This and That

Various things that may be of interest:

  • MSRI in Berkeley has announced a \$70 million dollar gift from Jim and Marilyn Simons, and Henry and Marsha Laufer. This gift will make up the bulk of a planned endowment increase of \$100 million and is the largest endowment gift ever made to a US-based math institute. The success of the Renaissance Technologies hedge fund is what has made gifts on this scale possible. This summer MSRI will be renamed the “Simons Laufer Mathematical Sciences Institute”, and the directorship will pass from David Eisenbud to Tatiana Toro.
  • The journal Inference has just published an article by Daniel Jassby, which gives a highly discouraging view of the prospects for magnetic confinement fusion devices. Jassby, who worked for many years at the Princeton Plasma Physics Lab, argues that performance of magnetic confinement fusion systems has not much advanced in a quarter century, making for very bleak prospects that such designs will lead to a workable power plant in the forseeable future. He sees inertial confinement fusion systems like the National Ignition Facility at Livermore as making some progress, but ends with:

    The technological hurdles for implementing an ICF-based power system are so numerous and formidable that many decades will be required to resolve them—if they can indeed be overcome.

  • I’ve been spending some time reading Grothendieck’s Récoltes et Semailles, which is a simultaneously fascinating and frustrating experience. I’ve made it almost to the end of the first part, except that there will be another forty pages or so of notes to go. To get to the first part involved starting by reading through about two hundred pages of four layers of introduction. It seems that basically Grothendieck did no editing. Once he was done writing the first part, as he thought of more to say he’d add notes. He distributed copies to various other mathematicians, and then kept adding new introductions, with various references to how this fit in with more technical mathematical documents he was working on (La “Longue Marche” à Travers la Théorie de Galois, À la poursuite des champs).

    After the first part, looking ahead there’s the daunting prospect of 1500 pages with the theme of examining his deepest mathematical ideas and what he felt was the “burial” that he and his ideas had been subjected to after his leaving active involvement with the math research community in 1970. Quite a few years ago I did spend some time looking through this part to try and learn more about Grothendieck’s mathematical ideas. I’ll see if I can try again, with the advantage of now knowing somewhat more about the mathematical background.

    Besides the frustrating aspects, what has struck me most about this is that there are many beautifully written sections, capturing Grothendieck’s feeling for the beauty of the deepest ideas in mathematics. One gets to see what it looked like from the inside to a genius as he worked, often together with others, on a project that revolutionized how we think about mathematics. This material is really remarkable, although embedded in far too much that is extraneous and repetitive. The text desperately needs an editor.

    There are various places online one can find parts of the book and other related material, sometimes translated. Two places to look are the Grothendieck Circle, and Mateo Carmona’s site.

  • For an up-to-date project on reworking foundations of mathematics (with an eye to eliminating analysis…), Dustin Clausen and Peter Scholze are now teaching a course on Condensed Mathematics and Complex Geometry, lecture notes here.
  • I noticed that the Harvard math department website now has an article on Demystifying Math 55. The past couple years this course has been taught by Denis Auroux, and one can find detailed course materials including lecture notes at his website.

    The current version of the course tries to cover pretty much a standard undergraduate pure math curriculum in two semesters, with the first semester linear algebra, group theory and finite group representations, the second real and complex analysis. The course has gone through various incarnations over a long history, and has its own Wikipedia page. For various articles written about the course over the years, see here, here (about a Pavel Etingof version) and here (about a Dennis Gaitsgory version).

    I took the course in 1975-76, when the fall semester was taught by mathematical physicist Konrad Osterwalder, who covered some linear algebra and analysis rigorously, following the course textbook Advanced Calculus by Loomis and Sternberg. The spring semester was rather different, with John Hubbard sometimes following Hirsch and Smale, sometimes giving us research-level papers about dynamical systems to read, and then telling us to read and work through Spivak’s Calculus on Manifolds over reading period.

    My experience with the course was somewhat different than that described in the articles above, partly due to the particular instructors and their choices, partly due to the fact that I was more focused on learning as much advanced physics as possible. I don’t remember spending excessive amounts of time on the course, nor do I remember anyone I knew or ran into being especially interested in or impressed by my taking this particular course. What was a new experience was that it was clear the first semester that I was a rather average student in the class, not like in my high school classes. The second semester about half the students had dropped and I guess I was probably distinctly less than average. The current iteration of the course looks quite good for the kind of ambitious math student it is aimed at, and it would be interesting if a new textbook ever gets written.

Update: One more related item. This week Chapman University is hosting a conference about Grothendieck. Kevin Buzzard has posted his slides here.

Posted in Uncategorized | 16 Comments

Two Titans

If one tried to pick a single most talented and influential figure of the past 100 years in each of the fields of pure mathematics and of theoretical physics, I’d argue that you should pick Alexander Grothendieck in pure math and Edward Witten in theoretical physics. This afternoon I’ve run across two excellent sources of information about each of them.

