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 Reviewtoday, 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?

John Baez has written something here, and there’s an obituary notice here.

**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.

Last Updated on

RH advised me for a time when I was a math grad student at Rutgers. Later, as a grad student in physics at Brandeis, I moseyed into Cambridge in hopes of sounding out Glashow on my division algebra ideas. I was allowed to speak with Sheldon, but he confessed that abstract math was not his specialty, and I left shortly thereafter somewhat flustered. I ran into Robert Hermann in the science building cafe after I left and we sat together. I had just started explaining my ideas to him when Sheldon walked by. He spotted us, came over, and said that Robert was just the person I should be talking to. Of course, he was right. Sheldon sat with us, and I lost what little coherence I had left and made a poor showing of myself until I left a little while later. That, unfortunately, was the last interaction I remember having with Robert H, a really good guy. Had I known he was in town I would have gone straight to him. Alas … (About 18 years later I taught an undergrad course in particle physics with Sheldon G at Harvard; also a really good guy.)

Every time I think of Hermann, I always tend to relate him to Spivak, in view of their unique and somewhat idiosyncratic career paths.

Many things puzzled me over the years about Hermann, one of which being what is mentioned in this post. Namely how he did not end up getting due credit for having attempted to drawn connections between gauge field theory and differential geometry on fibre bundles even before Wu-Yang-Simons. I once heard rumors that I. M. Singer had actually noted Hermann’s contributions in this area.

Another strange anecdotal mystery to me is Hermann’s relationship with Sternberg. “Being critical” is perhaps an understatement about his BAMS review of Sternberg’s 1964 book, in which Hermann was in fact acknowledged by the author. (In all fairness though, Hermann’s book review was safely towered in its acrimonious level by Barry Mitchell’s on “Module Theory” written by a romannian mathematician.) In a subsequent year, Hermann appeared to explain at pains in one of his papers on Cartan’s EDS how his work was independent but perhaps no less valuable than the opus published at a slightly earlier time by Singer and Sternberg.

If one were to be pessimistically cynical, one would likely not be surprised at the tremendous amount of latent ego right underneath the thin skins of apparently humble mathematicians.

xyz abc is referring to this 1965 review

https://www.ams.org/journals/bull/1965-71-02/S0002-9904-1965-11286-1/

I don’t know what relations were like between Sternberg and Hermann, note that Hermann left Harvard (and says he wasn’t very happy there) about the time Sternberg arrived (1959).

It’s interesting to compare this review to this 1982 one of a Hermann book

https://www.ams.org/journals/bull/1982-06-03/S0273-0979-1982-15019-4/S0273-0979-1982-15019-4.pdf

which is also a quite critical review. Hermann and Sternberg shared many of the same interests, as well as each writing quite a few expository books. While in many ways different, their books to me seem to share some similar features. These include a mixture of beautiful, deep and insightful exposition of material not available elsewhere (although sometimes very demanding of the reader), together with some idiosyncratic material of less value. Maybe this is generically what happens if you spend a lot of your life absorbing the ideas of Elie Cartan, and then start writing a lot of books…

I just read the recommended selection from his book “Reflections” and I am more than a little impressed at his insights, at his clarity of thinking, and just how well he understood the problems our fields were facing. (I have a PhD in physics, but a strong interest in mathematics for its own sake and have given myself a crude first-year course in University mathematics via textbooks and supplementary online reading. It’s no replacement for a real degree, but it makes me less of an idiot when dealing with reading things like that book.)

Hermann’s review of Sternberg’s book is positive, bordering on enthusiastic. I’m not sure how anyone reading it could conclude that phrases such as “In this chapter Sternberg gives us, with great expository skill and taste, a glimpse of vast research areas …” are anything else. Criticisms are made, but they are constructive, not tendentious, and they are offered in a friendly way, not a destructive one.

D.F.,

The summary is

“In summary, this is a book that contains much useful material and that is, in general, well written, but that is marred by inattention to detail.”

but, in any case, I hadn’t heard that there was any particular problem between Sternberg and Hermann.

Hermann made public some of the negative grant referee reports, and they seem to me to reflect not personal hostility but an unsurprising negative take on a proposal to fund someone not playing by the conventional rules (i.e. quitting his job and writing not journal articles but self-published mainly expository books).

In the beginning pages of the Hermann link above, Hermann must have pounded on Kac’s “dehydrated elephants” half a dozen times. What exactly was all that about?

Anonyrat,

He was referring to this essay by Mark Kac (of Rockefeller University, where Hermann had been for a while)

https://www.jstor.org/stable/43636233

It’s pretty much a ferocious attack on the sort of mathematical physics practiced by Hermann, starting off with “it seems self-evident that mathematics is not likely to be much help in discovering laws of nature”. Like Hermann, I strongly disagree (but don’t want to start here the same argument about this that has often featured on this blog, most recently over Sabine Hossenfelder and “Lost in Math”).

Kac’s specialty was probability and analysis. He had little interest in geometry, even as mathematics, so not surprising he didn’t see it as important for physics. Note that the essay was written in 1972, same year Weinberg’s gravitation textbook was published, which also took the attitude that geometry was not important for physics, writing

“the passage of time has taught us not to expect that the strong, weak, and electromagnetic interactions can be understood in geometrical terms”

This didn’t age well, with 1973 seeing the triumph of gauge theory, the mid-late 1970s the dominance of ideas about geometry and topology in the subject, and by the 1980s Weinberg himself was working on such things, see discussion here

https://www.math.columbia.edu/~woit/wordpress/?p=529

So, the historical context for Hermann’s comments about this was that he was living through a period (late 1970s) of vindication of his point of view, a few years after Kac’s attack on it.

I’m very sorry to hear this. Finding an old, beautiful paper of Hermann was key to my being able to get over an obstacle in my thesis. I always hoped to contact him and thank him, and even wrote an email or two trying to get his address, but none of it led anywhere, and I’m sorry I wasn’t more persistent. I now have a number of his books, and look forward to mining them for insights. (I had the opportunity to own a very large number of his books, but unfortunately, the amount of shelf space they would take was ponderous, so I took only those that seemed best.)

RIP.

I bought a copy of Hermann’s “Lie Algebras and Quantum Mechanics” as an undergraduate at UC Berkeley, back when pterodactyls blotted out the sun. I learned a lot of basic ideas from the book (although there was significant overlap with other books I was reading at that time). There is a good discussion of affine Lie algebras and Schwinger terms, although issues concerning regularization (which is the hardest part of QFT) are not really discussed. From my current standpoint I’d characterize the book as very broad in a good way, but not especially deep. Nonetheless, for a first exposure to the topics, it was accessible and fun.

Thanks Peter,

That’s one book of Hermann’s I haven’t looked at but should. The topics you mention got a lot of attention in the 1980s. That Hermann was writing about them in 1969 is yet another example of him being way ahead of his time.

I just ran into an interesting review from 1973 that covers many of his books, will add a link as an update.