Shlomo Sternberg

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.{{Cite web |title=Shlomo Sternberg – The Mathematics Genealogy Project |url=https://www.mathgenealogy.org/id.php?id=10465 |access-date=2022-06-25 |website=mathgenealogy.org}}

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.{{Cite web|url=http://www.math.harvard.edu/people|title = Harvard Mathematics Department Alumini, Faculty, Staff, Students & More}}

Sternberg was awarded a Guggenheim fellowship in 1974{{Cite web |title=Shlomo Sternberg |url=https://www.gf.org/fellows/all-fellows/shlomo-sternberg/ |access-date=2022-06-25 |website=John Simon Guggenheim Memorial Foundation |language=en-US}} and an honorary doctorate by the University of Mannheim in 1991.{{Cite web |title=Honors |url=https://www.uni-mannheim.de/en/getting-involved/honors/ |access-date=2022-06-25 |website=Universität Mannheim |language=en}}{{Cite web |title=Historical List |url=https://www.uni-mannheim.de/en/getting-involved/honors/historical-list/ |access-date=2022-06-25 |website=Universität Mannheim |language=en}} He delivered the AMS Colloquium Lecture in 1990{{Cite web |title=Colloquium Lectures |url=https://www.ams.org/meetings/lectures/meet-colloquium-lect |access-date=2022-06-26 |website=American Mathematical Society |language=en}} and the Hebrew University's Albert Einstein Memorial Lecture in 2006.{{Cite web |title=The Annual Albert Einstein Memorial Lecture |url=https://academy.ac.il/Index/Entry.aspx?nodeId=936&entryId=19671}}

Sternberg was elected member of the American Academy of Arts and Sciences in 1969,{{Cite web |title=Shlomo Zvi Sternberg |url=https://www.amacad.org/person/shlomo-zvi-sternberg |access-date=2022-06-25 |website=American Academy of Arts & Sciences |language=en}} of the National Academy of Sciences in 1986,{{Cite web |title=Shlomo Sternberg |url=http://www.nasonline.org/member-directory/members/40329.html |access-date=2022-06-25 |website=nasonline.org}} of the Spanish Royal Academy of Sciences In 1999,{{Cite web |year=2003 |title=Relación de académicos desde el año 1847 hasta el 2003 |trans-title=List of academics from 1847 to 2003 |url=https://rac.es/ficheros/doc/00186.pdf |website=Real Academia de Ciencias Exactas, Físicas y Naturales |language=es}} and of the American Philosophical Society in 2010.{{Cite web |title=APS Member History |url=https://search.amphilsoc.org/memhist/search?creator=sternberg&title=&subject=&subdiv=&mem=&year=&year-max=&dead=&keyword=&smode=advanced |access-date=2022-06-25 |website=search.amphilsoc.org}}

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.{{Cite journal |last=Sternberg |first=Shlomo |year=1958 |title=On the Structure of Local Homeomorphisms of Euclidean n-Space, II |url=https://www.jstor.org/stable/2372774 |journal=American Journal of Mathematics |volume=80 |issue=3 |pages=623–631 |doi=10.2307/2372774 |jstor=2372774 |issn=0002-9327}}{{Cite journal |last=Sternberg |first=Shlomo |year=1957 |title=Local Contractions and a Theorem of Poincare |url=https://www.jstor.org/stable/2372437 |journal=American Journal of Mathematics |volume=79 |issue=4 |pages=809–824 |doi=10.2307/2372437 |jstor=2372437 |issn=0002-9327}}{{Cite journal |last=Bruhat |first=François |author-link=François Bruhat |date=1960–1961 |title=Travaux de Sternberg |url=https://eudml.org/doc/109610 |journal=Séminaire Bourbaki |volume=6 |pages=179–196 |issn=0303-1179}}

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.{{Cite journal |last1=Singer |first1=I. M. |author-link=Isadore Singer |last2=Sternberg |first2=Shlomo |date=December 1, 1965 |title=The infinite groups of Lie and Cartan Part I, (The transitive groups) |journal=Journal d'Analyse Mathématique |language=en |volume=15 |issue=1 |pages=1–114 |doi=10.1007/BF02787690 |doi-access=free |s2cid=123124081 |issn=1565-8538}} 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.{{Cite journal |last1=Guillemin |first1=V. |author-link=Victor Guillemin |last2=Quillen |first2=D. |author-link2=Daniel Quillen |last3=Sternberg |first3=S. |year=1966 |title=The classification of the complex primitive infinite pseudogroups |journal=Proceedings of the National Academy of Sciences |language=en |volume=55 |issue=4 |pages=687–690 |doi=10.1073/pnas.55.4.687 |issn=0027-8424 |pmc=224211 |pmid=16591345|doi-access=free |bibcode=1966PNAS...55..687G }}

