Gauge Theory and Langlands Duality

Posted on July 21, 2008 by woit

At the KITP in Santa Barbara there’s a wonderful program on Gauge Theory and Langlands Duality starting up this week, with some of the talks beginning to become available. The main topic will be the relations between S-duality in quantum field theory and geometric Langlands duality that Witten and collaborators have been working on the past few years.

I started trying to watch the talks, but the fact that the video quality is such that one can almost but not quite tell what is being written on the blackboard makes this a bit of a trial. I’m hoping that David Ben-Zvi or someone else will make available notes, which would help a lot. I did very much like Edward Frenkel’s description of the Langlands story as a “Grand Unified Theory of Mathematics”, and was interested to hear that he still feels that there are two different stories about the relation to QFT here, whose relationship is not at all understood (S-duality in 4d QFT is one, 2d CFT and vertex algebras is the other). It seems that A.J. Tolland is there, maybe he or someone else will do some blogging. As I get time to take in the lectures, I hope to write some more about them here.

Update: Notes for the talks are now also being posted, making following them on-line much more feasible. The quality of the talks is excellent, with Ed Frenkel so far giving a beautiful introduction to the roots of the Langlands program in number theory, David Ben-Zvi explaining the structures in topological quantum field theory that mathematicians are trying to exploit, David Morrison and Paul Aspinwall explaining mirror symmetry, D-branes, and the relation to N=(2,2) superconformal field theory, with examples, and Anton Kapustin starting on the 4d N=4 TQFT used to turn S-duality into a mirror symmetry.

Posted in Langlands | 11 Comments

The Emperor’s Last Clothes?

Posted on July 21, 2008 by woit

Bert Schellekens has posted on the arXiv an extended 87 page argument for the anthropic string theory landscape, entitled The Emperor’s Last Clothes? While most string theorists find the existence of the landscape and the corresponding inability to get any predictions out of the theory about particle physics rather discouraging, Schellekens instead sees this as an argument in its favor:

Initially, when string theory was touted as the “theory of everything” around 1984, there were hopes it would lead to exactly the opposite: a unique derivation of all the laws of physics. Evidence that quite the opposite was true started emerging almost immediately after 1984, but most people chose to ignore it. In 2003, after important additional evidence had been found, Leonard Susskind published a paper [2] entitled “The Anthropic Landscape of String Theory”, which finally started a debate that should have started fifteen years earlier. What is at stake in this debate is not only the uniqueness of our universe, but also the fate of string theory as a fundamental theory of all interactions.

In my opinion string theory gives the right answer, and the fact that it does adds to the evidence in its favour. I can say this without being accused of trying to put a positive spin on the recent developments, because I actually wrote in 1998 [3] that I hoped string theory would ultimately lead to a huge number of possible choices for the laws of physics, a point of view I have been advocating since the late eighties. I reached that conclusion after having been involved in one of the first papers [4] pointing out that the number of possibilities was humongous…

We all hope to live during a time when big things are happening in our field, and I have never doubted that this is one of those things. I have spent the last twenty years trying to convey my sense of excitement to my colleagues, but with little success. But in the last few years I have been delighted to see more evidence coming in supporting this point of view, so that the mood has started to change. I hope this is the right time to make one more attempt.

Schellekens describes in great detail the anthropic argument and the arguments for the string theory landscape. He addresses some of the counter-arguments, especially in three appendices. He doesn’t explictly deal with the main counter-argument that I’ve made repeatedly here: the anthropic landscape is not science (since it is not testable), rather it is just an elaborate excuse for the failure of the speculative idea of getting the SM out of a 10/11d string/M-theory.

Schellekens has the following comments about “string phenomenology”, noting that he worked in the area around 1987 and recently, finding not much has changed:

I have been active in this are around 1987 (which led me to the conclusions presented here) and again in the last few years, and to me the similarities are more striking than the differences. There has certainly been progress: we can obtain string solutions that are more similar to the Standard Model than twenty years ago, and we have more methods to construct them. There has been major progress in moduli stabilization and supersymmetry breaking. There is more interest in “landscape statistics”. But very little seems to have changed in the way many people view the problem we are facing. Although many of my string phenomenology colleagues claim that it was clear to them a long time ago that there are many solutions, I cannot help noticing that they still talk about their most recent “model” as if it would actually have a chance to be the Standard Model. And even nowadays one still hears the occasional expression of hope for the unknown and elusive dynamical principle that will select the vacuum. The most common way of dealing with the large vacuum degeneracy is to say “I do not care about the other 10500vacua, I only care about the one that describes our universe”. That may sound reasonable, and fact it may sound like the very definition of phenomenology, but it is actually an escape from reality.

