This Week’s Hype

Nanopoulos and co-authors have predictions from superstring theory that are “in strong agreement with NANOGrav data.” He has been at this now for almost 40 years. See for instance Experimental Predictions from the Superstring from 1985, where the superstring predicted a top mass of 55 GeV and 360 GeV squarks.

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Understanding Confinement

This week and next there’s an interesting summer school going on at the IAS, with topic Understanding Confinement. Videos of talks are available here or at the IAS video site.

Taking a look at some of the first talks brings back vividly my graduate student years, which were dominated by thinking about this topic. When I arrived in Princeton in 1979, the people there had been working for several years on trying to understand confinement semi-classically, in terms of instantons and other solutions to the Yang-Mills equations (e.g. merons). By 1979 it had become clear that such semi-classical calculations were not sufficient to understand confinement and people were looking for other ideas. There were quite a few around, including the idea that there was some sort of string theory dual to pure Yang-Mills theory, and I spent quite a lot of time reading up on efforts of Migdal, Polyakov and others to find a formulation of string theory that would provide the needed dual. I ended up writing my thesis on lattice gauge theory, an approach which had the great advantage that you could at least put the calculation on a computer and start trying to get a reliable result for pure Yang-Mills numerically. Some of the calculations I did were done at the IAS, with Nati Seiberg and others. The other thing I spent a lot of time thinking about was how to put spinor fields on the lattice, the beginning of my interest in the geometry of spinors.

I strongly recommend watching Witten’s talk on Some Milestones in the Study of Confinement. His career started a few years before mine, with the early part very much dominated by the problem of how to make sense of Yang-Mills theory non-perturbatively, and this has has always been a motivating problem behind much of his work. In his talk he explains clearly the approaches to the problem (lattice gauge theory, 1/N, dual Meissner) that appeared very soon after the advent of QCD in 1973. He emphasizes how each of these approaches shows indications of a possible string theory dual, while frustratingly not leading to a string model that has the right properties, summarizing (41:30) the situation with:

The string theory we want is probably quite unlike any that we actually know, as of now. We don’t know how to make a string theory with the short distance behavior of asymptotic freedom.

In his talk he discusses later developments, in particular the Seiberg-Witten solution to N=2 SYM and the AdS/CFT duality between a string theory and N=4 SYM, explaining how these advances still don’t provide a viable approach to the confinement problem in pure Yang-Mills.

I’m looking forward to seeing the rest of the talks, and finding out more about some things that have happened over the years since I was most actively paying attention to what was happening with the confinement problem.

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Relative Langlands Duality

For several years now, David Ben-Zvi, Yiannis Sakellaridis and Akshay Venkatesh have been working on a project involving a relative version of Langlands duality, which among many other things provides a perspective on L-functions and periods of automorphic forms inspired by the quantum field theory point of view on geometric Langlands. For some talks about this, see quite a few by David Ben-Zvi (for example, talks here, here, here and here, slides here and here), the 2022 ICM contribution from Yiannis Sakellaridis, and Akshay Venkatesh’s lectures at the 2022 Arizona Winter School (videos here and here, slides here and here). Also helpful are notes from Ben-Zvi’s Spring 2021 graduate course (see here and here).

A paper giving details of this work has now appeared, with the daunting length of 451 pages. I’m looking forward to going through it, and learning more about the wide range of ideas involved. A recent post advertised James’s Arthur’s 204 page explanation of the original work of Langlands, and the ongoing progress on the original number field versions of his conjectures. It’s worth noting that while there are many connections to the ideas originating with Langlands, this new work shows that the “Langlands program” has expanded into a striking vision relating different areas of mathematics, with a strong connection to deep ideas about quantization and quantum field theory. The way in which these ideas bring together number theory and quantum field theory provide new evidence for the deep unity of fundamental ideas about mathematics and physics.

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Million Dollar Prize for Scholze and Stix

At a news conference in Tokyo today there evidently were various announcements made about IUT, the most dramatic of which was a 140 million yen (roughly one million dollar) prize for a paper showing a flaw in the claimed proof of the abc conjecture. It is generally accepted by experts in the field that the Scholze-Stix paper Why abc is still a conjecture conclusively shows that the claimed proof is flawed. For a detailed discussion with Scholze about the problems with the proof, see here. For extensive coverage of the IUT story on this blog, see here.

