Shlomo Sternberg

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Shlomo Sternberg
Born(1936-01-20)January 20, 1936
DiedAugust 23, 2024(2024-08-23) (aged 88)
Alma mater Johns Hopkins University
Awards Guggenheim Fellowship, 1974
Scientific career
FieldsMathematics
Institutions Harvard University
New York University
University of Chicago
Thesis Some Problems in Discrete Nonlinear Transformations in One and Two Dimensions  (1955)
Doctoral advisor Aurel Friedrich Wintner
Doctoral students Victor Guillemin
Ravindra Kulkarni
Yael Karshon
Steve Shnider
Israel Michael Sigal
Sandy Zabell  [ de ]
Website https://www.math.harvard.edu/people/sternberg-shlomo/

Shlomo Zvi Sternberg (January 20, 1936 – August 23, 2024) was an American mathematician known for his work in geometry, particularly symplectic geometry and Lie theory.

Contents

Education and career

Sternberg earned his PhD in 1955 from Johns Hopkins University, with a thesis entitled Some Problems in Discrete Nonlinear Transformations in One and Two Dimensions, supervised by Aurel Wintner. [1]

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 was George Putnam Professor of Pure and Applied Mathematics until 2017. Since 2017, he was Emeritus Professor at the Harvard Mathematics Department. [2]

Sternberg was awarded a Guggenheim fellowship in 1974 [3] and an honorary doctorate by the University of Mannheim in 1991. [4] [5] He delivered the AMS Colloquium Lecture in 1990 [6] and the Hebrew University's Albert Einstein Memorial Lecture in 2006. [7]

Sternberg was elected member of the American Academy of Arts and Sciences in 1969, [8] of the National Academy of Sciences in 1986, [9] of the Spanish Royal Academy of Sciences In 1999, [10] and of the American Philosophical Society in 2010. [11]

Research

Sternberg's first well-known published result, based on his PhD thesis, is 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. He also proved generalizations of the Birkhoff canonical form theorems for volume preserving mappings in n-dimensions and symplectic mappings, all in the smooth case. [12] [13] [14]

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. [15] Together with Victor Guillemin and Daniel Quillen, he extended this classification to a larger class of pseudogroups: the primitive infinite pseudogroups. As a by-product, they also obtained the "integrability of characteristics" theorem for over-determined systems of partial differential equations. [16]

Sternberg provided contributions also to the topic of Lie group actions on symplectic manifolds, in particular involving various aspects of the theory of symplectic reduction.[ citation needed ] For instance, together with Bertram Kostant he showed how to use reduction techniques to give a rigorous mathematical treatment of what is known in the physics literature as the BRST quantization procedure. [17] Together with David Kazhdan and Bertram Kostant, he showed how one can simplify the analysis of dynamical systems of Calogero type by describing them as symplectic reductions of much simpler systems. [18] Together with Victor Guillemin he gave the first rigorous formulation and proof of a hitherto vague assertion about Lie group actions on symplectic manifolds, namely the Quantization commutes with reduction conjecture. [19] This last work 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 proved by Guillemin-Sternberg [20] and Michael Atiyah [21] 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," [22] "Symplectic Techniques in Physics", [23] and "Semi-Classical Analysis". [24] His "Lectures on Differential Geometry" [25] 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". [26]

Sternberg worked with Yuval Ne'eman on supersymmetry in elementary particle physics, exploring from this perspective the Higgs mechanism, the method of spontaneous symmetry breaking and a unified approach to the theory of quarks and leptons. [27]

Religion

Sternberg was Jewish and an orthodox rabbi. [8]

Death

Sternberg died in the old city of Jerusalem, on August 23, 2024. His funeral took place at Eretz Hachayim Cemetery in Beit Shemesh Israel on August 25, 2024. [28]

