Robert Hermann (mathematician)

Robert C. Hermann (April 28, 1931 – February 10, 2020) was an American mathematician and mathematical physicist. In the 1960s Hermann worked on elementary particle physics and quantum field theory, and published books which revealed the interconnections between vector bundles on Riemannian manifolds and gauge theory in physics, before these interconnections became "common knowledge" among physicists in the 1970s.

Biography

Born in Brooklyn, Hermann studied in Paris and at Princeton University, where he attended lectures by Charles Ehresmann and where in 1955 under Donald Spencer he received his PhD with thesis The Differential geometry of homogeneous spaces. He was a Benjamin Peirce Instructor at Harvard University in 1957, and researcher at MIT Lincoln Laboratory 1959 to 61 when he lectured at University of California Berkeley. In 1962 he became an associate professor at Northwestern University and was raised to full professor. In 1967 he taught at University of California Santa Cruz until moving to Rutgers University (1970 to 75). Subsequently he did research primarily with financial support from the Ames Research Center of NASA.^[1] In the academic year 1969/1970 he was at the Institute for Advanced Study.

Following the French school of Élie Cartan, Hermann published numerous books on differential geometry and Lie group theory and their applications to differential equations, integrable systems, control theory, and physics. Most of these books were published in Brookline, Massachusetts by Math Sci Press, which he founded. Two series were published: a green series "Interdisciplinary Mathematics" and a blue series "Lie Groups: History, Frontiers and Applications".

It is a deeply regrettable fact that the flow of information back and forth between "modern" geometric and algebraic mathematics and classical applied mathematics has been so minimal, even though there is clearly a solid basis for such interaction. One of my overall motives in writing my series of books "Interdisciplinary Mathematics" was to facilitate this flow...[despite] high structural and mental barriers to such cross-fertilization.^[2]

The blue series considered history of differential geometry and Lie theory, and edited, with extensive new commentary, the work of Sophus Lie,^[3] Gregorio Ricci-Curbastro and Tullio Levi-Civita,^[4] Felix Klein's Vorlesungen über Mathematikgeschichte,^[5] Élie Cartan,^[6] Georges Valiron^[7] and the contributions to invariant theory by David Hilbert.^[8]

Robert Hermann died on February 10, 2020.^[9]

Early works

1966: Lie Groups for Physicists, Benjamin^[10]
1968: Differential geometry and the calculus of variations, Academic Press, second edition, Brookline 1977
1969: Fourier analysis on groups and partial wave analysis, Benjamin
1970: Lie algebras and quantum mechanics, Benjamin^[11]
1970: Lectures in mathematical physics, Benjamin
1970: Vector Bundles in mathematical physics, Benjamin 1970^[11]
1973: Geometry, Physics and Systems, Dekker
1973: Differential geometric methods and ideas in physics and engineering, Rutgers University Press
1974: Physical Aspects of Lie group theory, Montreal, Presse Universitaire de Montreal

Published by Math Sci Press, Brookline, Massachusetts

1973, 1977: Topics in the mathematics of quantum mechanics
1973: Algebraic topics in systems theory
1973: General algebraic ideas
1973: Topics in General Relativity
1973: Energy-Momentum Tensors
1973: Linear and tensor algebra
1974: Spinors, Clifford and Cayley Algebras
1974: Linear systems and introductory algebraic geometry
1974: Geometric structure theory of systems-control theory and physics
1975: Gauge fields and Cartan–Ehresmann Connections
1977: The geometry of non-linear differential equations, Bäcklund transformations, and solitons
1977: (with Clyde Martin): Algebro-geometric and Lie theoretic techniques in systems theory
1977: Toda lattices, cosymplectic manifolds, Bäcklund transformations, and kinks
1977: Quantum and fermion differential geometry
1978: (with contributions by Frank Estabrook, Hugo Wahlquist) Yang–Mills, Kaluza–Klein, and the Einstein program
1979: Cartanian geometry, nonlinear waves, and control theory, Brookline, 2 parts: Part A, Part B 1980^[12] (Cartanian meant in the sense of Élie Cartan)
1979: "Kleinian mathematics from an advanced standpoint", appendices to Felix Klein (M. Ackerman translator) Development of Mathematics in the 19th Century ISBN 0-915692-28-7
1980: (with Norman Hurt) Quantum statistical mechanics and Lie group harmonic analysis
1984: Topics in the geometric theory of linear systems
1984: Topics in the geometric theory of integrable dynamical systems
1988: Topics in physical geometry
1991: Geometric computing science – first steps
1991: Geometric structures in nonlinear systems, Brookline 1991 (including hydrodynamics, deformation structures, with list of publications by Hermann to 1991)
1992: Constrained mechanics and Lie theory
1993: Lie–Cartan–Ehresmann Theory
1994: Lie-theoretic ordinary differential equations, numerical analysis, mechanics, and differential systems
1994: C–O–R generalized functions, current algebras and control

