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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. He also wrote some well-known textbooks.
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.
Sternberg was awarded a Guggenheim fellowship in 1974 and an honorary doctorate by the University of Mannheim in 1991. He delivered the AMS Colloquium Lecture in 1990 and the Hebrew University's Albert Einstein Memorial Lecture in 2006.
Sternberg was elected member of the American Academy of Arts and Sciences in 1969, of the National Academy of Sciences in 1986, of the Spanish Royal Academy of Sciences In 1999, and of the American Philosophical Society in 2010.
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 , and of relating Cartan's results to recent results in the theory of and supplying rigorous (by present-day standards) proofs of his main theorems. 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.
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. 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. Together with David Kazhdan and Bertram Kostant, he showed how one can simplify the analysis of of Calogero type by describing them as symplectic reductions of much simpler systems. 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. 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 on Compact space symplectic manifolds and the theory of . This theorem, the "AGS convexity theorem," was simultaneously proved 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," "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, Lie theory and the geometry of . He also published the more recent "Curvature in mathematics and physics".
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.
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