Mikhail Leonidowitsch Gromov

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Mikhail Leonidowitsch Gromov

Mikhail Leonidovich Gromov (also Michael or Misha Gromov , Russian Михаил Леонидович Громов , mostly Mikhail Gromov and Mikhail Gromov cited; * 23. December 1943 in Boksitogorsk , RSFSR , Soviet Union ) is a Russian - French mathematician who mainly to differential geometry , Analysis and Group theory researches. Gromow has been a French citizen since 1992. He is considered to be one of the leading surveyors and has been awarded the Abel Prize 2009 , among other things .


Gromov attended school 217 ( Petri School ) in Leningrad. He then studied at the local university , where he graduated in 1965 and received his doctorate in 1969 under Vladimir Rochlin . From 1967 he was assistant professor there, in 1973 he completed his habilitation (Soviet doctoral degree) in Leningrad. In 1974 he became professor at the University of Stony Brook in New York , 1981 at the University of Paris and from 1982 until today at the IHES in Bures-sur-Yvette near Paris, where he is still a permanent member. He also became Jay Gould Professor at the Courant Institute of Mathematical Sciences of New York University in 2008 .


In the geometric group theory (as the founder, he in expansion of the true end of the 1980s, combinatorial group theory , in particular from the work of Max Dehn was the beginning of the 20th century) studied Gromov groups of polynomial growth order and introduced the concept of hyperbolic groups a . In symplectic topology , he established the term pseudoholomorphic curve . His homotopy principle for differential relations is important in the theory of partial differential equations ; Gromow expanded it older approaches, including that of John Nash .

Especially in Riemannian geometry , Gromow opened up many new perspectives, for example by examining asymptotic and global aspects and formulating them in inequalities. He concepts come as fast flatness ( almost flatness ) of metrics and contexts and simplicial volume . He also dealt with scrolling , sub-Riemannian manifolds and index theory of operators .

The laudation for the 2009 Abel Prize named the following contributions from Gromow to mathematics:

  • played a crucial role in the creation of modern global Riemannian geometry. His solutions to important problems of global geometry were based on new general concepts named after him, such as the convergence of Riemannian manifolds and the principle of compactness.
  • one of the founders of symplectic geometry. Holomorphic curves , previously an important tool in the geometry of complex manifolds, were generalized by Gromow in his famous 1985 work to J-holomorphic curves in symplectic manifolds . This led to the theory of the Gromow-Witten invariants , now an extremely active area with links to modern quantum field theory and the creation of symplectic topology, and it permeated and changed many other areas of mathematics.
  • his work on groups of polynomial growth , which with the ideas introduced there led to a completely different view of discrete infinite groups. He discovered the geometry of the finally generated groups and solved various outstanding problems. Its geometrical approach made complicated combinatorial arguments much more natural and effective.

In 1970 ( A topological technique for the construction of solutions of differential equations and inequalities ), 1978 ( Synthetic geometry in Riemannian manifolds ) and 1983 ( Infinite groups as geometric objects ) he was invited speaker at the International Congress of Mathematicians (ICM). In 1986 in Berkeley he gave a plenary lecture on "Soft and Hard Symplectic Geometry" at the ICM . In 2012 he gave a plenary lecture at the European Congress of Mathematicians (ECM) in Krakow ( In a search for a structure ).


In 2018, Gromow was one of the 200 signatories of an appeal in the newspaper Le Monde , in which it was warned of drastic consequences such as the extinction of human species, unless a quick rethinking of problem areas such as climate change and the extinction of species and wider planetary limits occurs.

Awards and memberships

He is an honorary doctor from the Universities of Geneva and Tel-Aviv.

He is a member of the Académie des Sciences (1997), the Academia Europaea , the American Academy of Arts and Sciences (1989), the National Academy of Sciences , the Russian Academy of Sciences (2011), the Norwegian Academy of Sciences and an honorary member of London Mathematical Society .



  • Metric structures for Riemannian and non-Riemannian spaces (attachments by M. Katz, P. Pansu, S. Semmes), Birkhäuser 1999
  • Partial Differential Relations , Springer Verlag, Results of Mathematics and its Frontier Areas, 1986
  • Asymptotic invariants of infinite groups. Geometric group theory, Vol. 2 (Sussex, 1991), London Math. Soc. Lecture Note Ser., 182, Cambridge Univ. Press, Cambridge, 1993.
  • Spaces and Questions , in Noga Alon u. a. (Editor) Visions in Mathematics , Geometric and functional analysis, special volume, GAFA 2000, Birkhäuser, Volume 1, pp. 118-161
  • with Werner Ballmann , Viktor Schroeder: Manifolds of non positive curvature , Birkhäuser 1985

Publications (selection)

  • Stable mappings of foliations into manifolds. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 33 1969 707-734.
  • with Wladimir Rochlin : Imbeddings and immersions in Riemannian geometry. (Russian) Uspehi Mat. Nauk 25 1970 no. 5 (155), 3-62.
  • with Blaine Lawson : The classification of simply connected manifolds of positive scalar curvature. Ann. of Math. (2) 111 (1980) no. 3, 423-434.
  • Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. No. 53: 53-73 (1981).
  • with Jeff Cheeger , Michael E. 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.
  • Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. No. 1982, 56: 5-99 (1983).
  • Filling Riemannian manifolds. J. Differential Geom. 18 (1983) no. 1, 1-147.
  • with Blaine Lawson : Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Inst. Hautes Études Sci. Publ. Math. No. 58: 83-196 (1984) (1983).
  • Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82 (1985) no. 2, 307-347.
  • with Jeff Cheeger : L 2 -cohomology and group cohomology. Topology 25 (1986) no. 2, 189-215.
  • Hyperbolic groups. Essays in group theory, 75-263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987.
  • with Juri Burago , Grigori Perelman : AD Aleksandrov spaces with curvatures bounded below. (Russian) Uspekhi Mat. Nauk 47 (1992), no.2 (284), 3--51, 222
  • with Richard Schoen : Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one. Inst. Hautes Études Sci. Publ. Math. No. 1992, 76: 165-246.
  • Carnot-Carathéodory spaces seen from within. Sub-Riemannian geometry, 79–323, Progr. Math., 144, Birkhäuser, Basel, 1996.
  • Random walk in random groups. Geom. Funct. Anal. 13 (2003), no. 1, 73-146.


From October 21, 2011 to March 18, 2012, the Paris Fondation Cartier hosted the exhibition Mathematics: A Beautiful Elsewhere , for which the film director and artist David Lynch, among others, contributed exhibits in collaboration with Gromow.

See also

Web links

Some papers available online:

Individual evidence

  1. https://www.lemonde.fr/idees/article/2018/09/03/le-plus-grand-defi-de-l-histoire-de-l-humanite-l-appel-de-200-personnalites -pour-sauver-la-planete_5349380_3232.html