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.{{cn|date=September 2024}} 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.{{Cite journal |last1=Kostant |first1=Bertram |author-link=Bertram Kostant |last2=Sternberg |first2=Shlomo |date=May 15, 1987 |title=Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras |url=https://dx.doi.org/10.1016/0003-4916%2887%2990178-3 |journal=Annals of Physics |language=en |volume=176 |issue=1 |pages=49–113 |doi=10.1016/0003-4916(87)90178-3 |bibcode=1987AnPhy.176...49K |issn=0003-4916}} 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.{{Cite journal |last1=Kazhdan |first1=D. |author-link=David Kazhdan |last2=Kostant |first2=B. |author-link2=Bertram Kostant |last3=Sternberg |first3=S. |year=1978 |title=Hamiltonian group actions and dynamical systems of Calogero type |url=https://onlinelibrary.wiley.com/doi/10.1002/cpa.3160310405 |journal=Communications on Pure and Applied Mathematics |language=en |volume=31 |issue=4 |pages=481–507 |doi=10.1002/cpa.3160310405}} 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.{{Cite journal |last1=Guillemin |first1=V. |author-link=Victor Guillemin |last2=Sternberg |first2=S. |date=October 1, 1982 |title=Geometric quantization and multiplicities of group representations |url=https://doi.org/10.1007/BF01398934 |journal=Inventiones Mathematicae |language=en |volume=67 |issue=3 |pages=515–538 |doi=10.1007/BF01398934 |bibcode=1982InMat..67..515G |s2cid=121632102 |issn=1432-1297}} 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{{Cite journal |last1=Guillemin |first1=V. |author-link=Victor Guillemin |last2=Sternberg |first2=S. |date=October 1, 1982 |title=Convexity properties of the moment mapping |url=https://doi.org/10.1007/BF01398933 |journal=Inventiones Mathematicae |language=en |volume=67 |issue=3 |pages=491–513 |doi=10.1007/BF01398933 |bibcode=1982InMat..67..491G |s2cid=189830182 |issn=1432-1297}} and Michael Atiyah{{Cite journal |last=Atiyah |first=M. F. |author-link=Michael Atiyah |year=1982 |title=Convexity and Commuting Hamiltonians |url=http://doi.wiley.com/10.1112/blms/14.1.1 |journal=Bulletin of the London Mathematical Society |language=en |volume=14 |issue=1 |pages=1–15 |doi=10.1112/blms/14.1.1}} 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,"{{cite book |first=Shlomo |last=Sternberg|authorlink=Shlomo Sternberg|title=Geometric Asymptotics|date=December 31, 1977 |publisher=American Mathematical Society|isbn=0821816330}} "Symplectic Techniques in Physics",{{cite book |first=Shlomo |last=Sternberg|authorlink=Shlomo Sternberg|title=Symplectic Techniques in Physics|date=May 25, 1990 |publisher=Cambridge University Press|isbn=0521389909}} and "Semi-Classical Analysis".{{cite book |first=Shlomo |last=Sternberg|authorlink=Shlomo Sternberg|title=Semi-Classical Analysis|date= September 11, 2013|publisher=International Press of Boston|isbn=978-1571462763}} His "Lectures on Differential Geometry"{{cite book |first=Shlomo |last=Sternberg|authorlink=Shlomo Sternberg|title=Lectures on Differential Geometry|date=March 11, 1999|publisher=American Mathematical Society|isbn=0821813854}} 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".{{cite book |first=Shlomo |last=Sternberg |authorlink=Shlomo Sternberg|title=Curvature in mathematics and physics|date=August 22, 2012 |publisher=Dover Books on Mathematics|isbn=978-0486478555 }}

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.{{Cite journal |last1=Ne'eman |first1=Yuval |author-link=Yuval Ne'eman |last2=Sternberg |first2=Shlomo |year=1980 |title=Internal supersymmetry and unification |journal=Proceedings of the National Academy of Sciences |language=en |volume=77 |issue=6 |pages=3127–3131 |doi=10.1073/pnas.77.6.3127 |issn=0027-8424 |pmc=349566 |pmid=16592837|doi-access=free |bibcode=1980PNAS...77.3127N }}

Sternberg was Jewish and an orthodox rabbi.

