The n-Category Café

Robert Hermann, a student of Charles Ehresmann, understood the unity of physics and differential geometry before this understanding became widespread. As an undergraduate I profited immensely from his book Lie Groups for Physicists, but this was just one of many books he wrote, first for the publisher Benjamin but later at his own press, Mathematical Science Press, based in Brookline, Massachusetts.

In his book Cartanian Geometry, Nonlinear Waves, and Control Theory he wrote:

I began in 1970 to write this series of books in order to develop a unified mathematical science and technology. After all, if subjects like category theory, logic, differential topology are accepted and integrated into the mathematical world, why not system theory, mathematical elementary particle theory, relativity, etc.? I had no master plan, but intended to write down what I could, as best I could, and see where it led.

Twenty volumes are now completed and I can say more definitively that the unifying theme is the role that geometry plays in physics and engineering. ‘Applied mathematics’ is usually thought of as involving the more concrete parts of analysis and certain areas like numerical analysis and combinatorics, which interface computer science; but my vision is quite different. To a large extent I am inspired by the historical example of the 19th century, where the basis of much of the fruitful interchange between mathematics and physics was precisely in the area we call ‘geometry’ or ‘the geometric theory of differential equations.’

Here are some of the books he wrote:

• Lie Groups for Physicists, Benjamin 1966

• Differential Geometry and the Calculus of Variations, Academic Press 1968, 2nd edn, Brookline 1977

• Fourier Analysis on Groups and Partial Wave Analysis, Benjamin 1969

• Lie Algebras and Quantum Mechanics, Benjamin 1970

• Lectures in Mathematical Physics, Benjamin 1970

• Vector Bundles in Mathematical Physics, Benjamin 1970

• Geometry, Physics and Systems, Dekker 1973

• Differential Geometric Methods and Ideas in Physics and Engineering, Rutgers University Press, 1973

• Algebraic Topics in Systems Theory, Brookline 1973

• General Algebraic Ideas, Brookline 1973

• Topics in General Relativity, Brookline 1973

• Energy-Momentum Tensors, Brookline 1973

• Linear and Tensor Algebra, Brookline 1973

• Physical Aspects of Lie Group Theory, Montreal, Presse Universitaire de Montreal, 1974

• Geometric Structure Theory of Systems - Control Theory and Physics, Brookline 1974

• Linear Systems and Introductory Algebraic Geometry, Brookline 1974

• Gauge Fields and Cartan–Ehresmann Connections, Brookline 1975

• with Clyde Martin: Algebro-geometric and Lie theoretic techniques in control theory, Brookline 1977

• Topics in the Mathematics of Quantum Mechanics, Brookline, 1973, 1977

• Quantum and Fermion Differential Geometry, Brookline 1977

• Toda Lattices, Cosymplectic Manifolds, Bäcklund Transformations, and Kinks, Brookline 1977

• The Geometry of Non-linear Differential equations, Bäcklund Transformations, and Solitons, Brookline 1977

• Yang–Mills, Kaluza–Klein, and the Einstein Program, Brookline 1978 (with contributions by Frank Estabrook, Hugo Wahlquist)

• Cartanian Geometry, Nonlinear waves, and Control Theory, Brookline, 2 parts: Part A 1979, Part B 1980

• Topics in the Geometric Theory of Linear Systems, Brookline 1984

• Topics in the Geometric Theory of Integrable Dynamical Systems, Brookline 1984

• Topics in Physical Geometry, Brookline 1988

• with Norman Hurt: Quantum Statistical Mechanics and Lie Group Harmonic Analysis, Brookline 1980

• Geometric Structures in Nonlinear Systems, Brookline 1991 (including hydrodynamics, deformation structures, with list of publications by Hermann to 1991)

• Lie–Cartan–Ehresmann Theory, Brookline 1993

• Geometric Computing Science — First Steps, Brookline 1991

• Constrained Mechanics and Lie Theory, Brookline 1992

• Lie-theoretic Ordinary Differential Equations, Numerical Analysis, Mechanics, and Differential Systems, Brooklyn 1994

• C–O–R Generalized Functions, Current Algebras and Control, Brookline 1994