References

^ Letters by Hermann published in Cartanian Geometry, Nonlinear Waves and Control Theory (1979)
^ R. Hermann (1979) Cartanian Geometry, Nonlinear Waves and Control Theory, page 51, Math Sci Press
^ Sophus Lie's 1884 differential invariant paper, 1976; Sophus Lie's 1880 transformation group paper, 1975
^ Ricci and Levi-Civita's tensor analysis paper, 1975
^ The Development of mathematics in the 19th Century, 1979
^ Geometry of Riemannian Spaces, 1983
^ The geometric theory of ordinary differential equations, 1984; Classical differential geometry of curves and surfaces, 1986
^ Hilbert's Invariant Theory Papers, 1978
^ "DR. ROBERT C. HERMANN". Legacy.com. February 12, 2020. Retrieved February 16, 2020.
^ Biedenharn, Lawrence C. (1967). "Review of Lie Groups for Physicists by Robert Hermann". Physics Today. 20 (10): 98–99. doi:10.1063/1.3033996.
^ ^a ^b Chernoff, P. R.; Marsden, J. E. (1973). "Review: Lie Algebras and Quantum Mechanics and Vector Bundles in Mathematical Physics and their relationship to the other works of Robert Hermann". Bull. Amer. Math. Soc. 79 (6): 1150–1162. doi:10.1090/s0002-9904-1973-13356-7.
^ Givens, Clark R.; Millman, Richard S. (1982). "Review: Cartanian geometry, nonlinear waves, and control theory". Bull. Amer. Math. Soc. (N.S.). 6 (3): 467–478. doi:10.1090/s0273-0979-1982-15019-4.

External links

Books from Math-sci Press
Robert Hermann at the Mathematics Genealogy Project
Peter Cains & 7 others (2021) Memorial for Robert Hermann, Notices of the American Mathematical Society

[1] Letters by Hermann published in Cartanian Geometry, Nonlinear Waves and Control Theory (1979)

[2] R. Hermann (1979) Cartanian Geometry, Nonlinear Waves and Control Theory, page 51, Math Sci Press

[3] Sophus Lie's 1884 differential invariant paper, 1976; Sophus Lie's 1880 transformation group paper, 1975

[4] Ricci and Levi-Civita's tensor analysis paper, 1975

[5] The Development of mathematics in the 19th Century, 1979

[6] Geometry of Riemannian Spaces, 1983

[7] The geometric theory of ordinary differential equations, 1984; Classical differential geometry of curves and surfaces, 1986

[8] Hilbert's Invariant Theory Papers, 1978

[9] "DR. ROBERT C. HERMANN". Legacy.com. February 12, 2020. Retrieved February 16, 2020.

[10] Biedenharn, Lawrence C. (1967). "Review of Lie Groups for Physicists by Robert Hermann". Physics Today. 20 (10): 98–99. doi:10.1063/1.3033996.

[Chernoff&Marsden-11] Chernoff, P. R.; Marsden, J. E. (1973). "Review: Lie Algebras and Quantum Mechanics and Vector Bundles in Mathematical Physics and their relationship to the other works of Robert Hermann". Bull. Amer. Math. Soc. 79 (6): 1150–1162. doi:10.1090/s0002-9904-1973-13356-7.

[12] Givens, Clark R.; Millman, Richard S. (1982). "Review: Cartanian geometry, nonlinear waves, and control theory". Bull. Amer. Math. Soc. (N.S.). 6 (3): 467–478. doi:10.1090/s0273-0979-1982-15019-4.

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