I was sorry to hear today of the recent death of Robert Hermann, at the age of 88. While I unfortunately never got to meet him, his writing had a lot of influence on me, as it likely did for many others with an overlapping interest in mathematics and fundamental physics. Early in my undergraduate years during the mid-1970s I first ran across some of Hermann’s books in the library, and found them full of fascinating and deep insights into the relations between geometry and physics. Over the years I’ve often come back to them and learned something new about one or another topic. The main problem with his writings is just that there is so much there that it is hard to know where to start.
While the relations between Riemannian geometry and general relativity were well-understood from Einstein’s work in the beginning of the subject, the relations between geometry and Yang-Mills theory were not known by Yang, Mills or other physicists working on the subject during the 1950s and 1960s. The understanding of these relations is conventionally described as starting in 1975, with the BPST instanton solutions and Simons explaining to Yang at Stony Brook about fiber bundles (leading to the “Wu-Yang dictionary” paper). But if you look at Hermann’s 1970 volume Vector Bundles in Mathematical Physics, you’ll find that it contains an extensive treatment of Yang-Mills theory in terms of connections and curvature in a vector bundle. While I don’t know if Hermann had written about the sort of topologically non-trivial gauge field configurations that got attention starting in 1975, he had at that point for a decade been writing in depth about the details of the relations between geometry and physics that were news to physicists in 1975.
Being ahead of your time and mainly writing expository books is unfortunately not necessarily good for a successful academic career. Looking through his writings this afternoon, I ran across a long section of this book from 1980, entitled “Reflections” (pages 1-82). I strongly recommend reading this for Hermann’s own take on his career and the problems faced by anyone trying to do what he was doing (the situation has not improved since then).
A general outline of his early career, drawn from that source is:
1948-50: undergraduate in physics, University of Wisconsin.
1950-52: undergraduate in math, Brown University.
1952-53: Fulbright scholar in Amsterdam.
1953-56: graduate student in math, Princeton. Thesis advisor Don Spencer.
1956-59: instructor at Harvard (“Harvard hired me as an instructor in the mistaken belief that I must be a topologist since I came from Princeton”).
1953-59: “My real work from 1953-59 was studying Elie Cartan!”
1959-61: position at MIT Lincoln Lab, taught course at Berkeley in 1961.
Hermann ultimately ended up at Rutgers, which he left in 1973, because he was not able to teach courses there in his specialty, and felt he had too little time to conduct the research he wanted to work on. It appears he expected to get by with some mix of grant money and profits from running a small publishing operation (Math Sci Press, which mainly published his own books). The “Reflections” section of the book mentioned above also contains some of his correspondence with the NSF, expressing his frustration at his grant proposals being turned down. At the end of a letter from late 1977 (which was at the height of excitement in the physics community over applying ideas from geometry and topology to high energy physics) he writes in frustration:
However, when I look in the Physical Review today, all the subjects which people in your position so enthusiastically supported ten years ago are now dead as the Phlogiston theory – and good riddance – while the topics I was working on then are now everywhere dense. Does one get support from the NSF by being right or by being popular?
Update: I’ve been reading some more of the essays Hermann published in the “Reflections” section of this book. Especially recommended is the section on Mathematical Physics of this 1979 essay (pages 30-38). His evaluation of the situation of the time I think was extremely perceptive.
Update: For more about Hermann, see some of the comments at this old blog posting. Also, on the topic of his book reviews, see this enthusiastic review of the Flanders book Differential forms with applications to the physical sciences.
Update: For an interesting review covering many of Hermann’s books, at the Bulletin of the AMS in 1973, see here.