Alexander Grothendieck

This week’s New Yorker has The Mysterious Disappearance of a Revolutionary Mathematician by Rivka Galchen. It’s a very well-done survey of Grothendieck’s life and work, aimed at a popular audience. If, like many mathematicians, you’ve always been fascinated by Grothendieck’s story, you won’t find too much in the article you haven’t seen before. But, if you’ve never delved into this story, you should read the article. On a related note, a copy of Récoltes et semailles that I ordered recently has just arrived in the mail, and I’m looking forward to spending some time with that this summer.

Edward Witten

In theoretical physics a very different but equally off-the-scale talented and influential figure is Edward Witten, who is the subject of a recent long and in-depth interview by David Zierler as part of the Oral Histories program at the Niels Bohr Library and Archives.

I first met Witten probably in 1977, when I was an undergraduate at Harvard and he was a Junior Fellow, recently arrived from Princeton. Over the years since then he has done a mind-blowing quantity of highly impressive work which I’ve done my best to try to follow. You can find many places where I’ve written about this here on the blog, and there’s also a lot in my book Not Even Wrong. Much of what he discussed in the interview was familiar to me, but I learned quite a bit new from his recollection of the details of how his work came about and how he thought about it. On some of the specifics of what happened many decades ago one should keep in mind that memory is imperfect. For instance, he describes a short period as a graduate student in economics at the University of Michigan, which surprised me since in research for my book I’d read that this was at the University of Wisconsin. Maybe I got this wrong, but if so I’m not the only one (see for instance here).

Witten’s work in the area where pure mathematics and quantum field theory overlap has had an overwhelming influence on those like myself who are fascinated by both subjects and their interaction. The landscape of this area would be completely different (and highly impoverished) without him. At the same time, his equally large influence in the area of attempts to unify physics I believe has been much more problematic.

I’ll quote here with a little commentary some of the passages from the interview that I found striking or where I learned something new.

About his early years:

I was very interested in astronomy when I was growing up. Well, I was not an exception; these were the days of the Space Race, so everybody was interested in astronomy. I was given a small telescope when I was about nine or ten. That’s certainly a vivid memory. Another vivid memory is learning calculus when I was eleven. My father sort of taught me calculus or gave me materials from which I could learn it. But I didn’t advance very much in math beyond that for quite a few years…

And then [after college at Brandeis] initially you thought you would go and become an economist?


What were your interests there? Did you think that your mathematical abilities would be applied well in that field?

It’s again hard to remember reliably, but I might have thought that. And I might have also thought that I could make a contribution to international development. But I realized- well, I came to the same realization I had come to when I was working on the McGovern campaign, that it wasn’t a good match for me. I remember being very embarrassed when I told the people in the department at Michigan who had been quite kind to me, that I had decided to leave. But in hindsight, I understand something that wasn’t that clear to me at the time, that if a given graduate program isn’t a good match for a given student, the department and the student are both better off if that’s realized sooner rather than later. If I had understood that at the time, I would have been less embarrassed, probably, with what I told them.

How to learn general relativity in ten days:

Was general relativity considered popular or interesting at Princeton at the time that you were a graduate student?

Well, I was certainly interested in it. I learned general relativity in a very exciting period of about ten days, from the book of Steve Weinberg. I mean, I tried to learn more from the book of Misner, Thorne, and Wheeler, and I did learn more from it, but my opinion of the book was what it remains now, which is that it’s got a lot of great stuff in it, but it’s a little bit hard to use it to learn systematically. The book I found useful for studying systematically was by Steve Weinberg.

The Harvard Society of Fellows:

Ed, did you enjoy the Harvard Society of Fellows, the social aspects of it?

Well, I enjoyed it up to a point, but let’s just say that many other people thrive on that more than I did.

On how he experienced the First Superstring Revolution.

I’m not exactly sure what I would have said if you had asked me. There’s no interview, so there’s no record of my thinking in 1982 or 1983, and I won’t be able to remember very well. But as I was telling you, I was interested enough to spend a whole summer reading John Schwarz’s review article, but a little bit wary of becoming too involved in it…

Something that was obvious to me but wasn’t immediately completely obvious to everybody was this. Green and Schwarz had put string theory in the form where there was a very strong case that there was a consistent quantum theory that described gravity together with other forces. And the other forces could be gauge fields, somewhat like in the Standard Model. But there was something extremely conspicuous that was wrong in terms of phenomenology, and that was that the weak interactions couldn’t violate parity…

And as it existed in 1982 and 1983, string theory was a consistent theory of gravity unified with other forces, but it completely missed the chiral structure. So, to me, that was a huge siren blaring. Anyway, to set the stage, I want to just point out to you that it was clear by 1982 or 1983 that there were an incredible variety of delicate things that fit together perfectly to make it possible to have a theory of quantum gravity based on string theory. It was unbelievable that it could all be a coincidence. Yet it was markedly wrong for describing the real world because of this question of the chiral nature of the fermions. But then in 1984, Green and Schwarz discovered a more general method of anomaly cancellation, and everything changed…

