Shlomo Zvi Sternberg (born 1936), is an American mathematician known for his work in geometry, particularly

symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...

and

Lie theory In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject i ...

Education and career

Sternberg earned his PhD in 1955 from

Johns Hopkins University Johns Hopkins University (Johns Hopkins, Hopkins, or JHU) is a private research university in Baltimore, Maryland. Founded in 1876, Johns Hopkins is the oldest research university in the United States and in the western hemisphere. It consiste ...

, with a thesis entitled "''Some Problems in Discrete Nonlinear Transformations in One and Two Dimensions''", supervised by Aurel Wintner. After postdoctoral work at

New York University New York University (NYU) is a private research university in New York City. Chartered in 1831 by the New York State Legislature, NYU was founded by a group of New Yorkers led by then- Secretary of the Treasury Albert Gallatin. In 1832, ...

(1956–1957) and an instructorship at

University of Chicago The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private university, private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park, Chicago, Hyde Park neighborhood. The University of Chic ...

(1957–1959), Sternberg joined the Mathematics Department at

Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of high ...

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. Among other honors, Sternberg was awarded a

Guggenheim fellowship Guggenheim Fellowships are grants that have been awarded annually since by the John Simon Guggenheim Memorial Foundation to those "who have demonstrated exceptional capacity for productive scholarship or exceptional creative ability in the ar ...

in 1974 and a honorary doctorate by the

University of Mannheim The University of Mannheim (German: ''Universität Mannheim''), abbreviated UMA, is a public research university in Mannheim, Baden-Württemberg, Germany. Founded in 1967, the university has its origins in the ''Palatine Academy of Sciences'', ...

in 1991. He delivered the AMS in 1990 and the

Hebrew University The Hebrew University of Jerusalem (HUJI; he, הַאוּנִיבֶרְסִיטָה הַעִבְרִית בִּירוּשָׁלַיִם) is a public university, public research university based in Jerusalem, Israel. Co-founded by Albert Einstein ...

's Albert Einstein Memorial Lecture in 2006. Sternberg was elected member of the

American Academy of Arts and Sciences The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, ...

in 1969, of the

National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nat ...

in 1986, of the Spanish Royal Academy of Sciences In 1999, and of the

American Philosophical Society The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communi ...

in 2010.

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. In the 1960s Sternberg became involved with Isadore Singer in the project of revisiting

Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry ...

'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, together with

Victor Guillemin Victor William Guillemin (born 1937 in Boston) is an American mathematician working in the field of symplectic geometry, who has also made contributions to the fields of microlocal analysis, spectral theory, and mathematical physics. He is a t ...

and

Daniel Quillen Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 19 ...

, 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 In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

. Sternberg provided major contributions also to the topic of

Lie group action In differential geometry, a Lie group action is a group action adapted to the smooth setting: G is a Lie group, M is a smooth manifold, and the action map is differentiable. __TOC__ Definition and first properties Let \sigma: G \times M \to M, ( ...

s on

symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called s ...

s, 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. Together with David Kazhdan and Bertram Kostant, he showed how one can simplify the analysis of

dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...

s of Calogero type by describing them as

symplectic reduction In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. ...

s of much simpler systems. Together with

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 In mathematics, more specifically in the context of geometric quantization, quantization commutes with reduction states that the space of global sections of a line bundle ''L'' satisfying the quantization condition on the symplectic quotient of a c ...

conjecture. 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 action In algebraic geometry, a torus action on an algebraic variety is a group action of an algebraic torus on the variety. A variety equipped with an action of a torus ''T'' is called a ''T''-variety. In differential geometry, one considers an action of ...

s on

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

symplectic manifolds and the theory of convex polytopes. This theorem, the "AGS convexity theorem," was simultaneously proved by Guillemin-Sternberg and

Michael Atiyah Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded t ...

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

: "Geometric Asymptotics," "Symplectic Techniques in Physics", and "Semi-Classical Analysis". His "Lectures on Differential Geometry" is a popular standard textbook for upper-level undergraduate courses on differential manifolds, the

calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...

and the geometry of G-structures. He also published the more recent "

Curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...

in mathematics and physics". Sternberg has, in addition, played a role in recent developments in

theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experi ...

