The Week’s Anti-Hype

I never thought I would see this happen: a university PR department correcting media hype about its research. You might have noticed this comment here a week ago, about a flurry of media hype about neutrinos and parallel universes. A new CNN story does a good job of explaining where the nonsense came from. The main offender was New Scientist, which got the parallel universe business somehow from Neil Turok and from here.

The ANITA scientists and their institution’s PR people were not exactly blameless, having participated in a 2018 publicity campaign to promote the idea that they had discovered not a parallel universe, but supersymmetry. They reported an observation here, which led to lots of dubious speculative theory papers, such as this one about staus. The University of Hawaii in December 2018 put out a press release announcing that UH professor’s Antarctica discovery may herald new model of physics. One can find all sorts of stories from this period about how this was evidence for supersymmetry, see for instance here, or here.

It’s great to see that the University of Hawaii has tried to do something at least about the latest “parallel universe” nonsense, putting out last week a press release entitled Media incorrectly connects UH research to parallel universe theory. CNN quotes a statement from NASA (I haven’t seen a public source for this), which includes:

Tabloids have misleadingly connected NASA and Gorham’s experimental work, which identified some anomalies in the data, to a theory proposed by outside physicists not connected to the work. Gorham believes there are more plausible, easier explanations to the anomalies.

The public understanding of fundamental physics research and the credibility of the subject have suffered a huge amount of damage over the past few decades, due to the overwhelming amount of misleading, self-serving BS about parallel universes and failed speculative ideas put out by physicists, university PR departments and the journalists who mistakenly take them seriously. I hope this latest is the beginning of a new trend of people in all these categories starting to fight hype, not spread it.

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Various Items

A few short items:

  • Things had been going quite well at the LHC, they were ahead of schedule, starting to ramp up intensity for the new run. Then at 5:30 this morning a weasel decided to visit a 66kV transformer, which did not end well either for the weasel or for the LHC power grid. The machine and a lot of its cryogenics lost power, and recovery is going to take a week or so.
  • For some commentary on the excitement about the new run building up (pre-weasel), see Tommaso Dorigo (at least I’m guessing he’s the author) here. He points to the twitter #MoarCollisions hashtag.
  • In various Breakthrough Prize related news, first there’s an announcement from Terry Tao about the new IMU Graduate Breakout Fellowships, funded by him and some of the other math prize winners.

    On the physics front, Caltech has Glitz and Qubits, about Alexei Kitaev and John Schwarz’s experience with the prize. Schwarz still hopes for vindication of his string theory prize by a discovery of SUSY at the LHC, assigning a much higher probability to this than I think most other people would these days:

    I would say the probability is on the order of 50 percent or so that it will show up.

    As for the glitz:

    At the 2014 award ceremony, Schwarz says, he and his wife, Patricia, were “both struck by the fact that the Hollywood types showed no interest in mingling with scientists.” And the media coverage also seemed to focus on the movie stars rather than the award winners.

    Kitaev points out a major positive effect of the $3 million: more respect from one’s family:

    But for Kitaev, the biggest impact of awards like the Breakthrough Prize in Fundamental Physics is on his family. “They don’t really understand what I’m working on,” he says. But thanks to these awards, they at least realize his research is a pretty big deal. “It helps me do more work because they have more respect for it,” he says. “My wife is really proud of me.”

  • In other big money news the Perimeter Institute and the Stavros Niarchos Foundation have announced a new professorship for Asimina Arvanitaki, funded by $8 million. More about her here.
  • At Nautilus you can read an interview with IAS director, theoretical physicist Robbert Dijkgraaf.
  • At Edge, there’s a conversation with Frank Wilczek. I’m quite curious what the following is about, have no idea:

    What I’ve been thinking about today specifically is something of a potential breakthrough in understanding our fundamental theories of physics. We have something called a standard model, but its foundations are kind of scandalous. We have not known how to define an important part of it mathematically rigorously, but I think I have figured out how to do that, and it’s very pretty. I’m in the middle of calculations to check it out.

