## Lucien Szpiro 1941-2020

I’m very sorry to hear (via Michael Harris) of the death this morning in Paris of Lucien Szpiro, of heart failure. Szpiro was a faculty member here at Columbia for a few years, livening up the place at a time when the department was smaller and quieter than it is now. He then went on to a position at the CUNY Graduate Center and was often at the department here for number theory related talks. The Graduate Center has a short bio of him here, and on his website you can find more about his work, including some very nice short and lucid lecture notes on arithmetic geometry (see here and here). Some pictures of him and other mathematicians at his 70th birthday conference in 2012 can be found here.

What Szpiro is probably most famous for is the “Szpiro Conjecture” about elliptic curves which he first formulated in 1981. This is essentially equivalent to the later abc conjecture that has been the topic of recent controversy, so we really should have been all this time arguing about Szpiro, not abc. In a 2007 blog post I put out the news that Szpiro had announced a proof of abc at a talk he gave at Columbia (at Dorian Goldfeld’s 60th birthday conference). Alas, a flaw in that proof was quickly found.

Update: Something about Szpiro from Christian Peskine:

Lucien Szpiro est décédé d’une crise cardiaque samedi 18 avril. Ceux qui l’ont bien connu souhaitent d’abord saluer un homme d’exception. Lucien était tout à la fois un solitaire, un collaborateur passionné et un patron aimé et respecté. Un homme solitaire, intransigeant sur sa liberté, sur ses choix et sur la considération qu’il attendait. Un collaborateur passionnément ouvert au partage des idées et des projets. Un leader entrainant ses amis dans des aventures scientifiques nouvelles et enrichissantes.

Recruté au CNRS après un cours passage comme assistant à la faculté des sciences de Paris, il y est resté jusqu’à son départ à City University (New York) au début du siècle. Le séminaire qu’il a animé pendant de nombreuses années a été pour beaucoup de collègues de tous ages un lieu d’étude et de formation. Son influence et ses recherches ont fait honneur au CNRS. Ses nombreux élèves en témoigneront de leur coté. Il était heureux à New York ou il avait trouvé une forme de sérénité.

Ayant collaboré intensément avec Lucien durant de nombreuses années, je comprends que je perds un ami avec qui j’ai partagé des moments d’une intensité et d’une beauté rares. Il aimait la vie, il aimait la science et il aimait la recherche mathématique.

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## Latest on abc

Davide Castelvecchi at Nature has the story this morning of a press conference held earlier today at Kyoto University to announce the publication by Publications of the Research Institute for Mathematical Sciences (RIMS) of Mochizuki’s purported proof of the abc conjecture.

This is very odd. As the Nature subheadline explains, “some experts say author Shinichi Mochizuki failed to fix fatal flaw”. It’s completely unheard of for a major journal to publish a proof of an important result when experts have publicly stated that the proof is flawed and are standing behind that statement. That Mochizuki is the chief editor of the journal and that the announcement was made by two of his RIMS colleagues doesn’t help at all with the situation.

For background on the problem with the proof, see an earlier blog entry here. In the Nature article Peter Scholze states:

My judgment has not changed in any way since I wrote that manuscript with Jakob Stix.

and there’s

“I think it is safe to say that there has not been much change in the community opinion since 2018,” says Kiran Kedlaya, a number theorist at the University of California, San Diego, who was among the experts who put considerable effort over several years trying to verify the proof.

I asked around this morning and no one I know who is well-informed about this has heard of any reason to change their opinion that Mochizuki does not have a proof.

Ivan Fesenko today has a long article entitled On Pioneering Mathematical Research, On the Occasion of Announcement of Forthcoming Publication of the IUT Papers by Shinichi Mochizuki. Much like earlier articles from him (I’d missed this one), it’s full of denunciations of anyone (including Scholze) who has expressed skepticism about the proof as an incompetent. There’s a lot about how Mochizuki’s work on the purported proof is an inspiration to the world, ending with:

In the UK, the recent new additional funding of mathematics, work on which was inspired by the pioneering research of Sh. Mochizuki, will address some of these issues.

which refers to the British government decision discussed here.

There is a really good inspirational story in recent years about successful pioneering mathematical research, but it’s the one about Scholze’s work, not the proof of abc that experts don’t believe, even if it gets published.

