Marcel Berger | |
---|---|

Born | Paris, France | 14 April 1927

Died | 15 October 2016 89) | (aged

Nationality | France |

Known for | Berger–Kazdan comparison theorem |

Awards | Leconte Prize (1978) |

Scientific career | |

Fields | Mathematics |

Institutions | Institut des Hautes Études Scientifiques |

Doctoral advisor | André Lichnerowicz |

Doctoral students | Jean-Pierre Bourguignon Yves Colin de Verdière Sylvestre Gallot François Labourie Pierre Pansu |

**Marcel Berger** (14 April 1927 – 15 October 2016) was a French mathematician, doyen of French differential geometry, and a former director of the Institut des Hautes Études Scientifiques (IHÉS), France. Formerly residing in Le Castera in Lasseube, Berger was instrumental in Mikhail Gromov's accepting positions both at the University of Paris and at the IHÉS.^{ [1] }

- 1956 Prix Peccot, Collège de France
- 1962 Prix Maurice Audin
- 1969 Prix Carrière, Académie des Sciences
- 1978 Prix Leconte, Académie des Sciences
- 1979 Prix Gaston Julia
- 1979–1980 President of the French Mathematical Society.
^{ [2] } - 1991 Lester R. Ford Award
^{ [3] }

- Berger, M.: Geometry revealed. Springer, 2010.
- Berger, M.: What is... a Systole? Notices of the AMS 55 (2008), no. 3, 374–376. online text
- Berger, Marcel (2003).
*A Panoramic View of Riemannian Geometry*. Springer-Verlag. ISBN 3-540-65317-1 xxiv+824 pp.CS1 maint: postscript (link)^{ [4] }^{ [5] } - Berger, Marcel (Feb 2000). "Encounter with a Geometer, Part I" (PDF).
*Notices of the AMS*.**47**(2): 183–194. - Berger, Marcel (Mar 2000). "Encounter with a Geometer, Part II" (PDF).
*Notices of the AMS*.**47**(3): 326–340. - Berger, Marcel; Gauduchon, Paul; Mazet, Edmond: Le spectre d'une variété riemannienne. (French) Lecture Notes in Mathematics, Vol. 194 Springer-Verlag, Berlin-New York 1971.
- Berger, Marcel: Sur les groupes d'holonomie homogène des variétés à connexion affine et des variétés riemanniennes. (French) Bull. Soc. Math. France 83 (1955), 279–330.
- Berger, Marcel: Les espaces symétriques noncompacts. (French) Ann. Sci. École Norm. Sup. (3) 74 1957 85–177.
- Berger, Marcel; Gostiaux, Bernard: Differential geometry: manifolds, curves, and surfaces. Translated from the French by Silvio Levy. Graduate Texts in Mathematics, 115. Springer-Verlag, New York, 1988. xii+474 pp. ISBN 0-387-96626-9
- Berger, Marcel: Geometry. II. Translated from the French by M. Cole and S. Levy. Universitext. Springer-Verlag, Berlin, 1987.
- Berger, M.: Les variétés riemanniennes homogènes normales simplement connexes à courbure strictement positive. (French) Ann. Scuola Norm. Sup. Pisa (3) 15 1961 179–246.
- Berger, Marcel: Geometry. I. Translated from the French by M. Cole and S. Levy. Universitext. Springer-Verlag, Berlin, 1987. xiv+428 pp. ISBN 3-540-11658-3
- Berger, Marcel: Systoles et applications selon Gromov. (French) [Systoles and their applications according to Gromov] Séminaire Bourbaki, Vol. 1992/93. Astérisque No. 216 (1993), Exp. No. 771, 5, 279–310.
- Berger, Marcel: Geometry. I. Translated from the 1977 French original by M. Cole and S. Levy. Corrected reprint of the 1987 translation. Universitext. Springer-Verlag, Berlin, 1994. xiv+427 pp. ISBN 3-540-11658-3
- Berger, Marcel: Riemannian geometry during the second half of the twentieth century. Reprint of the 1998 original. University Lecture Series, 17. American Mathematical Society, Providence, Rhode Island, 2000. x+182 pp. ISBN 0-8218-2052-4
- Besse, A.L.: Einstein Manifolds. Springer-Verlag, Berlin, 1987. ISBN 0-387-15279-2

**Riemannian geometry** is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a *Riemannian metric*, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.

In mathematics, **Gromov–Hausdorff convergence**, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence.

In differential geometry, a ** G_{2} manifold** is a seven-dimensional Riemannian manifold with holonomy group contained in

**Mikhael Leonidovich Gromov** 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.

In mathematics, **systolic geometry** is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations. See also a slower-paced Introduction to systolic geometry.

