Shlomo Sternberg

Shlomo Sternberg
Born 1936
Fields Mathematics

Harvard University,
New York University

University of Chicago
Alma mater Johns Hopkins University
(PhD 1955)
Doctoral advisor Aurel Friedrich Wintner
Doctoral students Victor Guillemin (Harvard, 1962)
Ravindra Kulkarni (Harvard, 1968)
Hubert Goldschmidt (Harvard, 1967)
Eric Grinberg (Harvard, 1983)
Yael Karshon (Harvard, 1993)

Shlomo Zvi Sternberg (born 1936), is an American mathematician known for his work in geometry, particularly symplectic geometry and Lie theory.

Sternberg earned his PhD in 1955 from Johns Hopkins University where he wrote a dissertation under Aurel Wintner. This became the basis for his first well-known published result known as the "Sternberg linearization theorem" which asserts that a smooth map near a hyperbolic fixed point can be made linear by a smooth change of coordinates provided that certain non-resonance conditions are satisfied. Also proved were generalizations of the Birkhoff canonical form theorems for volume preserving mappings in n-dimensions and symplectic mappings, all in the smooth case. (An account of these results and of their implications for the theory of dynamical systems can be found in Bruhat’s exposition “Travaux de Sternberg”, Seminaire Bourbaki, Volume 8. 1961).

After postdoctoral work at New York University (1956-1957) and an instructorship at University of Chicago (1957–1959) Sternberg joined the Mathematics Department at Harvard University in 1959, where he is currently George Putnam Professor of Pure and Applied Mathematics.

In the 1960s Sternberg became involved with Isadore Singer in the project of revisiting Élie Cartan’s papers from the early 1900s on the classification of the simple transitive infinite Lie pseudogroups, and of relating Cartan’s results to recent results in the theory of G-structures and supplying rigorous (by present-day standards) proofs of his main theorems. Also, in a sequel to this paper written jointly with Victor Guillemin and Daniel Quillen, he extended this classification to a larger class of pseudogroups: the primitive infinite pseudogroups. (One important by-product of the GQS paper was the “ integrability of characteristics” theorem for over-determined systems of partial differential equations. This figures in GQS as an analytical detail in their classification proof but is nowadays the most cited result of the paper.)

Many of Sternberg’s other papers have been concerned with Lie group actions on symplectic manifolds. Among his contributions to this subject are his paper with Bertram Kostant on BRS cohomology, his paper with David Kazhdan and Bertram Kostant on dynamical systems of Calogero type and his paper with Victor Guillemin on the “[Q,R] equals zero” conjecture. All three of these papers involve various aspects of the theory of symplectic reduction. In the first of these papers Bertram Kostant and Sternberg show how reduction techniques enable one to give a rigorous mathematical treatment of what is known in the physics literature as the BRS quantization procedure; in the second, the authors show how one can simplify the analysis of complicated dynamical systems like the Calogero system by describing these systems as symplectic reductions of much simpler systems, and the paper with Victor Guillemin contain the first rigorous formulation and proof of a hitherto vague assertion about group actions on symplectic manifolds; the assertion that “quantization commutes with reduction”.

The last of these papers was also the inspiration for a result in equivariant symplectic geometry that disclosed for the first time a surprising and unexpected connection between the theory of Hamiltonian torus actions on compact symplectic manifolds and the theory of convex polytopes. This theorem: the “ AGS convexity theorem” was simultaneously discovered by Guillemin-Sternberg and Michael Atiyah in the early 1980s.

Sternberg’s contributions to symplectic geometry and Lie theory have also included a number of basic textbooks on these subjects, among them the three graduate level texts with Victor Guillemin: “Geometric Asymptotics,”[1] “Symplectic Techniques in Physics”, [2] and "Semi-Classical Analysis". [3] His “Lectures on Differential Geometry” [4] is a popular standard textbook for upper-level undergraduate courses on differential manifolds, the calculus of variations, Lie theory and the geometry of G-structures. He also published the more recent "Curvature in mathematics and physics".[5]

Sternberg has, in addition, played a role in recent developments in theoretical physics: He has written several papers with Yuval Ne'eman on the role of supersymmetry in elementary particle physics in which they explore from this perspective the Higgs mechanism, the method of spontaneous symmetry breaking and a unified approach to the theory of quarks and leptons.

Among the honors he has been accorded as recognition for these achievements are a Guggenheim fellowship in 1974, election to the American Academy of Arts and Sciences in 1984, election to the National Academy of Sciences in 1986 and election to the American Philosophical Society in 2010. He has also been made an honorary member of the Academy of Arts and Sciences of the Royal Academy of Spain and awarded an honorary doctorate by the University of Mannheim. Sternberg delivered the Hebrew University’s Albert Einstein Memorial Lecture in 2006.

Selected books

See also


  1. Sternberg, Shlomo (December 31, 1977). "Geometric Asymptotics". American Mathematical Society.
  2. Sternberg, Shlomo (May 25, 1990). "Symplectic Techniques in Physics". Cambridge University Press.
  3. Sternberg, Shlomo (September 11, 2013). "Semi-Classical Analysis". International Press of Boston.
  4. Sternberg, Shlomo (March 11, 1999). "Lectures on Differential Geometry". American Mathematical Society.
  5. Sternberg, Shlomo (August 22, 2012). "Curvature in mathematics and physics". Dover Books on Mathematics.
  6. Humphreys, James E. (1995). "Review: Group theory and physics by S. Sternberg" (PDF). Bull. Amer. Math. Soc. (N.S.). 32 (4): 455–457. doi:10.1090/s0273-0979-1995-00612-9.
  7. Duistermaat, J. J. (1988). "Review: Symplectic techniques in physics by Victor Guillemin and Shlomo Sternberg" (PDF). Bull. Amer. Math. Soc. (N.S.). 18 (1): 97–100. doi:10.1090/s0273-0979-1988-15620-0.
  8. Hermann, R. (1965). "Review: Lectures on differential geometry by S. Sternberg" (PDF). Bull. Amer. Math. Soc. 71 (1): 332–337. doi:10.1090/S0002-9904-1965-11286-1.
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