: Two more:

  • Jim Baggott has a post on Status Anxiety, with more thoughts about the Munich conference and the use of the term “theory”.
  • Discover magazine has an article in the upcoming June issue on The Fall and Rise of String Theory. The story seems to be that String theory for some reason ran into a little trouble in 2006, but now it’s back, because Strominger has done some “string-inspired” black hole calculations, and some people are claiming inspiration from AdS/CFT for an approximate calculational method in some condensed matter models. The idea now seems to be that, starting from this, string theory is on its way to again finding a theory of everything. No comment on its hype problem.

Update: A special 2016 physics Breakthrough Prize has been awarded to the LIGO people. \$1 million split by Drever, Thorne and Weiss, \$2 million for the rest of the collaboration.

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Oral Histories

I recently ran across a recent interview with Mary K. Gaillard, which encouraged me to look again at the AIP’s oral histories site. For a review of her autobiographical book, see here.

She has the following comments on the current state of HEP theory:


What do you see as the future of theoretical physics? Where is the field headed?


Well, I think it’s headed towards insanity [laugh] by itself. I mean, no, if we don’t have experiments, people can let their imaginations run wild, and invent anything without it being verified or disproven. So I think it—I mean, if we want to understand more about what happens at higher energies, we have to have higher energy colliders. I don’t think—well, cosmology is tied to particle physics, and that’s probably something from—I mean, there is a lot of data coming from cosmology. And there is some data that will be coming from very low energy precision physics. But I don’t think that theory by itself—it needs to be kept in line by [laugh] experiments.


And so what advice do you have sort of globally for people entering the field in terms of the kinds of things they should study, and the way they should study those things?


That’s a [laugh]—actually, I often advise people to go into astro-particle physics just because I think that it has more promise of getting data because I don’t—I mean, I strongly believe you can’t go forward without good data, and unless—well, of course, if they do have another generation of colliders, that would be great. I just don’t know if that’s going to happen…

Another very recent interview I found interesting was that of Vipul Periwal. Periwal arrived as a Ph.D. student at Princeton around the time I was on my way out, starting his career right about when string theory hit in late 1984. He worked as a string theorist for quite a few years, ending up in a tenure-track faculty position at Princeton, but then left the HEP field completely, starting a new career in biology. Here are some extracts from his interview:


And what was David [Gross]’s research at this point? What was he pursuing?


String theory. He was just 100 percent in string theory. Right? They just did the heterotic string, and so everyone was — every seminar at Princeton at that time was all string theory. It was all string theory. Curt was working on it, David was working on it. Edward was working on it. Larry Yaffe was probably the only person — no, two people, Larry Yaffe and Ian Affleck were not doing string theory. Not that they couldn’t, but they just would not do it.


So you mean, despite at this point all of the work on string theory, there were still existential questions about what string theory was, that remained to be answered?


There still are.




No one has ever figured out what is string theory. I mean, if you go ask all the eminent string theorists, none of them can answer for you this one simple question. Can you show me a consistent string theory, where supersymmetry is broken?


Was it good for your research? Was it a good time for you [his postdoc at the IAS]?


I don’t think I did particularly interesting research. I did — I mean, I did okay, but I’m not particularly proud of anything I did there, except for one little paper I wrote, in which [laughs] — see, this is called the contrarian part — is I showed — people were very excited about the large N limit, so I took this toy model, and I showed that in the large N limit, it actually produced something nonanalytic, as in like, you could not, in any order of 1 over an expansions, ever see what the answer was that was exact at N equals infinity. So, in other words, it was to me a cautionary tale. Like, you think you’re doing large N and then getting an intuition for finite N. But here’s this very simple model where you can do the calculation exactly, and you can do all your 1 over N expansion as far as you want, and it’ll never tell you [laughs] about what’s going to happen at N equals infinity. But you know, it’s a — at this point, string theory was already at that time pretty much a sociological thing.


What do you mean “sociological”?


So, it’s something that was borne home to me gradually, that there’s no experimental proof. Like, are you a good physicist or a bad physicist? Who’s going to tell? How do you know? Right?