In differential geometry, a **quaternion-Kähler manifold** is a Riemannian 4n-manifold whose Riemannian holonomy group is a subgroup of Sp(*n*)·Sp(1) for some . Here Sp(*n*) is the sub-group of consisting of those orthogonal transformations that arise by **left**-multiplication by some quaternionic matrix, while the group of unit-length quaternions instead acts on quaternionic -space by **right** scalar multiplication. The Lie group generated by combining these actions is then abstractly isomorphic to .

In mathematics, a **Spin(7)-manifold** is an eight-dimensional Riemannian manifold with the exceptional holonomy group Spin(7). Spin(7)-manifolds are Ricci-flat and admit a parallel spinor. They also admit a parallel 4-form, known as the Cayley form, which is a calibrating form for a special class of submanifolds called Cayley cycles.

In Riemannian geometry, **Gromov's (pre)compactness theorem** states that the set of compact Riemannian manifolds of a given dimension, with Ricci curvature ≥ *c* and diameter ≤ *D* is relatively compact in the Gromov–Hausdorff metric. It was proved by Mikhail Gromov in 1981.

In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality

In geometry, **Alexandrov spaces with curvature ≥ k** form a generalization of Riemannian manifolds with sectional curvature ≥

**Systolic geometry** is a branch of differential geometry, a field within mathematics, studying problems such as the relationship between the area inside a closed curve *C*, and the length or perimeter of *C*. Since the area *A* may be small while the length *l* is large, when *C* looks elongated, the relationship can only take the form of an inequality. What is more, such an inequality would be an upper bound for *A*: there is no interesting lower bound just in terms of the length.

**Arthur Besse** is a pseudonym chosen by a group of French differential geometers, led by Marcel Berger, following the model of Nicolas Bourbaki. A number of monographs have appeared under the name.

In differential geometry, **systolic freedom** refers to the fact that closed Riemannian manifolds may have arbitrarily small volume regardless of their systolic invariants. That is, systolic invariants or products of systolic invariants do not in general provide universal lower bounds for the total volume of a closed Riemannian manifold.

**Spectral geometry** is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators. The case of the Laplace–Beltrami operator on a closed Riemannian manifold has been most intensively studied, although other Laplace operators in differential geometry have also been examined. The field concerns itself with two kinds of questions: direct problems and inverse problems.

**Mikhail "Mischa" Gershevich Katz** is an Israeli mathematician, a professor of mathematics at Bar-Ilan University. His main interests are differential geometry, geometric topology and mathematics education; he is the author of the book *Systolic Geometry and Topology*, which is mainly about systolic geometry. The Katz–Sabourau inequality is named after him and Stéphane Sabourau.

* Metric Structures for Riemannian and Non-Riemannian Spaces* is a book in geometry by Mikhail Gromov. It was originally published in French in 1981 under the title

**Edmond Bonan** is a French mathematician, known particularly for his work on special holonomy.

**Sylvestre F. L. Gallot** is a French mathematician, specializing in differential geometry. He is an emeritus professor at the Institut Fourier of the Université Grenoble Alpes, in the Geometry and Topology section.

In mathematics, the **Cartan–Hadamard conjecture** is a fundamental problem in Riemannian geometry and Geometric measure theory which states that the classical isoperimetric inequality may be generalized to spaces of nonpositive sectional curvature, known as Cartan–Hadamard manifolds. The conjecture, which is named after French mathematicians Élie Cartan and Jacques Hadamard, may be traced back to work of André Weil in 1926.

**Paul Gauduchon** is a French mathematician, known for his work in the field of differential geometry. He is particularly known for his introduction of *Gauduchon metrics* in hermitian geometry. His textbook on spectral geometry, written with Marcel Berger and Edmond Mazet, is a standard reference in the field.

- ↑ "Archived copy". Archived from the original on 2016-10-22. Retrieved 2016-10-19.CS1 maint: archived copy as title (link)
- ↑ "Archived copy". Archived from the original on 2016-10-24. Retrieved 2016-10-23.CS1 maint: archived copy as title (link)
- ↑ Berger, Marcel Y. (1990). "Convexity".
*Amer. Math. Monthly*.**97**: 650–678. doi:10.2307/2324573. - ↑ Osserman, Robert (2005-01-01). "Review of A Panoramic View of Riemannian Geometry".
*SIAM Review*.**47**(1): 186–188. doi:10.1137/SIREAD000047000001000163000001. JSTOR 20453617. - ↑ Giblin, Peter (2005-01-01). "Review of A Panoramic View of Riemannian Geometry".
*The Mathematical Gazette*.**89**(514): 162–163. doi:10.1017/s0025557200177289. JSTOR 3620688.

- Claude LeBrun (Editor and Translator). "Marcel Berger Remembered",
*Notices of the American Mathematical Society*, December 2017, Volume 64, Number 11, pp. 1285–1295.

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