Mikhael Gromov (mathematician)

Source: Wikipedia, the free encyclopedia.
For other people with the same name, see Gromov.

Mikhael Gromov
Михаил Громов
Vladimir Rokhlin
Doctoral studentsDenis Auroux
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 Russian-French mathematician known for his work in geometry, 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

pathologists.[1] His mother was the cousin of World Chess Champion Mikhail Botvinnik, as well as of the mathematician Isaak Moiseevich Rabinovich.[2] 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.[3] When Gromov was nine years old,[4] 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.[3]

Gromov studied mathematics at

Vladimir Rokhlin.[5]

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.[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 to move 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 to 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.[6]

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 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.[8] He adopted French citizenship in 1992.[9]

Work

Gromov's style of geometry often features a "coarse" or "soft" viewpoint, analyzing asymptotic or large-scale properties.

mathematical biology,[10] the structure of the brain and the thinking process, and the way scientific ideas evolve.[6]

Motivated by

holomorphic maps.[G89] His work initiated a renewed study of the Oka–Grauert theory, which had been introduced in the 1950s.[13][14]

Gromov and Vitali Milman gave a formulation of the concentration 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 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 manifolds 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.[15]

Since the seminal 1964 publication of James Eells and Joseph Sampson on harmonic 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 into metric spaces would imply new results on discrete groups, following 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.[19] A sample application of Gromov and Schoen's methods is the fact that lattices in the isometry group of the quaternionic hyperbolic space are arithmetic.[GS92]

Riemannian geometry

In 1978, Gromov introduced the notion of almost flat manifolds.[G78] 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 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.[20][21][22]

In 1979,

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.[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.[24]

In 1981, Gromov identified topological restrictions, based upon Betti numbers, on manifolds which admit Riemannian metrics of nonnegative sectional curvature.[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 comparison theorem, together with the Bishop–Gromov inequality on volume of geodesic balls.[25] 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 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]

In

injectivity radius estimate for closed manifolds.[27] Cheeger, Gromov, and Michael Taylor localized Cheeger's estimate, showing how to use Bishop−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 that Grigori Perelman's "noncollapsing theorem" for Ricci flow, which controls volume, is sufficient to allow applications of Richard Hamilton's compactness theory.[31][32][33] Cheeger, Gromov, and Taylor applied their injectivity radius estimate to prove Gaussian control of the heat kernel, although these estimates were later improved by Li and Yau as an application of their gradient estimates.[26]

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"[G83] Gromov proved that every essential manifold with a Riemannian metric contains a closed non-contractible geodesic of length at most .[34]

Gromov−Hausdorff convergence and geometric group theory

In 1981, Gromov 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 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.[G93]

Gromov's compactness theorem had a deep impact on the field of

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.[35][36]

Another consequence 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]

Symplectic geometry

Gromov's theory of

Gromov-Witten theory, which is a widely studied topic reaching into string theory, algebraic geometry, and symplectic geometry.[39][40][41] From a different perspective, Gromov's work was also inspirational for much of Andreas Floer's work.[42]

h-principle to hold for the problem of constructing certain symplectomorphisms. They also introduced analogous notions in contact geometry; the existence of convex contact structures was later studied by Emmanuel Giroux.[43]

Prizes and honors

Prizes

Honors

See also

Publications

Books

BGS85.
Zbl 0591.53001.[50]
G86.
Gromov, Mikhael (1986). Partial differential relations. Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Vol. 9. Berlin:
Zbl 0651.53001.[51]
G99a.
Gromov, Misha (1999).
Zbl 0953.53002.[52]
G18.
Gromov, Misha (2018). Great circle of mysteries. Mathematics, the world, the mind.
Zbl 1433.00004
.

Major articles

G69.
Gromov, M. L. (1969). "Stable mappings of foliations into manifolds".
Zbl 0205.53502
.
G78.
Gromov, M. (1978). "Almost flat manifolds".
Zbl 0432.53020
.
GL80a.
Gromov, Mikhael;
Zbl 0445.53025
.
GL80b.
Gromov, Mikhael;
Zbl 0463.53025
.
G81a.
G81c.
CGT82.
G82.
Gromov, Michael (1982). "Volume and bounded cohomology".
Zbl 0515.53037
.
G83.
Gromov, Mikhael (1983). "Filling Riemannian manifolds".
Zbl 0515.53037
.
GL83.
Gromov, Mikhael;
Zbl 0538.53047
.
GM83.
Gromov, M.;
Zbl 0522.53039
.
CG86a.
CG86b.
G87.
Gromov, M. (1987). "Hyperbolic groups" (PDF). In
Zbl 0634.20015
.
EG91.
BGP92.
GS92.
Gromov, Mikhail;
Zbl 0896.58024
.
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:
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:
Zbl 0864.53025
.
G00.
Gromov, Misha (2000). "Spaces and questions" (PDF). In
Zbl 1006.53035
.
G03b.
Gromov, Mikhaïl (2003). "On the entropy of holomorphic maps" (PDF).
Zbl 1080.37051
.