I mean, I’d go and give a talk somewhere, and I remember this very clearly. I went and gave a talk at SUNY Stony Brook, what’s now called, I guess, Stony Brook University. And at the end of the talk, I was talking to one of the faculty there who’d invited me. And he said, “So, what does XYZ think of this work?” And I was just taken aback. I was like, wait, you’re a physicist. I’m a physicist. Why do we need to know what XYZ thinks of this?




Right? That’s what I mean by sociology.


I see. It’s as much about what a certain group of peers thinks about the theory.


Yeah, and this really perturbed me. As far as I was concerned, after the string perturbation theory diverges thing, I was not interested in doing perturbative calculations. So, what the solution was that people did was: okay, we’ll work on various supersymmetric theories where there is no higher contribution, and under the assumption that there is supersymmetry, you can use holomorphicity to deduce things from the structure of the fact that there’s so much supersymmetry. And this really bothered me, as in okay, there’s this really amazingly beautiful structure, and lots of very pretty mathematical results that are coming out — mathematical results that are suggested by these correlations. But I just don’t get — as a physicist, I don’t to want to have to worry about, “What does XYZ think about what I’m doing?”


Yeah, because you’re pursuing a truth, and it’s either true or it’s not. It doesn’t really matter what other people think about it.


Right. I really don’t care. I mean, no matter how much I respect — and I do — Edward, or David, or whoever, I really don’t need to know what they think about my work. Right? I just — anyhow —


How does that attitude serve you in an academic setting, though? Right?


It doesn’t.


How does that attitude affect you in terms of tenure considerations and things like that?


Yeah, so when I was — no, so I actually — I mean, when I was — well, I have no — I’m really stupid sociologically, as in, I have no instinct for self-preservation. So, I could see I had role models in front of me of how people with tenure…




…succeed, not just getting tenure at Princeton, but getting tenure at very good places after Princeton, too. And I paid zero attention to all this. So, while I was at Princeton, I tried doing some lattice gauge theory.

With this attitude, it’s not surprising that in Periwal didn’t get tenure at Princeton. He didn’t soon get job offers elsewhere in HEP theory, and decided in 2001 better to try another field than keep going in the one he was in. The interview ends with:


Alright. So, really, the last question. What does the big breakthrough moment look like for you? How would you conceptualize this in terms of putting all of this together? What does that big breakthrough look like?


If I could make a prediction that was clinically testable, that would make me very happy.


Do you think you’ll get there? It’s the thing that motivates you.


Yeah. I want — you know, I said this once. We had someone visiting when I was managing the physics seminar at Princeton once, as an assistant professor. So, this guy asked me, “So, Vipul, what are you working on?” And I was very jaundiced at that time about making a prediction. So, I said, “Well, lattice gauge theory,” which, you know, nobody at Princeton did lattice gauge theory. You were all supposed to be doing string theory. I said, “Yeah, I want a number before I die.” [laughs] People are looking at me like, “What kind of lunatic is this?” But you know, a number. That would be nice.

Looking through the old interviews, I found one of very personal interest, that of Gerald Pearson, who worked with my grandfather Gaylon Ford at Bell Labs. Some of his stories mentioned work with my grandfather (whose main expertise was in the design and construction of vacuum tubes) at Bell Labs during the 1930s. During this period both studied at Columbia, where my grandfather got a master’s degree in physics.


Gaylon Ford worked with Johnson. When Kelly was head of the tube department, he worked in that area. And then they had a big shakeup after which the job was no longer available. Much against his desires, he came over to work with us.


In 1938 you were moved over from Johnson’s group into Becker’s. In fact, you and Sears seem to have changed places.


Before that took place, I remember Johnson called me into his office one day and he wanted to know if I would like to work on… well, Buckley had sent a memorandum asking for temperature regulators for buried cable. Johnson wanted to know if I would like to work in this area. Of course, no one likes to change their jobs but I said, “Fine” and we agreed that I would spend a portion of my time on this problem and that’s where thermistors came from. This continued on and it was very successful. Then it was decided that the work fit in better with Becker’s area than it did with Johnson’s. And, well you asked me about Ford. He was the one who was brought from the tube shop to work on this. And then he later went to work on something else.