So, anyway, what was really problematical for Green and Schwarz was the combination of fermion chirality and anomalies. Taking these together, it seemed that string theory could not work. But then, in August 1984, Green and Schwarz discovered a new mechanism for anomaly cancellation, and everything changed…

So, it was immediately obvious to me, once they made their discovery, that you could make at least semi-realistic models of particle physics, in that framework. But also, to me, I had done kind of an experiment in the following sense. I had spent two years watching this, wondering, could it be? Can it be that all the coincidences that had been discovered that made string theory possible were just coincidences? As far as I was concerned, the discovery they made in 1984 was an empirical answer of “no” to that question. If the miraculous-looking things that had been discovered up to 1982 and 1983 were truly coincidences, you’d then predict there wouldn’t be any more such coincidences. That had proved to be wrong when they made this miraculous-looking discovery about anomalies that enabled the theory to be much more realistic.

In explaining this to you, I’m trying to help you understand why this had so much of an impact on my thinking, watching from the outside for a couple years, wondering if this subject was as amazing as it appeared to me. And a “no” answer would have predicted there shouldn’t be more miraculous discoveries. And that was, to my satisfaction, disproved in August 1984. So, after that, the hesitation that had kept me from becoming more heavily involved earlier evaporated. Now, I realized that in the physics world, there were plenty of people who hadn’t lived through this two years of uncertainty that I had lived through, and in many cases they had never heard of the whole thing until August 1984. And they hadn’t done the experiment I had done. So, they didn’t react as I did.

Here Witten explains how one very specific technical calculation triggered for him a dramatic vision of a possibility of a unified theory of everything, a vision that has stayed with him to this day, nearly four decades later.

About his evangelism for string theory unification starting in 1984:

How much cheerleading did you do among your colleagues, both near and far, after this revolution in 1984, that this is what people should concentrate on? That we can have this figured out in the near term?

I wasn’t intentionally cheerleading, but I was very enthusiastic. And I actually think I was right to be enthusiastic. I wasn’t intentionally cheerleading, but to the extent that I encouraged other people to get involved, I’ve got no regrets about it at all (laughter).

Another very interesting recent interview in the same series is one with Cumrun Vafa. Here’s what Vafa remembers about that time:

I remember I was at my office, I had come back from a trip, from I think the summer school in Europe, in Italy. Had come back to my office in Princeton on the fourth floor, and Ed’s office is on the third floor. And he rarely came to our floor, fourth floor, but here he was, coming and knocking at my door, and then saying, “Have you heard about the revolution?”…

I said, “What revolution?” He said, “The SO(32) revolution.” Okay, that was my first introduction to Green and Schwarz’s work. SO(32) revolution. I said, “No, what is it?” He said, and he was completely sure, confident, that physics is not going to be the same after this. He said, “Physics is going to change forever because of this, and now everybody is going to work on this.”

I had left Princeton for Stony Brook early that summer. During the next few years, reports I got from fellow postdocs who tried to talk to Witten about their work were pretty uniformly something like “he told me that what I’m doing is all well and good, but that I really should be working on string theory.”

Unlike the case in the interview with Vafa previously mentioned, Zierler doesn’t really try and pin Witten down on the subject of the problems of string theory. He does ask:


What was happening at the time or has happened since in the world of experimentation or observation that may get us closer to string theory being testable?

but lets Witten give a non-answer, which in effect is that the landscape means string theory unification is completely untestable, so he has pretty much given up:

So, if you talked to me in the 1980s, I’m sure I would have expressed some hope about seeing supersymmetry as part of the answer of the hierarchy problem. But I would have expressed a lot of confidence about observing something that would have explained the hierarchy problem. …

But ultimately, with the LHC, experiment has reached the point that it’s extremely problematic to have what’s called a natural explanation of the weak scale, a mechanism that would explain in a technically natural way why the Higgs particle is as light as it is, thus making all the particles light. It’s actually a baby version of the problem with the cosmological constant. So, to the extent that the multiverse is a conceivable interpretation of why the universe accelerates so slowly, it’s also a conceivable explanation of why the weak scale is so small. It might be the right interpretation. But if it is, it’s not very encouraging for understanding the universe. When the multiverse idea became popular around 1999, 2000, and so on, I was actually extremely upset, because of the feeling that it would make the universe harder to understand. I eventually made my peace with it, accepting the fact that the universe wasn’t created for our convenience.

You would think that having an untestable theory on your hands would mean that you would try something else, anything else, but Witten seems convinced that whatever its problems, it’s the only way forward:

[About the second superstring revolution and M-theory] It’s satisfying to know that there was only one candidate for superunification. There’s only one reasonable candidate now for the theory that combines gravity and quantum mechanics. Before 1995, there was more than one. It’s more satisfying to know that the theory seems to have a lot of possible manifestations, in terms of approximate vacuum states, but at a fundamental level, there’s only one fundamental theory or system of equations, that we admittedly don’t understand very well. That’s got to be an advance of some kind…

By the time he [Einstein] had the theory [GR], he had the right mathematical framework of Riemannian geometry. At least by the time the theory was invented, he had the ideas it was based on, and some of them he had had before.