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.{{cite web |title=Passing of Prof. Shlomo Z. Sternberg |url=https://networks.h-net.org/group/discussions/20041793/passing-prof-shlomo-z-sternberg |website=H-Judaic |access-date=26 August 2024 |date=24 August 2024}}

• Shlomo Sternberg (2019) A Mathematical Companion to Quantum Mechanics Dover Publications {{ISBN|9780486826899}} {{ISBN|0486826899}}
• 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
• 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}}{{cite web|url=https://www.maa.org/press/maa-reviews/curvature-in-mathematics-and-physics|date=November 8, 2012|title=Review of Curvature in Mathematics and Physics by Shlomo Sternberg|website=MAA Reviews, maa.org|author=Ruane, P. N.}}
• 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}}{{cite journal|author=Humphreys, James E.|authorlink=James E. Humphreys|title=Review: Group theory and physics by S. Sternberg|journal=Bull. Amer. Math. Soc. (N.S.)|volume=32|issue=4|year=1995|pages=455–457|url=http://www.ams.org/journals/bull/1995-32-04/S0273-0979-1995-00612-9/S0273-0979-1995-00612-9.pdf|doi=10.1090/s0273-0979-1995-00612-9|doi-access=free}}
• 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; reprint, 2006 Colloquium Publications {{ISBN|978-0821841846}}
• Paul Bamberg and Shlomo Sternberg (1988) A Course in Mathematics for Students of Physics Volume 1 1991 Cambridge University Press. {{ISBN|978-0521406499}}
• Paul Bamberg and Shlomo Sternberg (1988) A Course in Mathematics for Students of Physics Volume 2 1991 Cambridge University Press. {{ISBN|978-0521406505}}
• Victor Guillemin and Shlomo Sternberg (1984) Symplectic Techniques in Physics, 1990 Cambridge University Press {{ISBN|978-0521389907}}{{cite journal |author=Duistermaat, J. J. |year=1988 |title=Review: Symplectic techniques in physics by Victor Guillemin and Shlomo Sternberg |url=http://www.ams.org/journals/bull/1988-18-01/S0273-0979-1988-15620-0/S0273-0979-1988-15620-0.pdf |journal=Bull. Amer. Math. Soc. (N.S.) |volume=18 |issue=1 |pages=97–100 |doi=10.1090/s0273-0979-1988-15620-0 |doi-access=free |authorlink=Hans Duistermaat}}
• 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{{cite journal|author=Arnold, V.|authorlink=Vladimir Arnold|title=Review of Celestial Mechanics I, II by S. Sternberg|journal=Bull. Amer. Math. Soc.|year=1972|volume=78|issue=6|pages=962–963|url=http://www.ams.org/journals/bull/1972-78-06/S0002-9904-1972-13067-2/S0002-9904-1972-13067-2.pdf|doi=10.1090/s0002-9904-1972-13067-2|doi-access=free}}{{cite journal|journal=SIAM Review|year=1976|volume=18|issue=1|page=132|title=Review of Celestial Mechanics, Part I by Shlomo Sternberg|author=Pollard, Harry|authorlink=Harry Pollard (mathematician)|doi=10.1137/1018021}}
• 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
• 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}}.{{cite journal|author=Hermann, R.|authorlink=Robert Hermann (mathematician)|title=Review: Lectures on differential geometry by S. Sternberg|journal=Bull. Amer. Math. Soc.|year=1965|volume=71|issue=1|pages=332–337|url=http://www.ams.org/journals/bull/1965-71-02/S0002-9904-1965-11286-1/S0002-9904-1965-11286-1.pdf|doi=10.1090/S0002-9904-1965-11286-1|doi-access=free}}
• I. M. Singer and Shlomo Sternberg (1965) The infinite groups of Lie and Cartan. Part I. The transitive groups, Journal d'Analyse Mathématique 15, 1—114.

{{Portal|Biographies|Mathematics}}

• Symplectic manifold
• Symplectic topology

{{Reflist}}

• [https://people.math.harvard.edu/~shlomo/ Sternberg's home page] at Harvard has links to a half dozen on-line books
• {{MathGenealogy|id=10465}}
• Shiurim delivered by Rabbi Shlomo Sternberg - [http://www.sternberg-shiurim.org/ Sternberg-Shiurim.org]
• Personal documents and Halachic writings and recordings https://drive.google.com/drive/folders/1rq4XczLX_ia_2eVJJd0x-XEmBt22VjKH?usp=drive_link

{{Authority control}}

{{DEFAULTSORT:Sternberg, Shlomo}}

Category:1936 births

Category:2024 deaths

Category:20th-century American mathematicians

Category:21st-century American mathematicians

Category:Differential geometers

Category:American topologists

Category:Johns Hopkins University alumni

Category:Harvard University Department of Mathematics faculty

Category:Members of the United States National Academy of Sciences

Category:Members of the American Philosophical Society