Between paywalls and the limitations of Google translate, I’m not sure exactly what the process is for Scholze and Stix to collect their million dollars. Perhaps they just need to publish their paper, but it seems that the decision may be up to the businessman who is contributing the funds, and it’s unclear what his process will be.

For a few sources I’ve found, see here, here and here. If others have reliable and more detailed sources they can point to (especially anything in English), please do so.

Update: Press release here. Rules for the million dollars are

Nobuo Kawakami makes his own judgment as an individual.
The review method will not be disclosed, but the papers to be reviewed must be papers in mathematics that have been published on MathSciNet and have published more than 10 papers on arithmetic geometry in the past 10 years. Only papers that have been peer-reviewed and published in journals.

Scholze and Stix may not want to take time to submit their paper to a journal (Scholze has a history of turning down large prizes…). It occurs to me that there are quite a few arithmetic geometers who understand well the problem with the proof, could write something up and possibly get it published. Maybe a collaboration could be formed to do this.

Update: New Scientist has a story here. The quotes from Fumiharu Kato aren’t especially encouraging for IUT, who “estimates that fewer than 10 people in the world comprehend the concept.”

Kato believes that the controversy stems from the fact that Mochizuki doesn’t want to promote his theory, talk to journalists or other mathematicians about it or present the idea in a more easily digestible format, believing his work speaks for itself. Kato says that his current and former students are also reticent to do the same because they see him “as a god” in mathematics and don’t want to go against his wishes.

Because of this, most mathematicians are “at a loss” for a way to understand IUT, says Kato, who concedes that, despite earlier optimism about the idea, it is possible that the theory will eventually be disproven.

Ivan Fesenko is much more of a believer:

He told New Scientist that there is no doubt about the correctness of IUT and that it all hinges on a deep understanding of an existing field called anabelian geometry.

“All negative public statements about the validity of IUT have been made by people who do not have proven expertise in anabelian geometry and who have zero research track record in anabelian geometry,” he says.

Update: Scientific American has a news story about this, which summarizes the situation with the proof as:

So despite Mochizuki’s latest publication, there is still doubt among experts about the state of the abc conjecture. Most number theorists cannot make up their own mind because they are unable to follow the proof. And because both Scholze and Mochizuki enjoy an excellent reputation in their field, it is unclear who is right.

This gets the story quite wrong, and misunderstands how mathematics works. The problem is that there is no proof that anyone (Mochizuki himself included) can explain to anyone else, this is not about “do I believe this guy or the other guy?”. Yes, most mathematicians don’t have the technical knowledge to evaluate this sort of proof, but there are plenty who do, and they are either saying there is no proof, or, for the few supporting the proof, unable to explain it to anyone else.

Posted in abc Conjecture | 40 Comments

The Work of Robert Langlands

This is more of an advertisement than a blog post. This evening on the arXiv James Arthur has posted a wonderful 204 page document explaining the work of Robert Langlands, written in conjunction with the award of the Abel Prize to Langlands.

This isn’t an introduction to the subject, but if you have some idea of what the Langlands program is about, it provides a wealth of valuable explanations at a more detailed level of exactly what Langlands discovered. It ends with a discussion of the “Beyond Endoscopy” program of his later career.

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Physicists Prove That Parallel Worlds Cannot Be Extremely Different From Each Other

Stories about the latest prediction of superstring theory here and here, based on a Tsukuba University press release about this paper. Generally ignoring this kind of nonsense these days, but the new feature of this one is that the press release sure seems to have been written by ChatGPT.

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A Mathematics AI Factory?

A few days ago I read a fascinating article in New York magazine: Inside the AI Factory, which explained how the very large business of hiring humans to do tasks that generate training data for AI works. One reaction I had to this was “at least this means math is safe from AI, nobody is going to pay mathematicians to generate this sort of training data.” Yesterday though I ran across this tweet from Kevin Buzzard, which advertises work (see this link) that seems to be of this kind.

A company called Prolific is advertising work paying 20-25 pounds/hour doing tasks in Lean 3. This company is in exactly the business described in the NY mag articles, hiring people to do tasks as part of “studies”, which often are generating AI training data.

