In Memoriam

Shlomo Sternberg, a leading mathematician and founding member of the Association for Mathematical Research, passed away on August 23, 2024. Born on January 20, 1936, Sternberg was a true prodigy. He received his Ph.D. under Aurel Wintner at Johns Hopkins in 1956 at the age of 20. In 1958, he was ordained as an orthodox rabbi.

After postdoctoral work at the Courant Institute (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 his retirement in 2017.

In 1960, at Harvard, Sternberg created the highly inovative course, Math 55. The aim was to teach advanced calculus from a standpoint which exposed the students to modern concepts such as distributions, differential forms, functoriality, etc. One of us (J.C.) took the course during the first year it was given. As he can attest, Shlomo was a charismatic lecturer and the course, as a whole, was extraordinarily inspiring. (Math 55 still survives, and is sometimes billed on the internet as the world’s hardest undergraduate math course. However, the current version bears little resemblance to the original.)

Sternberg’s mathematical works go back to the 1950s and extend well into the 21st century. He had 125 publications, including 15 books. Among the most influential of his books are “Lectures on Differential Geometry”, “Group theory and physics”, “Supersymmetry and equivariant de Rham theory”, “Symplectic techniques in physics” and “Geometric Asymptotics”, the last three of which are coauthored with his former student and long time collaborator Victor Guillimen.

Below, we will mention several prominent papers from Sternberg’s extensive opus.

Sternberg’s first well-known result, based on his PhD thesis, is known as the ”Sternberg linearization theorem”. It states that near a hyperbolic fixed point, a smooth map can be made linear by a smooth change of coordinates if certain non-resonance conditions are satisfied.

In the 1960s, Sternberg and Isadore Singer provided rigorous proofs of Eli Cartan’s classification of the simple transitive infinite Lie pseudogroups and related them to the theory of G-structures. With Guillemin and Daniel Quillen, he extended the classification to the primitive infinite pseudogroups, As an application, they obtained the ”integrability of characteristics” theorem for over-determined systems of partial differential equations.

A central theme in Sternberg’s work concerns the role of symmetry in geometric models for physical phenomena. His long collaboration with the distinguished Israeli physicist, Yuval Ne’eman, lasted from 1962 until Ne’eman’s death in 2006. A guiding principle in their work is the use of supersymmetry to eliminate certain undetermined coefficients in the standard model of elementary particles. Their 1975 paper, “ Graded Lie algebras in mathematics and physics (Bose–Fermi symmetry)” is extensively cited by both mathematicians and physicists. Their 2005 review paper, derives physical properties (the electroweak isospin and hypercharge) of elementary particles from representations of the superalgebra su(m/1) and expresses the Higgs field as the degree zero component of a superconnection in the sense of Quillen.

With David Kazhdan and Bertram Kostant, Sternberg showed how one can simplify the analysis of completely integrable dynamical systems of Calogero-Moser type by describing them as symplectic reductions of much simpler systems.

Shlomo’s Ph.D. mentorship of Victor Guillemin evolved into a lifelong collaboration, yielding 50 joint publications including 6 books. They essentially created the field of Hamiltonian group actions, which model physical symmetries of phase space.

In “Convexity properties of the moment mapping”, they prove that for a Hamiltonian torus action on a compact symplectic manifold, the image of the momentum map (or “moment map”) is a convex polytope. This result, which was was proved independently and at the same time by Michael Atiyah, led to a flurry of activity, connecting symplectic geometry to geometry of polytopes.

In “Geometric quantization and multiplicities of group representations” they formulated the “Quantization commutes with reduction” principle, also known as “[Q, R] = 0”. For a compact Lie group G and a symplectic G-manifold X, the quantization Q(X) should then be a linear G representation. [Q, R] = 0 identifies the linear subspace of Q(X) that is fixed by G with the quantization of the symplectic quotient, X/G, In that paper, Guillemin and Sternberg proved [Q, R] = 0 rigorously when X is a Kähler manifold and Q(∆) is interpreted as the space of global holomorphic sections of a prequantization line bundle. In actuality, the quantization commutes with reduction principle goes well beyond the Kähler case and has inspired enormous activity.

Sternberg was an elected member of the American Academy of Arts and Sciences, 1969, National Academy of Sciences, 1986, and the American Philosophical Society, 2010.

We miss Shlomo. Though he is gone, his inspiration lives on.

Jeff Cheeger and Alan Weinstein

March 21, 2025.