String theory and M-theory have always been different. From the beginning, they were discovered by people who discovered formulas or bits and pieces of the theory without understanding what’s behind it at a more fundamental level. And what we understand now, even today, is extremely fragmentary, and I’m sure very superficial compared to what the real theory is. That’s the problem with the claim that supposedly I invented M-theory. It would make at least as much sense to say that M-theory hasn’t been invented yet. And you could also claim it had been invented before by other people. Either of those two claims is defensible (laughter). So I made some incremental advances in a subject that’s far from being properly understood.

This “we don’t know what the fundamental equations are, but we know that they are unique” argument has never made any sense to me.

On his relation to mathematics:

What did it feel like to win the Fields Medal as a physicist?

Well, it was a thrill, of course. It felt a little funny because I knew that obviously I was a non-standard selection. And I don’t like controversy about science, and I felt that I might have been a controversial choice in the math world. But on the other hand, I hadn’t selected myself, so I didn’t feel any controversy was my fault…

What’s a little funny about my relation to the math world is that although some of my papers are of mathematical interest, they rarely have the detail of math papers. And I can’t provide that detail. I simply don’t have the right background. What I bring to the subject is an ability to understand what quantum field theory or string theory have to say about a math question. But quantum field theory and string theory are not in the precise mathematical form where such statements can usually be rigorous.

The “I don’t like controversy about science” quote makes clear that Witten and I are temperamentally very different…

About the birth of geometric Langlands:

By the late 1980s- I’m probably forgetting bits of the story, I should tell you- but by the late 1980s, Sasha Beilinson and Vladimir Drinfeld had discovered what they called a geometric version of the Langlands program, and it involved ingredients of quantum field theory. Tantalizing. But it was tantalizing because they were using familiar ingredients of quantum field theory in a very unfamiliar way. It looked to me as if somebody had put the pieces at random on a chess board. The pieces were familiar, but the position didn’t look like it could happen in a real chess game. It just looked crazy. But anyway, it was clear it had to mean something in terms of physics. I even worked on that for a while at the time.

I think I’ve gotten this slightly out of order. I think when I worked on it was actually before the work of Beilinson and Drinfeld, driven by other clues. And the Beilinson and Drinfeld work was one of the things that made me stop, because I realized that A, I couldn’t understand what they were doing at the time, and B, there were too many things I didn’t know that they knew, and that seemed to be part of the story. Anyway, as you can see, my memories from whatever happened in the late 1980s are pretty scrambled.

They wrote a famous paper that was never finished and never published. It’s 500 pages long. You can find it online, if you like. They have an incredibly generous acknowledgement of what they supposedly learned from me, which is way exaggerated. Based on a hunch, I told them about a paper of Nigel Hitchin, but I didn’t understand anything of what they attributed to me. At any rate, regardless, even if I didn’t understand what they did with it, the fact that I was able to point them to the right paper was another sign of the fact that what they were doing had something to do with the physics I knew. But I couldn’t make sense of the connection. And this kept nagging at me off and on for a long time.

He then goes on to tell the story of the IAS workshop on geometric Langlands and how it led to his work on a QFT version of geometric Langlands.

In recent years Witten has continued to work on geometric Langlands and other topological quantum field theory related topics at the mathematics end of things. As far as physics goes, he is following the very popular “it from qubit”, quantum gravity from information theory, line of thinking:

And the third time [revolutions: first and second were two superstring revolutions] has been the last six or seven years. It’s actually hard to remember the evolution of my thinking (laughter). I reread an interview I had done in 2014 which told me what my thinking was in 2014 better than I could have remembered it reliably (laughter). And what I told the interviewer at that time was somewhat similar to what I’m telling you right now. So, this has gone on for a while, and despite that, I haven’t really found the right way to become involved myself. But I do suspect that something big is happening.

What has happened since 2014, when you initially got excited about this?

There have been various striking developments, but a particularly dramatic one came in 2019 when there was success in understanding what is known as the Page curve in black hole evaporation… Lots of things have happened that show that there’s a conspiracy between gravity and quantum mechanics. Somehow gravity at the classical level knows about quantum mechanics and statistical mechanics…

To bring the conversation right up to the present, as we discussed right at the beginning, your interest in quantum information. And you said you don’t yet know how you might break into the field. What might be some possible avenues?

Well, when I was a graduate student, I sat down one day with piles of paper preprints. We didn’t have the archive. I’d sit down with piles and piles of paper preprints, and go through them, trying to find something I might do. The most interesting calculation I did as a student- I told you about it- was this calculation of deep inelastic photon-photon scattering, which was inspired by a paper I saw by Roger Kingsley, who studied the question but not quite with the most modern QCD ideas. So, when I was a graduate student trying to break in, I would go through piles of preprints. I guess the equivalent now is to look at papers in the archive and try to see what I might do. And I have made some minor contributions, actually, but I don’t feel like I’ve fully become engaged with the subject, as I have with other subjects in the past.