Let’s see if we can date that time. Ford wasn’t working with you yet. Ford is here with you in 1934. But this move didn’t take place until ’38.


Yes, that’s what I was saying. He first came over to work on change of resistance with temperature. And he was working with a sulfide compound. And then, let’s see, what happened to him. He went someplace else and Johnson called me into his office and asked me if I would like to carry on Ford’s work and we agreed that I should do it part time and still work on noise. But I said I didn’t want to work with sulphur, it smelled too bad. I said if I work in that area, I’m going to use some other materials. So I made a study of that. First I worked on boron and then on a combination of oxides. A lot of my patents are on such materials and devices. These devices are still used today in the buried cable system as volume regulators.

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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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Final Fourier Analysis Notes

Our semester here at Columbia is finally over, and I’ve put the lecture notes on Fourier analysis that I wrote up in one document here. A previous blog posting explained the origin of the notes: they cover the second half of this semester’s course, from the point at which the course became an online course due to the COVID-19 situation.

Not much blogging going on here, mainly since everyone staying home seems to have kept news of much interest to a minimum.

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Some Philosophy of Science

Defenders of certain failed speculative theories like to accuse those who point to their failure of being “Popperazzi”, relying on mistaken and naive notions about predictions and falsifiability due to Karl Popper. That’s never been the actual argument for failure, and two excellent pieces have just appeared that explain some of the real issues.

  • Sabine Hossenfelder’s latest blog entry is Predictions are overrated, a critique of the naive view that you can evaluate a physical theory simply by the criterion “Does it make predictions?”. She goes over several important aspects of the underlying issues here, making clear this is a complex subject that resists people’s desire for a simple, easy to use criterion for evaluating a scientific theory.
  • Over at Aeon, Jim Baggott writes about this under the headline How science fails, focusing on the life and work of philosopher of science Imre Lakatos. I wish I had been aware of the ideas of Lakatos when I wrote a chapter about the complexities of evaluating scientific success or failure in my book Not Even Wrong, since he was concerned with exactly the sorts of issues I was grappling with there.

    One of the main ideas of Lakatos is that you should conceptualize the problem in terms of characterizing a research program as “progressive” or “degenerating”. As relevant new experimental and theoretical results come in, is the research program showing progress towards greater explanatory power or is it instead losing explanatory power, for instance by adding new complex features purely to avoid conflict with experiment? One way I like to think of this is that it’s hard to come up with an absolute measure of success of a research program, but you can more easily evaluate the derivative: is some new development positive or negative for the program?

    I don’t think there’s any question but that supersymmetry, GUTs, and string theory are classic examples of degenerating research programs. In 1984-5 there was great hope for a certain idea about how to get a unified theory out of string theory (compactification on Calabi-Yaus), but everything we have learned since then has made this hypothesis one with less and less explanatory power.

    The Lakatos framework has the feature that there is no absolute notion of failure. It always remains possible that the derivative will change: for instance the LHC will find superpartners, or a simple compactification scheme that looks just like the real world will be found. The not so easy question to answer is when to give up on a degenerating research program. I think right now prominent string theorists are taking the attitude that it’s past time to give up work on the idea of string theory unification (and they already have), but not yet time to admit failure publicly (since, you never know, a miracle may happen…).

And now, for something completely different: If you want something more entertaining to read about particle physics, I highly recommend Tommaso Dorigo’s Anomaly! (see review here). The one problem with that book was that it stopped in the middle of the story (end of Tevatron Run 1). He now is making available some chapters (see here and here) he wrote that cover the later, Run 2, part of the story.

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Why the Szpiro Conjecture is Still a Conjecture

There has been a remarkable discussion going on for the past couple weeks in the comment section of this blog posting, which gives a very clear picture of the problems with Mochizuki’s claimed proof of the Szpiro conjecture. These problems were first explained in the 2018 Scholze-Stix document Why abc is still a conjecture.