Update: See the comment posted here from Peter Scholze further explaining the underlying problem with the Mochizuki proof.

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## Philip Anderson 1923-2020

I heard this morning about the death yesterday of Philip Anderson, at the age of 96. It’s not hard to make the case that Anderson was the most important condensed matter theorist of the twentieth century, with a huge influence on how we think about the subject. I believe he was even responsible for the name “condensed matter”. There are already obituaries at Princeton, and at the New York Times. For more about his work, Douglas Natelson has written something here.

Anderson’s career intersected with the field of high energy physics in several ways. Most importantly, what is often called the Higgs phenomenon really is a discovery of Anderson’s, and this should have been recognized by a second Nobel for him (he already had one for some of his work in condensed matter). I’ve written extensively about this story on the blog, see for example here, here and here. The story is a bit complicated, but it’s undeniable that in November 1962 Anderson submitted the paper Plasmons, Gauge Invariance and Mass to the Physical Review, and it was published on April 1, 1963. This was more than a year before the Higgs/Brout/Englert/Guralnik/Hagen/Kibble papers that HEP theorists always point to as the original ones. If you read Anderson’s paper, you’ll find a discussion of the “Higgs mechanism” which gets at the basic physics in much the same way we think of it today. There was no reason for HEP theorists to miss this paper, Anderson had written and published it not as a condensed matter paper, but as a contribution to current high energy theory. The only counter-argument I’ve gotten about this is that “Anderson’s explicit model was non-relativistic”, but this is a physical phenomenon for which relativity is not particularly relevant. Does it really make sense to argue that recognition should not go to a theorist who discovers a new phenomenon, but to others who later show that a possible problem (e.g. inconsistency with special relativity) not considered by the discoverer really isn’t there?

My time as a student at Princeton during 1979-84 (during which years Anderson split his time between Princeton and Bell Labs) was a high point of interaction between the condensed matter and HEP theory groups. HEP theorists trying to understand QCD investigated many examples of non-perturbative quantum field theory behavior that were of common interest with Anderson and others working on condensed matter. Anderson did have a major philosophical difference with the reductionist point of view of many HEP theorists, with his 1972 paper More is Different providing a strong critique of reductionism and emphasizing the importance of emergent behavior. In some sense, leading HEP theorists in recent years have come around to his point of view, often working on emergent models of space-time, with little interest in what microscopic physics space-time might be emerging from.

Anderson made no friends among the HEP community when he came out in 1987 against building the SSC (which was cancelled in 1993), for more about this, see this blog entry. He was also a skeptic about string theory, which perhaps made for some discomfort as Princeton HEP theory centered around this subject starting in 1984.

The two personal interactions with Anderson that I remember both involved him providing me with significant encouragement. When I took my general exams at Princeton, one component was an exam on condensed matter theory. This was not then and is not now a subject I know much about. After the exam there later was a gathering of students and faculty, and Anderson came up to me to tell me that he had graded my condensed matter exam. On one problem evidently I had gone about it wrong, but at some point had stopped and written that the result I was getting wasn’t sensible. Anderson complimented me on this, telling me that knowing when a calculation wasn’t making sense was an important skill. This was the nicest way imaginable to encourage a student who didn’t really know what he was doing.

Twenty years later, in early 2001, after I had written this article and had distributed it to several theorists including Anderson asking for comments, this is what he wrote back to me:

(Jan 19, 2001)
Dear Peter, I’m sorry to have been so slow to get back to you; my printer blew out when I tried to print out your attachment. When I finally got it it blew my mind–I loved seeing my vague misgivings made explicit. I would say that perhaps a stronger argument is the way of coping with black hole entropy, but since hearing about that i have never had it made clear to me that the story is unique to string theory except as a representative of sane quantum theories of gravity–the point made is that the gravitational theory provides its own cutoff.

I would hope but wouldn’t guarantee that P Today would publish it–I would be glad to introduce you as a guest columnist but they might not accept that.

When Lewontin and friends wanted to do a similar, but less well justified, job on sociobiology they wrote a broadside for the New York Review of Books.Since there are so many popular books I see no reason not to do that. If that doesn’t work I could perhaps insert you into John Brockman’s “third culture” chat room.. Anyhow, good luck–pwa

I wrote back to Anderson, telling him in particular that a prominent theorist had advised me not to publish the piece, since it would be counterproductive and the problems I discussed were well-known. He responded:

(Jan 23, 200)
Dear Peter, thanks for your reply. Of course [Prominent Theorist] would feel that the article would be counterproductive. Wasn’t that just the point? And to whom is it all well-known? The general public? Deans and department chairmen who make the hiring decisions? Science journalists who create the buzz? Politicians and bureaucrats who control the purse strings? Bullshit!