Witten and the interviewer discuss the difficulty of finding something to work on that is not too hard but still significant, and he comments that this is:

…the difficulty I’ve had getting involved with quantum information theory and gravity. I found a few things that I could do, but they were a little bit too narrow to really make me think that I was getting involved where I wanted to. And I haven’t quite found the right avenue. But I haven’t given up (laughter). I do have the feeling that’s the direction where something big is most likely to happen. You see, there isn’t a general understanding of what string/M-theory mean. And there’s something missing in the general understanding of quantum gravity. The biggest hope would be that those two would somehow make contact with each other.

I can understand why Witten hopes that the mystery of quantum information theory and gravity will give insight into and resolve the mystery of what M-theory is, finally vindicating his 1984 vision, but this looks to me like a very, very long shot.

Posted in Uncategorized | 49 Comments

Brian Conrad on the California Mathematics Framework

Brian Conrad has been doing the state of California a great service by taking a careful look at the drafts of the proposed California Mathematics Framework and the research they are supposedly based on.   He has recently created a website where he has been writing up commentary on what he has found. Conrad is known among his colleagues as one of the most careful and level-headed research mathematicians around, and these characteristics show through in what I have read on his new site.

I should make it clear that personally I have zero expertise on the topic of K-12 math education, so my own views on the matter aren’t worth much (and, I think anyone commenting on this should ask themselves about the same issue). Conrad has put a huge amount of his time and effort into learning about the subject and the extensive relevant math education research, and it is this that makes paying attention to the views of a university math professor a good idea in this case.

There’s commentary about this appearing elsewhere, including a Wall Street Journal article, and a blog entry by my Columbia colleague Andrew Gelman.

Posted in Uncategorized | 9 Comments

Three Quick Items

Just time for three quick items:

  • There’s a wonderful book out now published by the Simons Center at Stony Brook, with the title Crossings. It tells the story of the center and of various people involved with it through a large number of interesting pieces written by these people. The book is published by the center, available here.
  • Symmetry magazine has an article out today, with the title Can a theory ever die?. It’s largely about supersymmetry, with “No” the answer to the title question. There’s a story about Bruno Zumino I’d never heard before:

    In 1996 theorist Jonathan Feng attended a seminar about searches for new particles predicted by the mathematically elegant theory of Supersymmetry. The speaker was optimistic that researchers would find the particles at massive colliders such as the Tevatron, then in operation at the US Department of Energy’s Fermi National Accelerator Laboratory, or the Large Hadron Collider, then under construction at CERN.

    Feng noticed Bruno Zumino, one of the founders of Supersymmetry, in the audience. Zumino’s reaction to the talk confused Feng.

    “He left the seminar shaking his head,” says Feng, who is now a professor at the University of California, Irvine. “I thought he would be happy that an army of people were looking for his theory. So why was he shaking his head?”

    Feng caught up with the distinguished theorist during the coffee break. He still remembers what Zumino told him: “I never thought it would be this hard. If it’s this hard, then they’re never going to find it.”

    So, twenty-six years ago one of the leaders of the field thought the idea was going nowhere and likely doomed. In the years after that LEP and the Tevatron put much stronger limits on SUSY before closing down in 2000 and 2011 respectively.
    From 2010 to the present day, the LHC has again put far stronger limits on SUSY particles. Most now agree it is overwhelmingly unlikely that the rest of the LHC or HL-LHC runs will change the situation. And, prospects for a higher energy collider are very uncertain and many decades away. You’d think that would be the end of it.

    But the article quotes theorists determined to keep at it (no quotes from anyone who thinks SUSY is over) and there’s still an active community of people pursuing what Zumino thought was doomed multiple decades and accelerator generations ago. Large conferences continue to be scheduled, for example SUSY 22 this summer, which will be preceded by a pre-SUSY school designed to train a new generation to work on the failed ideas for many decades to come.

  • I’m leaving soon to spend a couple days in Texas, giving a colloquium talk at the University of Texas at Dallas math department.

Update: The slides from the talk at UT Dallas are here.

Posted in Uncategorized | 6 Comments

Various and Sundry

  • Last week a review of the Mochizuki IUT papers appeared at Math Reviews, written by Mohamed Saïdi. His discussion of the critical part of the proof is limited to:

    Theorem 3.11 in Part III is somehow reinterpreted in Corollary 3.12 of the same paper in a way that relates to the kind of diophantine inequalities one wishes to prove. One constructs certain arithmetic line bundles of interest within each theatre, a theta version and a q-version (which at the places of bad reduction arises essentially from the q-parameter of the corresponding Tate curve), which give rise to certain theta and q-objects in certain (products of) Frobenioids: the theta and q-pilots. By construction the theta pilot maps to the q-pilot via the horizontal link in the log-theta lattice. One can then proceed and compare the log-volumes of the images of these two objects in the relevant objects constructed via the multiradial algorithm in Theorem 3.11.