In order to make this discussion more legible, and provide a form for it that can be consulted and distributed outside my blog software, I’ve put together an edited version of the discussion. I’ll update this document if the discussion continues, but it seemed to me to now be winding down.

Depending on one’s background, one will be able to get less or more out of trying to follow this discussion, but it seems to me that it makes an overwhelmingly convincing case that Mochizuki’s articles do not contain a proof of the conjecture and should not be published by PRIMS. No one involved in the discussion claims that there is an understandable and convincing proof in the articles. The discussion is rather about Scholze’s argument that there is no way that the kind of thing Mochizuki is doing can possibly work. While Scholze may not have a fully rigorous, loophole-free argument (and given the ambiguous nature of many of Mochizuki’s claims, this may not be possible) the burden is not on him to do this.

To justify the PRIMS decision to publish the proof, one needs to assume that the referees have some understood and convincing counterargument to that of Scholze, one that nobody has made publicly anywhere. If this really is the case, the editors of PRIMS need to make public these counterarguments, and those mathematicians who find them convincing need to be able to explain them.

A note on comments: if someone has further technical comments on the mathematical issues being discussed at the earlier posting, they should be submitted there. For discussion of issues surrounding publication of the claimed Mochizuki proof, this would be the right place (and I’ve moved a couple recent ones to here). For comments about Szpiro and his conjecture, the posting about him would be an appropriate one.

Update: I hear that the editors of PRIMS are aware of the recent discussion of the problems with the Mochizuki proof, but have decided to go ahead with the publication of the proof anyway. They do not seem to intend to release any information about their editorial process, in particular what counter-arguments to Scholze’s they considered. In effect, they are taking the stand that they have convincing evidence that Scholze is wrong about the mathematics here, but cannot make it public for confidentiality reasons.

Note that the discussion in the comment thread itself has some later entries after the ones gathered in the pdf document I created.

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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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Various and Sundry

  • If like most other people you’re stuck at home, and having trouble concentrating on the projects you thought the current situation would cause you to finally find the time to complete, one thing you could do is watch a lot of talks about mathematics online. As far as I can tell, mathematicians are doing much better than any other field right now in dealing with this, since they have a wonderful site developed at MIT called It contains a fairly comprehensive set of listings of Math seminars now being run online.

    If there’s anything similar in the physics community, I’d be interested to hear about it.

  • A new book about the problems of fundamental physics has recently appeared, David Lindley’s The Dream Universe: How Fundamental Physics Lost its Way. I’ve been thinking for a while about whether to write about it here, have held off mainly because I felt I didn’t have much interesting to say. Today I see that Sabine Hossenfelder has written a review of the book which I mostly agree with, so you should read what she has to say.

    There are a couple places where I significantly disagree with her. For one thing, unlike Hossenfelder, I’m a great fan of Lindley’s much earlier book on this topic, his 1993 The End of Physics. This was written a very long time ago, at a time when writing for the public about fundamental physics was uniformly positive about the glories of string theory. Unlike all those books, this one has held up well. Reading it at the time it came out, it was remarkable to me to find someone else seeing the same problems with the field that seemed to me obvious, providing a very helpful indication that “no, I’m not crazy, there really is something wrong going on here.”

    It’s interesting to read Hossenfelder’s take on the way Lindley makes “mathematical abstraction” the villain in this story:

    The problem in modern physics is not the abundance of mathematical abstraction per se, but that physicists have forgotten mathematical abstraction is a means to an end, not an end unto itself. They may have lost sight of the goal, alright, but that doesn’t mean the goal has ceased existing.

    Here is where I definitely part company with Lindley, and to some extent with Hossenfelder. The current problems with fundamental physics have nothing to do with mathematical abstraction, but with the refusal to give up on bad physical ideas that don’t work. Thirty-six years ago Witten and many other leaders of the field fell in love not with a mathematical abstraction, but with a bad physical idea: replace fundamental particles with fundamental strings. One reason they fell in love with this idea was that it could be fit together with two other bad ideas they had been dallying with at the time, that there are new forces mixing leptons and quarks (GUTs), and that you can relate bosons and fermions with the square root of translation symmetry (SUSY).