Yes, i agree that the version you sent me was too technical for a general journal like the NY review, but actually not much–the Lewontin arguments were pretty technical. Anyhow, good luck with this one. I showed it to a knowledgeable colleague, not a stringy but one who collaborates with them, and he also liked it.–pwa

Anderson then put me in touch with an editor at Physics Today. They decided not to publish the piece, first suggesting that instead I write a letter to the editor, and finally rejecting even that as “too inflammatory”. Anderson’s positive response to the piece though provided significant encouragement for my decision to start writing publicly about this issue.

He was one of the greats, and will be missed.

Update: More about Anderson from Physics World, Horganism, and Science magazine.

Update: David Derbes reminds me to point out that Higgs himself always properly credited Anderson, generally referring to the “Anderson mechanism”, or more specifically describing what he had done as the “relativistic Anderson mechanism”. For instance, he wrote here:

I call this the relativistic Anderson mechanism because Anderson described it first: it was his misfortune not to do so explicitly enough.

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## This Week’s Hype

Maybe it’s because people are at home with nothing else to do, but somehow the COVID-19 pandemic seems to be having the side-effect of generating new infections of “test of string theory” hype, a disease common many years back that seemed to more recently be under control. The example of a few days ago has now spread widely (see for instance Popular Mechanics), sometimes mutating into tests of “sting theory”. Today there’s a new example out, on the middle of the front page at Scientific American: Will String Theory Finally Be Put to the Experimental Test?

Of course the answer is “No”, this is just one more in the Swampland strain of string theory hype. This latest example is based on a paper by Bedroya and Vafa, where they make a “Transplanckian Censorship Conjecture”. The weird aspect of this kind of string theory hype is that it’s not a “test of string theory”, because it really has nothing to do with string theory. The authors of this paper are making a conjecture about “any consistent theory of quantum gravity”. If their conjecture is true we shouldn’t see the kind of B-modes in the CMB that were mistakenly claimed in the BICEP2 fiasco of 2014. So, the “test” here is a claim of falsification if experiments do for real see these B-modes. But what is being tested is a conjecture about any consistent theory of quantum gravity (one with very weak evidence). If B-modes are seen by a future experiment, the two possible conclusions to be drawn will be:

• There is no consistent theory of quantum gravity.
• The Transplanckian Censorship Conjecture is wrong.

It’s pretty clear what the correct choice between these two will be, and none of this will “test string theory.”

Update: I should have also pointed to this paper. Will Kinney today gave a talk, It Came From the Swampland, which went over this subject seriously in detail. His conclusion, which seemed to be shared by a string theorist he was talking to at the end, was pretty much that these conjectures should not be taken seriously. It looks like they’re already in conflict with both experimental results as well as theoretical model-building.

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## Fourier Analysis Notes

This semester I’ve been teaching a course on Fourier Analysis, which has, like just about everything, been seriously disrupted by the COVID-19 situation. Several class sessions have been canceled, and future ones are supposed to resume online next week. To improve matters a bit, I’ve been writing up lecture notes for the material since in-person lectures were canceled, and we’ll see how long I have the energy to keep this up.

The website for the course is here, giving detailed information about what it covers. In terms of level of mathematical rigor, the concept is to use the course as an opportunity to give students some motivation for a conventional real analysis course. The only prerequisite for the course is our usual Calculus sequence, which is not proof-based. In this class students are expected to try and follow proofs given in the book and in class, but not expected to be very good at coming up with their own proofs in the assignments, which mostly are computational. The textbook (Stein-Shakarchi) is based on a Princeton course with a somewhat similar philosophy of providing an introduction to analysis, but it is very challenging for the students to follow. I’ve looked around, but not found a better alternative. Other books on the subject tend to be either books for mathematics students that are even more abstract and challenging, or books for engineers that focus on either signal analysis or PDEs. Since the math department already has a PDE class I want to emphasize other things you can do with the subject.