. He has worked with

Yuval Ne'eman Yuval Ne'eman ( he, יובל נאמן, 14 May 1925 – 26 April 2006) was an Israeli theoretical physicist, military scientist, and politician. He was Minister of Science and Development in the 1980s and early 1990s. He was the President ...

supersymmetry In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...

elementary particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) and b ...

, exploring from this perspective the

Higgs mechanism In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other bei ...

, the method of spontaneous symmetry breaking and a unified approach to the theory of

quarks A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly ...

and

leptons In particle physics, a lepton is an elementary particle of half-integer spin ( spin ) that does not undergo strong interactions. Two main classes of leptons exist: charged leptons (also known as the electron-like leptons or muons), and neut ...

Religion

Sternberg is Jewish and a

Rabbi A rabbi () is a spiritual leader or religious teacher in Judaism. One becomes a rabbi by being ordained by another rabbi – known as ''semikha'' – following a course of study of Jewish history and texts such as the Talmud. The basic form of ...

. He was among the mathematicians who debunked the mathematics foundations of Michael Drosnin's controversial claims in

The Bible Code ''The Bible Code'' is a book by Michael Drosnin, first published by Simon & Schuster in 1997. A sequel, ''Bible Code II: The Countdown'', was published by Penguin Random House in 2002, and also reached New York Times Best-Seller status. In 2010 ...

. Sternberg is described by rabbi Herschel Schachter of

Yeshiva University Yeshiva University is a private Orthodox Jewish university with four campuses in New York City."About YU
on the Yeshiva Universi ...

as "a big genius in learning and math" who played a role in establishing that

swordfish Swordfish (''Xiphias gladius''), also known as broadbills in some countries, are large, highly migratory predatory fish characterized by a long, flat, pointed bill. They are a popular sport fish of the billfish category, though elusive. Swordfi ...

is kosher.

Selected monographs and books

*Shlomo Sternberg (2019) A Mathematical Companion to Quantum Mechanics Dover Publications * Shlomo Zvi Sternberg and Lynn Harold Loomis (2014) Advanced Calculus (Revised Edition) World Scientific Publishing ; 978-981-4583-93-0 * Victor Guillemin and Shlomo Sternberg (2013) Semi-Classical Analysis International Press of Boston * Shlomo Sternberg (2012) Lectures on Symplectic Geometry (in Mandarin) Lecture notes of Mathematical Science Center of Tsingua University, International Press * Shlomo Sternberg (2012) Curvature in Mathematics and Physics Dover Publications, Inc. * Sternberg, Shlomo (2010). Dynamical Systems Dover Publications, Inc. * Shlomo Sternberg (2004), Lie algebras, Harvard University * Victor Guillemin and Shlomo Sternberg (1999) Supersymmetry and Equivariant de Rham Theory 1999 Springer Verlag * 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 * 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 * Paul Bamberg and Shlomo Sternberg (1988) A Course in Mathematics for Students of Physics Volume 1 1991 Cambridge University Press. * Paul Bamberg and Shlomo Sternberg (1988) A Course in Mathematics for Students of Physics Volume 2 1991 Cambridge University Press. * Victor Guillemin and Shlomo Sternberg (1984) Symplectic Techniques in Physics, 1990 Cambridge University Press * Guillemin, Victor and Sternberg, Shlomo (1977) Geometric asymptotics Providence, RI: American Mathematical Society. ; 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 * 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) . * I. M. Singer and Shlomo Sternberg (1965) The infinite groups of Lie and Cartan. Part I. The transitive groups,

Journal d'Analyse Mathématique The ''Journal d'Analyse Mathématique'' is a triannual peer-reviewed scientific journal published by Magnes Press ( Hebrew University of Jerusalem). It was established in 1951 by Binyamin Amirà. It covers research in mathematics, especially cla ...

15, 1—114.

References

External links