Shlomo Zvi Sternberg (born November 20, 1936), is an American mathematician known for his work in geometry, particularly
symplectic geometry and
Lie theory.
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 is Emeritus Professor at the Harvard Mathematics Department.[2]
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] Also, 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 major contributions also to the topic of
Lie group actions on
symplectic manifolds, in particular involving various aspects of the theory of symplectic reduction. 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 BRS
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]
Shlomo Zvi Sternberg and Lynn Harold Loomis (2014) Advanced Calculus (Revised Edition) World Scientific Publishing
ISBN978-981-4583-92-3; 978-981-4583-93-0
Victor Guillemin and Shlomo Sternberg (2013) Semi-Classical Analysis International Press of Boston
ISBN978-1571462763
Shlomo Sternberg (2012) Lectures on Symplectic Geometry (in Mandarin) Lecture notes of Mathematical Science Center of Tsingua University, International Press
ISBN978-7-302-29498-6
Shlomo Sternberg (2012) Curvature in Mathematics and Physics Dover Publications, Inc.
ISBN978-0486478555[32]
Sternberg, Shlomo (2010). Dynamical Systems Dover Publications, Inc.
ISBN978-0486477053
Shlomo Sternberg (2004), Lie algebras, Harvard University
Victor Guillemin and Shlomo Sternberg (1999) Supersymmetry and Equivariant de Rham Theory 1999 Springer Verlag
ISBN978-3540647973
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.
ISBN0-521-24870-1[33]
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
ISBN978-0821841846
Paul Bamberg and Shlomo Sternberg (1988) A Course in Mathematics for Students of Physics Volume 1 1991 Cambridge University Press.
ISBN978-0521406499
Paul Bamberg and Shlomo Sternberg (1988) A Course in Mathematics for Students of Physics Volume 2 1991 Cambridge University Press.
ISBN978-0521406505
Victor Guillemin and Shlomo Sternberg (1984) Symplectic Techniques in Physics, 1990 Cambridge University Press
ISBN978-0521389907[34]
Guillemin, Victor and Sternberg, Shlomo (1977) Geometric asymptotics Providence, RI: American Mathematical Society.
ISBN0-8218-1514-8; reprinted in 1990 as an on-line book
Shlomo Sternberg (1969) Celestial Mechanics Part I W.A. Benjamin[35][36]
Shlomo Sternberg (1969) Celestial Mechanics Part II W.A. Benjamin[35]
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)
ISBN0-8284-0316-3.[37]
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.[15]