The first set of lecture notes I wrote up were only loosely connected to Fourier analysis, through the Poisson summation formula. They dealt with theta functions and the zeta function, giving the standard proof of the functional equation for the zeta function that uses Poisson summation. I confess that one reason for covering this material is that I’ve always been fascinated by the connection between theta functions, quantization, representation theory (through the Heisenberg and metaplectic groups), and number theory. This subject contains a wealth of ideas that bring together fundamental physics and deep mathematics. On the mathematics side, this story was generalized by Tate in his thesis, where he developed what is essentially the GL(1) case of the modern theory of automorphic forms that underpins the Langlands program. On the physics side, one can think of what is going on as the standard canonical quantization of a finite-dim phase space, but with a lattice in the phase space giving a discrete subgroup of the usual Heisenberg group, and lots of new structure. For the details of this, one place to look is volume III of Mumford’s books on theta functions.

The second set of lecture notes, which I’ve just started on, are intended as an introduction to the theory of distributions, a topic that isn’t in Stein-Shakarchi. I highly recommend the book by Strichartz referred to in the notes for more details, with the notes maybe best used just as an introduction to that book.

I don’t want to turn this blog into yet one more place for discussion of the COVID-19 situation that just about all of us are obsessed with at the moment. If you’re interested in my personal experience, I’m doing fine. Almost all of us in New York are now pretty much confined to our apartments (it helps that the weather outside today is terrible), other than for short ventures out to get food or some exercise. I’d like to optimistically think that New York started taking action to stop the virus spread early enough to avoid disaster at the local hospitals. The best place I’ve seen to try and follow what is happening there is this web page. We’ll see in the next couple days if the problem has started to peak, or has much further to go.

I’m in much better shape than most people, having left town early for a spring break vacation. I had been planning a trip to Paris, at the very last minute instead rented a car and started out on a road trip in the general direction of New Orleans, consulting coronavirus report maps for where to avoid. Ended up in Memphis and then the Mississippi Delta (it had become clear New Orleans was a bad idea) before finally deciding that the situation was getting serious everywhere and it was time to head home. So, ended up back here in New York in relatively good mental shape for the confinement to come. Good luck to all of us in dealing with the coming challenges…

Update: The semester is now over, and I’ve put together all the notes I wrote up in one document, available here. This fixes mistakes/typos/etc. in earlier versions of the notes, so I’m changing the links to point to the final version.

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## This Week’s Hype

In this disturbing time of pandemic, it’s reassuring to see that some activities continue as usual. On the string theory hype front, yesterday NASA put out a press release announcing that Chandra Data Tests ‘Theory of Everything’, which starts by explaining that:

Despite having many different versions of string theory circulating throughout the physics community for decades, there have been very few experimental tests. Astronomers using NASA’s Chandra X-ray Observatory, however, have now made a significant step forward in this area.

This is based on a paper announcing limits on axions based on data from the Chandra X-ray telescope, which starts off with the dubious claim that axions “are generic within String Theory”. It seems to be very hard to get some people to understand that the number of “tests of string theory” is not “very few” but zero, for the simple reason that there are no predictions of string theory, generic or otherwise.

As usual, this kind of thing gets picked up by other news sources. In a sign of the times, the spin given to the bogus “test” is now often negative for string theory: This Galaxy Cluster May Have Just Dealt a Major Blow to String Theory.

Update: This is getting attention at The Daily Galaxy, under the headline “Mind of God?” –The Detection of ‘String-Theory’ Particles Would Change Physics Forever”.

For more on religion and string theory, there’s a new podcast featuring IAS theorist Tom Rudelius, entitled The Multiverse, the Polygraph, and the Resurrection. In an older podcast at Purpose Nation, Rudelius tells us this about the views of Nima Arkani-Hamed:

To quote preeminent theorist Nima Arkani-Hamed, who is certainly no theist: “The multiverse isn’t a theory. It’s a cartoon, right, it’s like this cartoon picture of something that we might think might be going on but we really don’t have any solid theory of how it would work.”

It seems that Arkani-Hamed shares my views on this.

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## Penrose at The Portal

Since last summer Eric Weinstein has been running a podcast entitled The Portal, featuring a wide range of unusual and provocative discussions. A couple have had a physics theme, including one with Garrett Lisi back in December.