    Saïdi gives no indication that any one has ever raised any issues about the proof of Corollary 3.12, with no mention at all of the detailed Scholze/Stix criticism that this argument is incorrect. In particular, in his Zentralblatt review Scholze writes:

    Unfortunately, the argument given for Corollary 3.12 is not a proof, and the theory built in these papers is clearly insufficient to prove the ABC conjecture….
    In any case, at some point in the proof of Corollary 3.12, things are so obfuscated that it is completely unclear whether some object refers to the q-values or the $\theta$-values, as it is somehow claimed to be definitionally equal to both of them, up to some blurring of course, and hence you get the desired result.

    After the Saïdi review appeared, I gather that an intervention with the Math Reviews editors was staged, leading to the addition at the end of the review of

    Editor’s note: For an alternative review of the IUT papers, in particular a critique of the key Corollary 3.12 in Part III, we refer the reader to the review by Scholze in zbMATH:

    Since the early days of people trying to understand the claimed proof, Mochizuki has pointed to Saïdi as an example of someone who has understood and vouched for the proof (see here). Saïdi is undoubtedly well aware of the Scholze argument and his decision not to mention it in the review makes clear that he has no counter-argument. The current state of affairs with the Mochizuki proof is that no one who claims to understand the proof of Corollary 3.12 can provide a counter-argument to Scholze. Saïdi tries to deal with this by pretending the Scholze argument doesn’t exist, while Mochizuki’s (and Fesenko’s) approach has been to argue that Scholze should be ignored since he’s an incompetent. The editors at PRIMS claim that referees have considered the argument, but say they can’t make anything public. This situation makes very clear that there currently is no proof of abc.

  • At one point the American Institute of Mathematics (founded in 1994 with financing from John Fry) was supposed to move from its location behind a Fry’s Electronics store to a castle in Morgan Hill modeled on the Alhambra (see here). This never worked out, and last year Fry’s Electronics declared bankruptcy. The latest news is that next year AIM will move to Caltech, for more see here.
  • I’ll never understand why places like MIT continue to teach undergraduate courses on a failed speculative idea about physics.
  • There has been a lot of coverage in the press of claims by a group analyzing old CDF data to have come up with a dramatically better value for the W mass (one seven sigma away from the SM value). While this would be really wonderful if it were true, unfortunately that doesn’t seem very likely. There isn’t a well-motivated theoretical reason for this discrepancy, this is a very challenging measurement, and the new value seriously disagrees with several previous measurements at CERN. For an informed discussion of this from someone who was on CDF and has worked on these sorts of analyses, see Tommaso Dorigo’s blog post.
  • It will be interesting to see how well the LHC experiments can ultimately do this measurement. The LHC is about to start up again after a long shutdown, with beam commissioning starting on Friday.
Posted in abc Conjecture, Uncategorized | 24 Comments

Two New Quantum Field Theory Books

I’ve recently noticed that two very good new books on quantum field theory have become available, one aimed more at mathematicians, one purely for physicists.

What Is a Quantum Field Theory?

Available online now from Cambridge University Press (actual printed books to come soon) is mathematician Michel Talagrand’s What Is a Quantum Field Theory?. While it’s subtitled “A First Introduction for Mathematicians” and definitely aimed more at mathematicians than physicists, it’s a wonderful resource for anyone who wants to understand exactly what a quantum field theory is.

Like many mathematicians, Talagrand tried to learn about quantum field theory first from physics textbooks, which tend to avoid any precise definition of even the basics of the subject. He soon found what was the best source for someone looking for more precision, Gerald Folland’s 2008 Quantum Field Theory: A Tourist Guide for Mathematicians. Folland’s book is extremely good, but also extremely terse. In 325 pages it covers more carefully the material of an old-style QFT book such as Schweber’s 900 page or so An Introduction to Relativistic Quantum Field Theory from 1961. Talagrand is covering much the same material, but with 742 pages to work with he is able (unlike Folland) to work out many topics in full detail, providing something previously unavailable anywhere else.

Both Folland and Talagrand have written books with much the same goal: to as precisely as possible explain the details of the renormalized perturbative expansion of QED. There is little overlap with the work of mathematical physicists who have aimed at rigorous non-perturbative constructions of quantum field theories. They are using canonical quantization methods and don’t overlap much with many of the more recent physics QFT textbooks, which are based on path integral quantization and aimed at getting to non-abelian gauge theories and non-perturbative techniques as quickly as possible.

When I was learning QFT not that long after the advent of the Standard Model, I had little patience for fat QFT books about perturbative QED and canonical methods. Why not just write down the path integral and start calculating? Over the years I’ve realized that things are not so simple, with canonical quantization and operator fields giving a perspective complementary to that of the path integral. Among the more modern books, volume 1 of Weinberg’s three-volume series is the one that best gives this different perspective, and is most closely related to what Talagrand is covering.

For mathematicians, Talagrand’s book is a great place to start. For physicists, Weinberg’s is an important perspective to get to know. If you’re reading Weinberg and want more detail about precisely what is going on, Talagrand’s new book would be a very good place to turn for help.