One unusual thing about this whole industry is that if you sign up for this work you often have no idea who your employer really is, or what your work will be used for, and you sign a non-disclosure agreement to not discuss what you do. In this case, a few things can be gleaned from discussions on the Lean Zulip server:

  • “it’s 40 dollars/hr now actually”
  • “I think everyone signed a consent form preventing them from disclosing any problems or even their participation.”
  • One problem was formalizing a proof of sin(x)=1 implies x=pi/2 (mod 2pi).
  • A representative of the company writes: “Some of our biggest partners are keen to work with Lean users because of its applicability as a theorem prover. They plan to launch numerous studies that require participants to have either a working or expert knowledge of Lean 3.” By “biggest partners”, presumably this is Microsoft/Google or something, not some small publishing organization.

If anyone knows anything about what’s up with this incipient possible math AI factory, please let us know. On the more general issues of math and AI, I’d rather not host a discussion here, partly because I’m pretty ignorant and not very interested in spending time changing that. The situation in mathematics is complicated, but as far as fundamental physics goes, theorem-proving is irrelevant, and applying “big data”/machine-learning/AI techniques to generate more meaningless calculations in a field drowning in them is pretty obviously unhelpful.

For a much better place to read about what is happening in this area, there’s an article in today’s New York Times by Siobhan Roberts: A.I. Is Coming for Mathematics, Too. At Silicon Reckoner, Michael Harris is in the middle of a series of posts about this past month’s workshop on “AI to Assist Mathematical Reasoning.”

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Fantasy, Faith and Physics

No blogging here the past few weeks, partly because I was away on vacation for a little while, but more because there hasn’t been anything I’ve seen worth writing about. Yesterday’s pulsar timing array and IceCube announcements unfortunately didn’t tell us anything about fundamental physics. In the past, such observational results pretty reliably led to absurd claims about evidence for string theory that I could complain about, but that phenomenon seems to be dying down. In this case, the only story that had such claims was one from Quanta Magazine, which explained that “the observations so far from NANOGrav and the other teams are consistent with what we’d expect to see from cosmic strings.”

I noticed that the people at the Institute of Art and Ideas have put together a program for Monday that includes a debate on the topic of “Fantasy, Faith and Physics.” The framing of the debate contrasts the conventional view of science with an alternative possibility: “should we accept that some beliefs, especially in the foundations of physics, are akin to religious beliefs dressed in mathematical language to give our theories meaning?” This kind of misses the point about the current problems in fundamental physics, since I doubt any of the panelists are going to defend such an alternative.

Very odd is what leads into this debate, an interview with Michio Kaku about his new book. Why promote such an atrociously bad book (see here and here) and broadcast Kaku’s absurd claims about this subject?

Maybe this debate will somehow lead to a substantive discussion of the main underlying problem, the nearly fifty-year dominance of a failed set (GUTs/SUSY/strings) of ideas about unification. A very powerful and influential part of the physics community, which will be represented in the debate by Juan Maldacena, continues to insist on the centrality of this set of ideas. To get a clear look at his arguments, see a recent IAI interview In defence of string theory and his colleague Edward Witten’s recent colloquium talk What Every Physicist Should Know About String Theory. The argument Maldacena and Witten are making is essentially the same one from the mid-eighties: string theory is the only possible consistent way to go beyond quantum field theory and get a consistent theory of quantum gravity. In my book and many other places, I’ve explained the many problems with this. Put simply, the problem is that there is no such thing as a well-defined string theory which successfully gives the SM and GR in four dimension. The claims about consistency are either about models that don’t reproduce the real world, or about still-unrealized hopes and dreams (which Penrose characterizes as “Fantasy”) rather than anything well-defined.

For a very clear statement of his point of view from Witten, see the question and answer section of the recent colloquium talk, starting around 1:20, where he starts by emphasizing the rigidity of the framework of relativistic quantum field theory. He then states:

My point of view is that string theory is the only significant idea that has emerged for any modification of the standard framework that makes any sense.

This is pretty much exactly the same argument he was making nearly forty years ago. I didn’t find it convincing then, since it seemed to me there was no reason to be so sure that a deeper understanding of relativistic QFT could not possibly lead to a consistent quantum theory with low energy limit GR. Witten had a good argument in 1984 that a possibly consistent generalization of relativistic QFT was worth studying, but the problem is that decades and tens of thousands of papers later, as far as unification goes, this study has been a failure, taking the field down paths (extra dimensions, SUSY) which lead to complex theories that don’t look at all like the real world.