One that I found completely fascinating was a recent interview with Roger Penrose. Penrose of course is one of the great figures of theoretical physics, and someone whose work has not followed fashion but exhibited a striking degree of originality. He and his work have often been a topic of interest on this blog: for one example, see a review of his book Fashion, Faith and Fantasy.

Over the years I’ve spent a lot of time thinking about Penrose’s twistors, becoming more and more convinced that these provide just the radical new perspective on space-time geometry and quantization that is needed for further progress on fundamental theory. For a long time now, string theorists have been claiming that “space-time is doomed”, and the recent “it from qubit” bandwagon also is based on the idea that space-time needs to be replaced by something else, something deeply quantum mechanical. Twistors have played an important role in recent work on amplitudes, for more about this a good source is a 2011 Arkani-Hamed talk at Penrose’s 80th birthday conference.

One of my own motivations for the conviction that twistors are part of what is needed is the “this math is just too beautiful not to be true” kind of argument that these days many disapprove of. There are many places one can read about twistors and the mathematics that underlies them. One that I can especially recommend is the book Twistor Geometry and Field Theory, by Ward and Wells. A one sentence summary of the fundamental idea would be

A point in space time is a complex two-plane in complex four-dimensional (twistor) space, and this complex two-plane is the fiber of the spinor bundle at the point.

In more detail, the Grassmanian G(2,4) of complex two-planes in $\mathbf C^4$ is compactified and complexified Minkowski space, with the spinor bundle the tautological bundle. So, more fundamental than space-time is the twistor space T=$\mathbf C^4$. Choosing a Hermitian form $\Omega$ of signature (2,2) on this space, compactified Minkowski space is the set of two-planes in T on which the form is zero. The conformal group is then the group SU(2,2) of transformations of T preserving $\Omega$ and this setup is ideal for handling conformally-invariant theories. Instead of working directly with T, it is often convenient to mod out by the action of the complex scalars and work with $PT=\mathbf{CP}^3$. A point in complexified, compactified space-time is then a $\mathbf{CP}^1 \subset \mathbf{CP}^3$, with the real Minkowski (compactified) points corresponding to $\mathbf{CP}^1$s that lie in a five-dimensional hypersurface $PN \subset PT$ where $\Omega=0$.

On the podcast, Penrose describes the motivation behind his discovery of twistors, and the history of exactly how this discovery came about. He was a visitor in 1963 at the University of Texas in Austin, with an office next door to Engelbert Schucking, who among other things had explained to him the importance in quantum theory of the positive/negative energy decomposition of the space of solutions to field equations. After the Kennedy assassination, he and others made a plan to get together with colleagues from Dallas, taking a trip to San Antonio and the coast. Penrose was being driven back from San Antonio to Austin by Istvan Ozsvath (father of Peter Ozsvath, ex-colleague here at Columbia), and it turned out that Istvan was not at all talkative. This gave Penrose time alone to think, and it was during this trip he had the crucial idea. For details of this, listen to what Penrose has to say starting at about 47 minutes before the end of the podcast. For a written version of the same story, see Penrose’s article Some Remarks on Twistor Theory, which was a contribution to a volume of essays in honor of Schucking.

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## This Week’s Hype

Sabine Hossenfelder already has this covered, but I wanted to add a few comments about this week’s hype, a new article in Quanta magazine by Philip Ball entitled Wormholes Reveal a Way to Manipulate Black Hole Information in the Lab (based on this paper). It’s the latest in a long tradition of bogus claims that studying relatively simple quantum systems is equivalent to studying string theory/quantum gravity. For an example from ten years ago, see here. The nonsensical idea back then (which got a lot of attention) was that somehow studying four qubits would “test string theory”.

A first comment would be that this is just profoundly depressing, because Ball is one of the best and most sensible science writers around (see my review of his excellent recent book on quantum mechanics) and Quanta magazine is about the the best semi-popular science publication there is. If this article were appearing in any one of the well-known examples of publications that traffic in misleading sensationalism, it wouldn’t be surprising and would best be just ignored.

Hossenfelder has pointed out one problem with the whole idea (we don’t live in AdS space), but a more basic problem is the obvious one pointed out by one of the first commenters at Quanta:

In the end, if an experiment is performed based on standard quantum mechanics, and verifies standard quantum mechanics as expected, then it is irrelevant that this aspect of standard quantum mechanics might be analogous to a vaguely-formulated and incomplete speculative idea about spacetime emergence — nor can it provide any experimental support whatsoever for that idea.

