Sciences]]
[[Category:Differential geometers]]
[[Category:Russian mathematicians]]
[[Category:20th-century French mathematicians]]
[[Category:21st-century French mathematicians]]
[[Category:French people of Russian-Jewish descent]]
[[Category:Group theorists]]
[[Category:New York University faculty]]
[[Category:Wolf Prize in Mathematics laureates]]
[[Category:Geometers]]
[[Category:Members of the French Academy of Sciences]]
[[Category:Members of the Norwegian Academy of Science and Letters]]
[[Category:Abel Prize laureates]]
[[Category:Foreign members of the Royal Society]]
[[Category:Soviet mathematicians]]
[[Category:University of Maryland, College Park faculty]]
[[Category:Russian scientists]]