Quantum Field Theory: An Integrated Approach

Over the years I’ve often consulted various parts of Eduardo Fradkin’s notes on quantum field theory on his web pages. On some basic topics I found these to give very clear explanations of things that were done in a confusing way elsewhere. After recently hearing that the notes are now a book from Princeton University Press, I ordered a copy, which recently arrived.

Fradkin’s book has not much overlap with the material in the Talagrand book described above, and is somewhat different than traditional high energy physics-oriented QFT books. It tries as much as possible to integrate the high energy physics point of view with that of condensed matter and statistical mechanics. Path integral methods are then fundamental. Unlike many other modern QFT textbooks that aim at getting to the details of perturbative Standard Model calculations, Fradkin is more oriented towards getting as quickly as possible to non-perturbative techniques and models of interest in statistical mechanics. He gives a good introduction to various of the modern non-perturbative QFT techniques that have been developed in recent decades, often motivated by the so far only partially successful attempt to come to terms with a strongly-interacting gauge theory like QCD.

While most of the book is quite good, the first few pages aren’t, and will immediately drive away mathematicians who might pick it up. The material in these pages about group theory uses bad terminology (for Fradkin, the “rank” of a Lie group is its dimension and the fundamental representation of SU(n) is the “spinor” representation) and sometimes is just completely wrong. On the second page of the first chapter after the introduction, he wants to explain why the Lorentz group is non-compact, in contrast to SO(3). To explain why SO(3) is compact he starts by mistakenly arguing that since it leaves the unit two-sphere invariant the points of SO(3) and of the unit two-sphere are in one-to-one correspondence, showing the volume of SO(3) is $4\pi^2$. This paragraph should be deleted in future editions of the book.

That this kind of thing can make it into a book like this is remarkable, but unfortunately relativistic QFT books and other sources (e.g. here) don’t always get right basic facts about the Lorentz and rotation groups. I once tried to do my part to remedy this, see here.

Update: John Collins has here an article that provides a careful discussion of scattering in QFT, starting with the basics, which could be thought of as part of a QFT book. This may be of interest to both physicists and mathematicians who want to see something less superficial than many text book discussions.

Posted in Book Reviews | 35 Comments


There will be a documentary broadcast tomorrow in Japan on Mochizuki’s claimed proof of the abc conjecture. I was interviewed for this by the filmmakers last year, but don’t know anything about whether and how that footage will be used. I’d be curious to hear reports from any Japanese-speaking readers who see the documentary tomorrow.

Over the years there has been a detailed coverage of this story here on the blog. To make it more accessible, I’ve added an abc conjecture category. In case the documentary doesn’t make this clear, the current consensus of experts in the field is that there is no proof. Peter Scholze and Jacob Stix identified a problem with Mochizuki’s proof in 2018 (discussed in detail by Scholze and others here), and Mochizuki has not provided a convincing answer to their objections. No one else (including the journal editors who published the proof in PRIMS) has been able to provide a clear explanation of the problematic part of the proof.

Update: NHK has two web pages summarizing the content of the program, see here and here for English translations.

Taylor Dupuy is still making implausible claims that Scholze’s criticism of the proof is invalid. To judge for yourself, see here a long detailed discussion of the issue between them involving several other experts.

Reports I’m seeing from those who have watched the program say that it does correctly explain that the proof is not accepted by many experts.

Posted in abc Conjecture | 16 Comments

The Anti-Science Movement

I noticed recently that Stony Brook is hosting next week a panel discussion devoted to

a conversation about one of the most grave challenges to confront humanity: the anti-science movement.

There is a truly grave challenge being referred to, but a serious mistake is being made about the nature of the challenge. In particular, there’s no evidence of an “anti-science” movement, quite the opposite. Across the globe, if you ask people what profession they respect the most, “scientist” comes out on top (see here). Likely the organizers have in mind climate denialists and anti-vaxxers as prime examples of “anti-science” behavior, but in my experience such people typically show a great devotion to pointing to scientists, scientific results and scientific papers to justify themselves. An example would be Lubos Motl, who has put out literally thousands of pages on his blog about climate and COVID science (by the way, his blog seems to have gone “by invitation only”, anyone know what that’s about?).

The problem isn’t “anti-science”, but bad science, promoted for ideological reasons. This is part of a larger truly grave challenge to humanity, that of our information environment being flooded with untruth, on a scale that dwarfs the output of the Ministry of Truth that Orwell foresaw. For years now we’ve been living with this in the form of phenomena like Trumpism, and the past few weeks have seen the Russian government exploiting these methods to conduct a campaign of brutal slaughter. I don’t know what the best way to address this challenge is, but unless something can be done, humanity has an ugly and disturbing future ahead of it.

Sticking to the problem of what to do about the promotion of bad science, there at least I have some experience trying to do something about one example of it (although with very limited success). This problem deserves attention and a panel discussion, but a panel in which four of six members have devoted a significant part of their careers to promoting a failed scientific research program is a really odd choice.

