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Mikhael Leonidovich Gromov (also Mikhail Gromov, Michael Gromov or Misha Gromov; russian: link=no|Михаи́л Леони́дович Гро́мов; born 23 December 1943) is a Russian-French mathematician known for his work in geometry, analysis and group theory. He is a permanent member of IHÉS in France and a Professor of Mathematics at New York University. Gromov has won several prizes, including the Abel Prize in 2009 "for his revolutionary contributions to geometry".

Biography

Mikhail Gromov was born on 23 December 1943 in Boksitogorsk, Soviet Union. His father Leonid Gromov and his Jewish mother Lea Rabinovitz were pathologists. His mother was the cousin of World Chess Champion Mikhail Botvinnik, as well as of the mathematician Isaak Moiseevich Rabinovich. 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.Newsletter of the European Mathematical Society, No. 73, September 2009, p. 19
/ref> When Gromov was nine years old, 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. 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. 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. 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. He changed his last name to that of his mother. When the request was granted in 1974, he moved directly to New York where a position had been arranged for him at Stony Brook. 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. He adopted French citizenship in 1992.

Work

Gromov's style of geometry often features a "coarse" or "soft" viewpoint, analyzing asymptotic or large-scale properties. He is also interested in mathematical biology,. the structure of the brain and the thinking process, and the way scientific ideas evolve. Motivated by Nash and Kuiper's C1 embedding theorem and Stephen Smale's early results, Gromov introduced in 1973 the method of convex integration and the h-principle, a very general way to solve underdetermined partial differential equations and the basis for a geometric theory of these equations. One application is the Gromov–Lees Theorem, named for him and Jack Alexander Lees, concerning Lagrangian immersions and a one-to-one correspondence between the connected components of spaces. In 1978, Gromov introduced the notion of almost flat manifolds. The famous quarter-pinched sphere theorem in Riemannian geometry says that if a complete Riemannian manifold has sectional curvatures which are all sufficiently close to a given positive constant, then 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. In 1979, Richard Schoen and Shing-Tung Yau showed that the class of smooth manifolds which admit Riemannian metrics of positive scalar curvature is topologically rich. In particular, they showed that this class is closed under the operation of connected sum and of surgery in codimension at least three. 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. They also showed how purely topological results such as Stephen Smale's h-cobordism theorem could then be applied to draw conclusions such as the fact that every closed and simply-connected smooth manifold of dimension 5, 6, or 7 has a Riemannian metric of positive scalar curvature. In 1981, Gromov formally introduced the Gromov–Hausdorff metric, which endows the set of all metric spaces with the structure of a metric space. 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. Gromov's compactness theorem had a deep impact on the field of geometric group theory. He applied it to understand the asymptotic geometry of the word metric of a group of polynomial growth, by taking the limit of well-chosen rescalings of the metric. By tracking the limits of isometries of the word metric, he was able to show that the limiting metric space has unexpected continuities, and in particular that its isometry group is a Lie group. As a consequence he was able to settle the Milnor-Wolf conjecture as posed in the 1960s, which asserts that any such group is virtually nilpotent. Using ultralimits, similar asymptotic structures can be studied for more general metric spaces. Important developments on this topic were given by Bruce Kleiner, Bernhard Leeb, and Pierre Pansu, among others. 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. 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. Along with Eliyahu Rips, Gromov introduced the notion of hyperbolic groups. 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" Gromov proved that every essential manifold M with a Riemannian metric contains a closed non-contractible geodesic of length at most $C\left(n\right)Vol\left(M\right)^$. Gromov founded the field of symplectic topology by introducing the theory of pseudoholomorphic curves. This led to Gromov–Witten invariants, which are used in string theory, and to his non-squeezing theorem.

Prizes and honors

Prizes

* 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) * Bolyai Prize in 2005 * Abel Prize in 2009 “for his revolutionary contributions to geometry”

Honors

* 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) * Member of the French Academy of Sciences (1997) * Delivered the 2007 Paul Turán Memorial Lectures.

* 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 δ-hyperbolic space * Filling area conjecture * Mean dimension

Publications

Books * Werner Ballmann, Mikhael Gromov, and Viktor Schroeder. Manifolds of nonpositive curvature. Progress in Mathematics, 61. Birkhäuser Boston, Inc., Boston, MA, 1985. vi+263 pp. ; * Misha Gromov. Metric structures for Riemannian and non-Riemannian spaces. Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, 152. Birkhäuser Boston, Inc., Boston, MA, 1999. xx+585 pp. ; * Mikhael Gromov. Partial differential relations. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 9. Springer-Verlag, Berlin, 1986. x+363 pp. ; * Misha Gromov. Great circle of mysteries. Mathematics, the world, the mind. Birkhäuser/Springer, Cham, 2018. vii+202 pp. ; Major articles

Notes

References

* 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