I understand that, for science journalists hearing that a large group of well-known physicists from Google, Stanford, Caltech, Princeton, Maryland and Amsterdam has figured out how to study quantum gravity in the lab (by teleporting things from one place to another via traversable wormholes!!), it’s almost impossible to resist the idea that this is something worth writing about. Please try.

Update: Philip Ball responds here.

Update: More from Philip Ball (and, if it appears, a response from me) at the Quanta article comment section, comments from one of the paper’s author’s also comments here.

Update: Commenter Anonyrat points out that the Atlantic is republishing this piece, as A Tiny, Lab-Size Wormhole Could Shatter Our Sense of Reality: How scientists plan to set up two black holes and a wormhole on an ordinary tabletop.

Update: In the future, I hope to as much as possible outsource coverage of this kind of thing to the Quantum Bullshit Detector. Today, see for instance this.

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## Why String Theory Is Both A Dream And A Nightmare (as well as a swamp…)

Ethan Siegel today has a new article at Starts With a Bang, entitled Why String Theory is Both a Dream and a Nightmare. For the nightmare part, he writes:

its predictions are all over the map, untestable in practice, and require an enormous set of assumptions that are unsupported by an iota of scientific evidence.

which I think just confuses the situation, which could be much more accurately and simply described as “there are no predictions”. The fundamental reason for this is also rather simply stated: the supposed unified theory is a theory in ten space-time dimensions, and no one has figured out a way to use this to get a consistent, predictive model with four space-time dimensions. If you don’t believe this, try watching the talks going on in Santa Barbara this week, which feature, after 17 years of intense effort, complete confusion about whether it is possible to construct such models with the right sign of the cosmological constant.

Siegel gets a couple things completely wrong, although this is not really his fault, due to the high degree of complexity and mystification which surrounds the 35 years of failed efforts in this area. About SUSY he writes

For one, string theory doesn’t simply contain the Standard Model as its low-energy limit, but a gauge theory known as N=4 supersymmetric Yang-Mills theory. Typically, the supersymmetry you hear about involves superpartner particles for every particle in existence in the Standard Model, which is an example of an N=1 supersymmetry. String theory, even in the low-energy limit, demands a much greater degree of symmetry than even this, which means that a low-energy prediction of superpartners should arise. The fact that we have discovered exactly 0 supersymmetric particles, even at LHC energies, is an enormous disappointment for string theory.

Like everything else, there’s no prediction from string theory about how many supersymmetries will exist. The special role of N=4 supersymmetric Yang-Mills theory has nothing to do with the problem of low energy SUSY, instead it occurs as the supposed dual to a very special 10d superstring background (AdS5 x S5). This is of interest for completely different reasons, one of which was the hope that this would provide a string theory dual to QCD, allowing use of string theory not to do quantum gravity, but to do QCD computations. This has never worked, with one main reason being that it can’t reproduce the asymptotic freedom property of QCD. Siegel tries to refer to this with

And when you look at the explicit predictions that have come out for the masses of the mesons that have been already discovered, by using lattice techniques, they differ from observations by amounts that would be a dealbreaker for any other theory.

including a table with the caption

The actual masses of a number of observed mesons and quantum states, at left, compared with a variety of predictions for those masses using lattice techniques in the context of string theory. The mismatch between observations and calculations is an enormous challenge for string theorists to account for.

He’s getting this from slide 31 of a talk by Jeff Harvey, but mixing various things up. The table has nothing to with lattice calculations, those are relevant to the other part of the slide, which is about string theory predictions for pure (no fermions) QCD glueballs. These are not physical objects, thus the comparison to lattice computer simulations, not experiment. The table he gives is from here and about real particles. The “predictions” are not made as he claims “using lattice techniques in the context of string theory.” There are no lattice techniques involved.

Normally Siegel does a good job of navigating complex technical subjects. The subject of string theory is now buried in a huge literature of tens of thousands of papers over forty years with all sorts of claims, many designed to obscure the fact that ideas haven’t worked out. It’s fitting that the name chosen for the kind of discussions going on at Santa Barbara this week is “The String Swampland”. String theory verily is now deep in a trackless swamp…

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## Robert Hermann 1931-2020

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?

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.

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