Marcel Berger

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Marcel Berger
Marcel Berger.jpeg
Marcel Berger in 1968
(photo from MFO)
Born(1927-04-14)14 April 1927
Paris, France
Died15 October 2016(2016-10-15) (aged 89)
NationalityFrance
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]

Contents

Awards and honors

Selected publications

See also

Related Research Articles

Riemannian geometry Branch of differential geometry

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 G2 manifold is a seven-dimensional Riemannian manifold with holonomy group contained in G2. The group is one of the five exceptional simple Lie groups. It can be described as the automorphism group of the octonions, or equivalently, as a proper subgroup of special orthogonal group SO(7) that preserves a spinor in the eight-dimensional spinor representation or lastly as the subgroup of the general linear group GL(7) which preserves the non-degenerate 3-form , the associative form. The Hodge dual, is then a parallel 4-form, the coassociative form. These forms are calibrations in the sense of Reese Harvey and H. Blaine Lawson, and thus define special classes of 3- and 4-dimensional submanifolds.

Mikhael Gromov (mathematician) Russian-French mathematician

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Systolic geometry

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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 ≥ k, where k is some real number. By definition, these spaces are locally compact complete length spaces where the lower curvature bound is defined via comparison of geodesic triangles in the space to geodesic triangles in standard constant-curvature Riemannian surfaces.

Introduction to systolic geometry

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.

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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.

<i>Metric Structures for Riemannian and Non-Riemannian Spaces</i> Book by Michail Gromov

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 Structures métriques pour les variétés riemanniennes, by CEDIC (Paris).

Edmond Bonan

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Sylvestre Gallot French mathematician

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 French mathematician

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.

References

  1. "Archived copy". Archived from the original on 2016-10-22. Retrieved 2016-10-19.CS1 maint: archived copy as title (link)
  2. "Archived copy". Archived from the original on 2016-10-24. Retrieved 2016-10-23.CS1 maint: archived copy as title (link)
  3. Berger, Marcel Y. (1990). "Convexity". Amer. Math. Monthly. 97: 650–678. doi:10.2307/2324573.
  4. 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.
  5. 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.

Further reading