    Unfortunately it seems to me that many theorists have now drawn the wrong conclusion from the sorry story of the last forty-some years, deciding that what they need to do is to stay away from unwholesome mathematics, and stick to the wholesome experimentally observable and testable. But what if the underlying reason you got in a bad relationship with a seriously flawed love interest was that there weren’t (and aren’t) any experimentally testable ones to be found? Maybe what you need to do is to work on yourself and why you stay in bad relationships: the mathematically abstract love of your life might still be out there.

  • Witten yesterday posted a definitely not mathematically abstract paper on the arXiv, Searching for a Black Hole in the Outer Solar System. It’s basically a proposal for finding a physical black hole we could then go and get into a relationship with. I can’t help thinking the probabilities are that getting into a healthy relationship with a new mathematical abstraction is more likely to work out than this.
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Proof of the abc Conjecture?

While I was traveling this past week, there was a conference held here entitled L-functions and Automorphic Forms, which was a celebration of the 60th birthday of my math department colleague Dorian Goldfeld. From all I’ve heard the conference was a great success, well attended, with lots of interesting talks. But by far the biggest excitement was due to one talk in particular, that of Lucien Szpiro on “Finiteness Theorems for Dynamical Systems”. Szpiro, a French mathematician who often used to be a visitor at Columbia, but is now permanently at the CUNY Graduate Center, claimed in his talk to have a proof of the abc conjecture (although I gather that, due to Szpiro’s low-key presentation, not everyone in the audience realized this…).

The abc conjecture is one of the most famous open problems in number theory. There are various slightly different versions, here’s one:

For each $\epsilon >0$ there exists a constant $C_\epsilon$ such that, given any three positive co-prime integers a,b,c satisfying a+b=c, one has

$$ c < C_ \epsilon R(abc)^{1+\epsilon}$$ where $R(abc)$ is the product of all the primes that occur in a,b,c, each counted only once.

The abc conjecture has a huge number of implications, including Fermat’s Last Theorem, as well as many important open questions in number theory. Before the proof by Wiles, probably quite a few people thought that when and if Fermat was proved it would be proved by first proving abc. For a very detailed web-site with information about the conjecture (which leads off with a quotation from Dorian “The abc conjecture is the most important unsolved problem in diophantine analysis”), see here. There are lots of expository articles about the subject at various levels, for two by Dorian, see here (elementary) and here (advanced).

As far as I know, Szpiro does not yet have a manuscript with the details of the proof yet ready for distribution. Since I wasn’t at the talk I can only relay some fragmentary reports from people who were there. Szpiro has been teaching a course last semester which dealt a bit with the techniques he has been working with, here’s the syllabus which includes:

We will then introduced the canonical height associated to a dynamical system on the Riemann Sphere. We will study such dynamical systems from an algebraic point of view. In particular we will look at the dynamics associated to the multiplication by 2 in an elliptic curve . We will relate these notions and the questions they raised to the abc conjecture and the Lehmer conjecture.

For more about these techniques, one could consult some of Szpiro’s recent papers, available on his web-site.

The idea of his proof seems to be to use a and b to construct an elliptic curve E, then show that if abc is wrong you get an E with too many torsion points over quadratic extensions of the rational numbers. The way he gets a bound on the torsion is by studying the “algebraic dynamics” given by the iterated map on the sphere coming from multiplication by 2 on the elliptic curve. I’m not clear about this, but it also seems that what Szpiro was proving was not quite the same thing as abc (his exponent was larger than 1+ε, something which doesn’t change many of the important implications).

Maybe someone else who was there can explain the details of the proof. I suspect that quite a few experts are now looking carefully at Szpiro’s arguments, and whether or not he actually has a convincing proof will become clear soon.

Update: I’m hearing from some fairly authoritative sources that there appears to be a problem with Szpiro’s proof.

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