Notes

  1. ISBN 978-3-642-39448-5. (also available on Gromov's homepage: link
    )
  2. ^ 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).
  3. ^ a b Newsletter of the European Mathematical Society, No. 73, September 2009, p. 19
  4. ^
    ISSN 1950-6244
    .
  5. ^ "Mikhael Gromov Receives the 2009 Abel Prize" (PDF). CIMS Newsletter. Courant Institute of Mathematical Sciences. Spring 2009. p. 1.
  6. ^ a b c Roberts, Siobhan (22 December 2014). "Science Lives: Mikhail Gromov". Simons Foundation.
  7. ISBN 9782701130538
    .
  8. ^ O'Connor, John J.; Robertson, Edmund F., "Mikhael Gromov (mathematician)", MacTutor History of Mathematics Archive, University of St Andrews
  9. ^ "Mikhail Leonidovich Gromov". abelprize.no.
  10. ^
    Notices of the AMS
    , 57 (3): 391–403, March 2010.
  11. MR 1660090
    .
  12. MR 1909245
    .
  13. S2CID 118671586
    .
  14. MR 3700709
    .
  15. ^ Talagrand, Michel A new look at independence. Ann. Probab. 24 (1996), no. 1, 1–34.
  16. ^ Eells, James, Jr.; Sampson, J. H. Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86 (1964), 109–160.
  17. ^ 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.
  18. ^ Kevin Corlette. Archimedean superrigidity and hyperbolic geometry. Ann. of Math. (2) 135 (1992), no. 1, 165–182.
  19. ^ 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.
  20. ^ 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.
  21. ^ Peter Buser and Hermann Karcher. Gromov's almost flat manifolds. Astérisque, 81. Société Mathématique de France, Paris, 1981. 148 pp.
  22. ^ 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.
  23. Zbl 0423.53032
    .
  24. Zbl 0688.57001
    .
  25. ^ Grove, Karsten; Shiohama, Katsuhiro A generalized sphere theorem. Ann. of Math. (2) 106 (1977), no. 2, 201–211.
  26. ^ a b Li, Peter; Yau, Shing-Tung. On the parabolic kernel of the Schrödinger operator. Acta Math. 156 (1986), no. 3-4, 153–201.
  27. ^ Cheeger, Jeff. Finiteness theorems for Riemannian manifolds. Amer. J. Math. 92 (1970), 61–74.
  28. ^ Anderson, Michael T. Ricci curvature bounds and Einstein metrics on compact manifolds. J. Amer. Math. Soc. 2 (1989), no. 3, 455–490.
  29. ^ 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.
  30. ^ Tian, G. On Calabi's conjecture for complex surfaces with positive first Chern class. Invent. Math. 101 (1990), no. 1, 101–172.
  31. ^ Grisha Perelman. The entropy formula for the Ricci flow and its geometric applications.
  32. ^ Hamilton, Richard S. A compactness property for solutions of the Ricci flow. Amer. J. Math. 117 (1995), no. 3, 545–572.
  33. ^ 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.
  34. ^ Katz, M. Systolic geometry and topology. With an appendix by J. Solomon. Mathematical Surveys and Monographs, volume 137. American Mathematical Society, 2007.
  35. ^ Pierre Pansu. Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. of Math. (2) 129 (1989), no. 1, 1–60.
  36. ^ 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.
  37. ^ Uhlenbeck, Karen K. Connections with Lp bounds on curvature. Comm. Math. Phys. 83 (1982), no. 1, 31–42.
  38. ^ Sacks, J.; Uhlenbeck, K. The existence of minimal immersions of 2-spheres. Ann. of Math. (2) 113 (1981), no. 1, 1–24.
  39. ^ 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.
  40. ^ 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.
  41. ^ Bourgeois, F.; Eliashberg, Y.; Hofer, H.; Wysocki, K.; Zehnder, E. Compactness results in symplectic field theory. Geom. Topol. 7 (2003), 799–888.
  42. ^ Floer, Andreas. Morse theory for Lagrangian intersections. J. Differential Geom. 28 (1988), no. 3, 513–547.
  43. ^ Giroux, Emmanuel. Convexité en topologie de contact. Comment. Math. Helv. 66 (1991), no. 4, 637–677.
  44. ^ Gromov Receives Nemmers Prize
  45. ^ "2009: Mikhail Leonidovich Gromov". www.abelprize.no.
  46. ^ Professor Mikhail Gromov ForMemRS | Royal Society
  47. ^ | National Academy of sciences of Ukraine, communication
  48. ^ Mikhaël Gromov — Membre de l’Académie des sciences
  49. ^ "Turán Memorial Lectures".
  50. doi:10.1090/s0273-0979-1987-15603-5
    .
  51. doi:10.1090/s0273-0979-1988-15654-6
    .
  52. doi:10.1090/s0273-0979-01-00904-1
    .
  53. doi:10.1090/s0273-0979-96-00669-6
    .

References

External links

Media related to Mikhail Leonidovich Gromov at Wikimedia Commons

Retrieved from "https://en.wikipedia.org/w/index.php?title=Mikhael_Gromov_(mathematician)&oldid=1220172798"