Shlomo Sternberg

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

• Shlomo Zvi Sternberg and Lynn Harold Loomis (2014) Advanced Calculus (Revised Edition) World Scientific Publishing ISBN 978-981-4583-92-3; 978-981-4583-93-0(pbk)
• Victor Guillemin and Shlomo Sternberg (2013) Semi-Classical Analysis International Press of Boston ISBN 978-1571462763
• Shlomo Sternberg (2012) Lectures on Symplectic Geometry (in Mandarin) Lecture notes of Mathematical Science Center of Tsingua University, International Press ISBN 978-7-302-29498-6
• Shlomo Sternberg (2012) Curvature in Mathematics and Physics Dover Publications, Inc.ISBN 978-0486478555
• Sternberg, Shlomo (2010). Dynamical Systems Dover Publications, Inc. ISBN 978-0486477053
• Shlomo Sternberg (2004), Lie algebras, Harvard University.
• Victor Guillemin and Shlomo Sternberg (1999) Supersymmetry and Equivariant de Rham Theory 1999 Springer Verlag ISBN 978-3540647973
• Victor Guillemin, Eugene Lerman, and Shlomo Sternberg, (1996) Symplectic Fibrations and Multiplicity Diagrams Cambridge University Press
• Shlomo Sternberg (1994) Group Theory and Physics Cambridge University Press.ISBN 0- 521-24870-1^[6]
• Steven Shnider and Shlomo Sternberg (1993) Quantum Groups. From Coalgebras to Drinfeld Algebras: A Guided Tour (Mathematical Physics Ser.) International Press
• Victor Guillemin and Shlomo Sternberg (1990) Variations on a Theme by Kepler 2006 Colloquium Publications ISBN 978-0821841846
• Paul Bamberg and Shlomo Sternberg -(1988) A Course of Mathematics for Students of Physics Volume 1 1991 Cambridge University Press.ISBN 978-0521406499
• Paul Bamberg and Shlomo Sternberg -(1988) A Course of Mathematics for Students of Physics Volume 2 1991 Cambridge University Press.ISBN 978-0521406505
• Victor Guillemin and Shlomo Sternberg (1984) Symplectic Techniques in Physics,^[7] 1990 Cambridge University Press ISBN 978-0521389907
• Guillemin, Victor and Sternberg, Shlomo (1977) Geometric asymptotics Providence, RI: American Mathematical Society. ISBN 0-8218-1514-8.; reprinted in 1990 as an on-line book
• Shlomo Sternberg (1969) Celestial Mechanics Part I W.A. Benjamin
• Shlomo Sternberg (1969) Celestial Mechanics Part II W.A. Benjamin
• Lynn H. Loomis, and Shlomo Sternberg (1968) Advanced Calculus Boston: (World Scientific Publishing Company 2014).; text available on-line (58 MBytes)
• Victor Guillemin and Shlomo Sternberg (1966) Deformation Theory of Pseudogroup Structures American Mathematical Society
• Shlomo Sternberg (1964) Lectures on differential geometry New York: Chelsea (1093) ISBN 0-8284-0316-3.^[8]
• I. M. Singer and Shlomo Sternberg (1960) The infinite groups of Lie and Cartan. Part I. The transitive groups J. Analyse Math. 15 1965 1114.

See also

• Isadore Singer
• Symplectic manifold
• Symplectic topology

References

External links

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