If you look at where things have ended up and the current research directions Maldacena and Witten are pushing, the odd thing is that they seem to have given up on unification, and for years now have been emphasizing the study of black holes in toy models with little to no connection to string theory. The most disturbing thing I heard in the Witten talk was at 1:24:16

If you had sufficient computing power, maybe with a quantum computer with a million qubits, I think you could simulate the dynamics of a quantum black hole…

Here Witten seems to be pointing to exactly the argument recently made by Juan Maldacena (see here), which has a specific claim about what you could do with a million qubits. This particular calculation would not in any way address the problems of the string theory program and is getting into Michio Kaku/wormhole publicity stunt territory.

Update: There’s an interview with Witten here, associated with his receipt of the Hamburg Prize for Theoretical Physics. About string theory, he explains that, despite 50 years of effort

We don’t understand it very well… In fact, I’d say we only understand a small part. So we’ve been struggling with that ever since the 70s and 80s trying to understand the intellectual framework that it should have been placed in.

He remains convinced though that alternative ideas are not the way to go:

… I find it implausible that physicists would discover a theory that is such a rich source of fruitful ideas about things that are definitely important in other fields by accident. And if we were not on the right track, I would say, it was a big accident. So my personal view is that it would be a cosmic conspiracy if string theory isn’t on the right track.

There is a growing number of critics who complain that string theory is very interesting, but hasn’t really delivered. Because we still have no idea whether it’s correct and we couldn’t make any experiments, which tells us if this is the case. According to you: To what extent is that criticism justified?

Well, not much, honestly. Where critics of string theory have had interesting ideas, they’ve tended to be absorbed as part of string theory. That’s happened several times. Twistor theory, black hole thermodynamics and noncommutative geometry are three examples of interesting ideas. They were by some regarded as alternatives or competitors of string theory, but actually in practice were absorbed as part of the picture in string theory.

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Zen University

The establishment of a new university in Japan has been announced, to be called ZEN University. One component of the new university will be the Inter Universal Geometry Center, with Fumiharu Kato as director, Ivan Fesenko as deputy director. The Center will offer an introductory course on IUTT. There’s a video here.

The website seems to be Japanese-only, here’s what I get via Google Translate:

If you pass all of our courses, you will be better equipped with IUT theory than any mathematics student in any university in the world. A student who blooms his talents that emerges from within. We plan to prepare prizes for such young people and encourage them to continue to participate in the community that seriously researches IUT theory…

Although it is difficult to understand, there are already more than 20 mathematicians in the world who understand and develop the IUT theory. I hope that you will boldly take on the challenge of researching IUT theory together with me so that you can be one of the next.

The problem with this subject though is not the number of people who understand IUTT, but the number who can explain to others in a convincing way the proof of corollary 3.12 in the third IUTT paper. From everything I have seen, that number has always been and remains zero.

Update: Another video here.

Posted in abc Conjecture | 17 Comments

From Quantum Mechanics to Number Theory via the Oscillator Representation

This past semester I taught our graduate class on Lie groups and representations, and spent part of the course on the Heisenberg group and the oscillator representation. Since the end of the semester I’ve been trying to clean up and expand this part of my class notes. I’m posting the current version, working title From Quantum Mechanics to Number Theory via the Oscillator Representation. This is still a work-in-progress, but I’ve decided today to step away from it a little while, work on other things, and then come back later perhaps with a clearer perspective on what I’d like to do with these notes. In a few days I’m heading off for a ten-day vacation in northern California, and one thing I don’t want to be thinking about then is things like how to get formulas involving modular forms correct.

There’s nothing really new in these notes, but this is material I’ve always found both fascinating and challenging, so writing it up has clarified things for me, and I hope will be of use to others. The basic relationship between quantum mechanics and representation theory explained here is something that I’ve always felt deserves a lot more attention than it has gotten.

In the past I’ve often made claims about the deep unity of fundamental physics and mathematics, One goal of this document is to lay out precisely one aspect of what I mean when making these claims. There are other much less well understood aspects of this unity, but the topic here is something well-understood.

One thing that struck me when thinking about this and teaching the class is that this is a central topic in representation theory, but one that often doesn’t make it into the textbooks or courses. Typically mathematicians develop theories with an eye to classifying all structures of a given kind. This case is a very unusual example where there is effectively a unique structure. The classification theorem here is that there is basically only one representation, but it is one with an unusually rich structure.

When I get back from vacation, I plan to get back to work on the ideas about twistors and unification that I’m still very excited about, but have set to the side for quite a few months while I was teaching the class and writing these notes. More about that in the next few months…

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