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

Sternberg earned his Ph.D in 1957 from Johns Hopkins University where he wrote a dissertation under Aurel Wintner. This became the basis for his first well-known published result: a generalization of the Birkhoff canonical form theorem to volume preserving mappings in n- dimensions. (A account of this result and of its implications for the theory of dynamical systems can be found in Bruhat’s expose “Travaux de Sternberg”, Seminaire Bourbaki, Volume 8. 1961. )

In the 1960’s Sternberg became involved with I. M. Singer in the project of revisiting Eli Cartan’s papers from the early 1900’s on the classification of the 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 Guillemin and Quillen, he extended this classification to a larger class of pseudogroups; the primitive infinite pseudogroups. (One important bi-product of the GQS paper was the “ integrability of characteristics” theorem for over-determined systems of 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 most frequently cited papers have been concerned with Lie group actions on symplectic manifolds. Among his major contrubutions to this subject are his paper with Bertram Kostant on BRS cohomology, his paper with Kazhdan and Kostant on dynamical systems of Calogero type and his paper with Guillemin on the “ equals zero” conjecture. All three of these papers involve various aspects of the theory of symplectic reduction. In the first of these papers 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 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 Atiyah in the early nineteen eighties.

Sternberg’s contributions to symplectic geometry and Lie theory have also included a number of basic text books on these subjects, among them the two graduate level texts with Guillemin: “Geometric Asymptotics,” and “Symplectic Techniques in Physics” and his “Lectures on Differential Geometry” (which is a popular standard text book for upper level undergraduate courses on differential manifolds, the calculus of variations, Lie theory and the geometry of G-structures.)

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

Among the many 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, awarded an honorary doctorate by the University of Mannheim and chosen to deliver Hebrew University’s prestigious Albert Einstein Memorial Lecture in 2006.

See also

• Isadore Singer
• Symplectic manifold
• Symplectic topology

References

Selected books:

• Singer, I. M. & Sternberg, S. (1985). The infinite groups of Lie and Cartan. Part I. The transitive groups. Cambridge, MA: Massachusetts Institute of Technology Press. OCLC 12880084.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Guillemin, Victor and Sternberg, Shlomo (1977). Geometric asymptotics. Providence, RI: American Mathematical Society. ISBN 0-8218-1514-8.{{cite book}}: CS1 maint: multiple names: authors list (link); reprinted in 1990 as an on-line book
• Sternberg, Shlomo (1983). Lectures on differential geometry. New York: Chelsea. ISBN 0-8284-0316-3.
• Bamberg, Paul and Sternberg, Shlomo (1988). A Course of Mathematics for Students of Physics. Cambridge: Cambridge University Press. ISBN 0-521-40649-8 (vol 1), ISBN 0-521-40650-1 (vol 2). {{cite book}}: Check |isbn= value: invalid character (help)CS1 maint: multiple names: authors list (link)
• Sternberg, Shlomo (1994). Group theory and physics. Cambridge: Cambridge University Press. ISBN 0-521-24870-1.
• Loomis, Lynn H. & Sternberg, Shlomo (1990). Advanced Calculus. Boston: Jones and Bartlett Publishers International.{{cite book}}: CS1 maint: multiple names: authors list (link); text available on-line (58 MBytes) .

Commentary on the so-called Bible codes

• On the Witztum-Rips-Rosenberg Sample of Nations, by Dror Bar-Natan, Brendan McKay, and Shlomo Sternberg
• The Bible Code: A Book Review by Allyn Jackson, followed by "Comments on the Bible Code" by Shlomo Sternberg, Notices of the AMS September 1997

External links

• Sternberg's home page at Harvard has links to a half dozen on-line books
• Shlomo Sternberg at the Mathematics Genealogy Project

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