Selected monographs and books

See also

Related Research Articles

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References

  1. ↑ "Shlomo Sternberg – The Mathematics Genealogy Project". mathgenealogy.org. Retrieved June 25, 2022.
  2. ↑ "Harvard Mathematics Department Alumini, Faculty, Staff, Students & More".
  3. ↑ "Shlomo Sternberg". John Simon Guggenheim Memorial Foundation. Retrieved June 25, 2022.
  4. ↑ "Honors". Universität Mannheim. Retrieved June 25, 2022.
  5. ↑ "Historical List". Universität Mannheim. Retrieved June 25, 2022.
  6. ↑ "Colloquium Lectures". American Mathematical Society. Retrieved June 26, 2022.
  7. ↑ "The Annual Albert Einstein Memorial Lecture".
  8. 1 2 "Shlomo Zvi Sternberg". American Academy of Arts & Sciences. Retrieved June 25, 2022.
  9. ↑ "Shlomo Sternberg". nasonline.org. Retrieved June 25, 2022.
  10. ↑ "Relación de académicos desde el año 1847 hasta el 2003" [List of academics from 1847 to 2003](PDF). Real Academia de Ciencias Exactas, Físicas y Naturales (in Spanish). 2003.
  11. ↑ "APS Member History". search.amphilsoc.org. Retrieved June 25, 2022.
  12. ↑ Sternberg, Shlomo (1958). "On the Structure of Local Homeomorphisms of Euclidean n-Space, II". American Journal of Mathematics. 80 (3): 623–631. doi:10.2307/2372774. ISSN   0002-9327. JSTOR   2372774.
  13. ↑ Sternberg, Shlomo (1957). "Local Contractions and a Theorem of Poincare". American Journal of Mathematics. 79 (4): 809–824. doi:10.2307/2372437. ISSN   0002-9327. JSTOR   2372437.
  14. ↑ Bruhat, François (1960–1961). "Travaux de Sternberg". Séminaire Bourbaki. 6: 179–196. ISSN   0303-1179.
  15. 1 2 Singer, I. M.; Sternberg, Shlomo (December 1, 1965). "The infinite groups of Lie and Cartan Part I, (The transitive groups)". Journal d'Analyse Mathématique . 15 (1): 1–114. doi: 10.1007/BF02787690 . ISSN   1565-8538. S2CID   123124081.
  16. ↑ Guillemin, V.; Quillen, D.; Sternberg, S. (1966). "The classification of the complex primitive infinite pseudogroups". Proceedings of the National Academy of Sciences. 55 (4): 687–690. Bibcode:1966PNAS...55..687G. doi: 10.1073/pnas.55.4.687 . ISSN   0027-8424. PMC   224211 . PMID   16591345.
  17. ↑ Kostant, Bertram; Sternberg, Shlomo (May 15, 1987). "Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras". Annals of Physics. 176 (1): 49–113. Bibcode:1987AnPhy.176...49K. doi:10.1016/0003-4916(87)90178-3. ISSN   0003-4916.
  18. ↑ Kazhdan, D.; Kostant, B.; Sternberg, S. (1978). "Hamiltonian group actions and dynamical systems of Calogero type". Communications on Pure and Applied Mathematics. 31 (4): 481–507. doi:10.1002/cpa.3160310405.
  19. ↑ Guillemin, V.; Sternberg, S. (October 1, 1982). "Geometric quantization and multiplicities of group representations". Inventiones Mathematicae. 67 (3): 515–538. Bibcode:1982InMat..67..515G. doi:10.1007/BF01398934. ISSN   1432-1297. S2CID   121632102.
  20. ↑ Guillemin, V.; Sternberg, S. (October 1, 1982). "Convexity properties of the moment mapping". Inventiones Mathematicae. 67 (3): 491–513. Bibcode:1982InMat..67..491G. doi:10.1007/BF01398933. ISSN   1432-1297. S2CID   189830182.
  21. ↑ Atiyah, M. F. (1982). "Convexity and Commuting Hamiltonians". Bulletin of the London Mathematical Society. 14 (1): 1–15. doi:10.1112/blms/14.1.1.
  22. ↑ Sternberg, Shlomo (December 31, 1977). Geometric Asymptotics. American Mathematical Society. ISBN   0821816330.
  23. ↑ Sternberg, Shlomo (May 25, 1990). Symplectic Techniques in Physics. Cambridge University Press. ISBN   0521389909.
  24. ↑ Sternberg, Shlomo (September 11, 2013). Semi-Classical Analysis. International Press of Boston. ISBN   978-1571462763.
  25. ↑ Sternberg, Shlomo (March 11, 1999). Lectures on Differential Geometry. American Mathematical Society. ISBN   0821813854.
  26. ↑ Sternberg, Shlomo (August 22, 2012). Curvature in mathematics and physics. Dover Books on Mathematics. ISBN   978-0486478555.
  27. ↑ Ne'eman, Yuval; Sternberg, Shlomo (1980). "Internal supersymmetry and unification". Proceedings of the National Academy of Sciences. 77 (6): 3127–3131. Bibcode:1980PNAS...77.3127N. doi: 10.1073/pnas.77.6.3127 . ISSN   0027-8424. PMC   349566 . PMID   16592837.
  28. ↑ "Passing of Prof. Shlomo Z. Sternberg". H-Judaic. August 24, 2024. Retrieved August 26, 2024.
  29. ↑ Ruane, P. N. (November 8, 2012). "Review of Curvature in Mathematics and Physics by Shlomo Sternberg". MAA Reviews, maa.org.
  30. ↑ 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 .
  31. ↑ 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 .
  32. 1 2 Arnold, V. (1972). "Review of Celestial Mechanics I, II by S. Sternberg" (PDF). Bull. Amer. Math. Soc. 78 (6): 962–963. doi: 10.1090/s0002-9904-1972-13067-2 .
  33. ↑ Pollard, Harry (1976). "Review of Celestial Mechanics, Part I by Shlomo Sternberg". SIAM Review. 18 (1): 132. doi:10.1137/1018021.
  34. ↑ 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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