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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Lucien Szpiro 1941-2020

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

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

Update: Something about Szpiro from Christian Peskine:

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

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

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

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This Week’s Hype

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

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

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

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

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

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This Week’s Hype

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

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

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

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

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

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

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

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

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Penrose at The Portal

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

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

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

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

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

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

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

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