The underlying thorny issue is that of how to evaluate scientific claims. Given the complexities of controversial science, non-experts generally have little choice but to try and identify experts and trust what they say. A major societal role of elite institutions is to provide such experts, ensuring that they provide trustworthy expertise, untainted by ideology or self-interest. A large part of what is going on these days seems to me to reflect a loss of faith in elite institutions, with an increasing perception that these are dominated by a well-off class pursuing not truth, but their own interests. As a product of such institutions I’m well aware of both their strengths and their weaknesses. We need them to do better, and in this case Stony Brook should come up with a better panel.

Update: I’ve heard that Lubos himself shutdown the blog, unwilling to agree to follow rules Google was now enforcing.

Posted in Fake Physics, Uncategorized | 42 Comments

Is Space-Time Really Doomed?

For many years now the consensus in a dominant part of the theoretical physics community has been that the center of attention should be on the problem of quantizing gravity, and that conventional notions of quantum theory and space-time geometry need to be abandoned in favor of something radically different. The slogan version of this is “Space-Time is Doomed.”

Ever since my student days long ago, I’ve spent a lot of time looking into the problems of quantum gravity and what people have tried to do to address these problems. The highly publicized attempts to get known physics out of radically different degrees of freedom that I’ve seen haven’t seemed to be making any progress, remaining very far from anything like known physics. In the case of string theory, which also claimed to be able to get particle physics, there was at one point a (highly over-hyped) relatively well-defined proposal that one could discuss, but that’s no longer the case.

Recently things have changed as I’ve become convinced of the promise of certain specific ideas about four-dimensional geometry involving twistors and Euclidean space-time signature. I’ve written about these here and on the blog, and have given some talks (see here and here). These ideas remain speculative and incomplete, but I think they provide some new ways of thinking about the problems of quantizing gravity and unifying it with the other forces.

The existence of a yearly essay competition gave me an excuse to write something about this which I just finished yesterday and sent in, with the title Is Space-Time Really Doomed?. After spending some time on a diversion into arithmetic geometry, I’ve been getting back to seriously thinking about this topic, looking forward to having time in coming months to concentrate on this. I hope the essay will encourage others to not give up on 4d geometry as doomed and unquantizable, but to realize that much is there still waiting to be explored.

Update: The essay is now on the arXiv here.

Update: Awards for this announced here. I got an honorable mention.

Posted in Euclidean Twistor Unification | 22 Comments

2022 Abel Prize to Dennis Sullivan

This year’s Abel Prize has gone to topologist Dennis Sullivan, for the announcement see here, with more information about Sullivan and his work here. There are press stories at Nature, the New York Times, Quanta, and elsewhere.

Sullivan was one of the leading figures in great advances in understanding the topology of manifolds in higher dimensions during the late 60s and 70s. Some of the best of his early work for many years was only available if you could find a copy of unpublished mimeographed notes from a 1970 MIT course. In 2005 a Tex’ed version of the notes was finally published (available here). This includes as a postscript Sullivan’s own description of this work, how it came about, and how it influenced his later work.

This was followed by wonderful work on rational homotopy theory, making use of differential forms. For this, see Sullivan’s 1977 Infinitesimal computations in topology, and lecture notes on this by Phil Griffiths and John Morgan. In later years Sullivan’s attention turned to subjects with which I’m not very familiar: topics in dynamical systems and the development of what he called “string topology”.

Since 1981 Sullivan has held the Einstein chair at the CUNY Graduate Center here in NYC, running a seminar each week that concentrates on the relation between topology and QFT. For many years these were held in a Russian style, going on for multiple hours, possibly with a break, until all participants were exhausted. There’s a remarkable collection of videos of these lectures at the seminar site, including many going way back into the 80s and 90s, with video recorded at a time when this was quite unusual (more recent ones are on Youtube).

When I first came to Columbia Sullivan was often here attending and giving lectures, for many years splitting his time between Paris (where he held a position at the IHES), New York and Rio. The Abel Prize biography explains

In 1981, Sullivan was made the Albert Einstein Chair in Science (Mathematics) at the Graduate School and University Center of The City University of New York. He kept his position at IHES and spent the next decade and a half shuttling between Paris and New York, often on Concorde.

Some of the various stories I heard about Sullivan’s lifestyle at the time involved his having multiple apartments in New York, which he used to host a variety of visiting mathematicians. Another story I heard directly from him was about how he survived an attempted car-jacking in Brazil, during which he was shot, but managed to escape and drive himself to a hospital for treatment. I had first heard about this from Mike Hopkins several years before. When I asked Hopkins why he had become a topologist, he said that one factor was the inspiring example of people like Sullivan who worked in the field, jokingly characterizing it as involving “real men who got into gun-fights”.

In 1997 Sullivan traded the IHES position for one at Stony Brook, and over the years has unfortunately been seen less often here at Columbia. Congratulations to him on the well-deserved prize!

Posted in Uncategorized | Comments Off on 2022 Abel Prize to Dennis Sullivan