First of all, if indeed there are 10500 vacua, it is highly unlikely that anyone will find “the Standard Model” in string theory. One should expect to find a huge number that satisfy all current experimental constraints. In addition, although we now have many techniques at our disposal to construct string theories in four dimensions, it is quite clear that we are just scratching the surface. Statistically speaking, our chances of finding even one of the expected huge number of Standard Model realizations is essentially zero. Furthermore, even if we do find one, we can only make predictions about novel phenomena if we know all the other solutions and their predictions for the same phenomena. This is a crucial change in comparison to the state of the art about ten years ago: with 1020 solutions (the largest number anyone may have expected), if one is found that agrees with all current data, the probability that there is a second one is extremely small. With 10500, the same probability is astronomical. So we should forget about the idea of finding the Standard Model and then making predictions based on it.

As for LHC predictions, Schellekens argues against the idea that it will see supersymmetry:

One could say that supersymmetry is a non-solution to a non-problem: the large weak scale hierarchy is already understood anthropically, and supersymmetry by itself does not even explain it…

With the start of the LHC just months away (at least, I hope so), this is more or less the last moment to make a prediction. Will low energy supersymmetry be found or not? I am convinced that without the strange coincidence of the gauge coupling convergence, many people (including myself) would bet against it. It just seems to have been hiding itself too well, and it creates the need for new fine-tunings that are not even anthropic (and hence more serious than the one supersymmetry is supposed to solve). But even if evidence for low energy supersymmetry emerges at the LHC, in the context of a landscape it will not be the explanation for the smallness of the weak scale. The explanation will in any case be anthropic. The landscape will undoubtedly allow a distribution of values for the weak scale, including values outside the anthropic window.

Schellekens ends up making the currently fashionable argument that it doesn’t matter that string theory doesn’t predict anything testable about particle theory, that the important thing is that it is a theory of quantum gravity:

During the last two decades there was some reason to hope that we might be able to do that [get experimental confirmation of string theory] by means of some prediction of a Standard Model feature. That hope is fading now. I am not saying that this will never happen, but I have seen too much wishful thinking to make an optimistic statement about this. Essentially, we came to that conclusion already in 1986 [4]. We are dealing with a theory of gravity. Getting information about it through the back door of particle physics is a luxury that we once had good reasons to hope for, but that may not exist. Rejecting a theory of gravity that makes no particle physics prediction may be like rejecting the theory of continental drift because it does not predict the shape of Mount Everest

He then goes on to acknowledge that we’re not going to get any experimental tests out of the quantum gravity aspect of string theory either:

One cannot count on any direct experimental check of a theory of quantum gravity, since any observable consequences it might have are extremely small, unless we are extremely lucky.

In the end, he seems to argue that the only evidence for string theory we may ever get is its consistency, something which is a very long ways from being shown. He does argue that string theory is in principle falsifiable, but the example he gives (that string theory would be wrong if coupling constants varied observably on astronomical scales) is not an uncontroversial one since other string theorists have argued that varying coupling constants would be evidence for string theory. There’s also the usual “who knows?” argument used against anyone who points out that evidence against an idea is overwhelming:

On longer timescales, it is clearly ridiculous to pretend that what we currently know will be the state of the art forever. When Darwin formulated his theory of evolution he was unaware of Mendel’s results on inheritance, and could not even have imagined DNA.

The truly peculiar thing about this is to see a scientist almost gleeful at the idea that a theory they have worked on their entire professional lives doesn’t predict anything:

To me, what is emerging looks very appealing. It fulfills and even exceeds the hopes I expressed in 1998. It is has been amazing to see this theory leading us in the right direction, sometimes even against the initial expectations of most of the people working on it. We should continue to follow its lead, and do everything in our power to strengthen its theoretical underpinnings. The emergence of a huge landscape” makes this more worthwhile then ever before.

Unfortunately, Schellekens is far from alone in this. At the FQXI web-site there’s an article about the string theory/cosmology couple Andrei Linde and Renata Kallosh entitled A Perfect Match (“How do you tie down the physics of the multiverse? With string.”) In it, Kallosh explains how “string cosmology” is now the hot topic:

These days, in fact, collaboration be-tween string people and cosmology people is all the rage.
“To give you a funny example, I had an invitation to give a talk at the Strings 2008 conference at CERN,” Kallosh says. “The way the invitation was writ-ten was, ‘Of course you are welcome to speak about any topic . . . but we would be very happy if you would give us a mini-review on string cosmology!’”

Suddenly, everyone is interested in their kind of union.

“I’m also working on other very formal, very stringy topics, which were always part of my skills,” Kallosh says. “But, at this moment, people want to know about string cosmology. I’m happily working on it . . . with Andrei’s help.”

Cosmology is now ascendant, with Kallosh arguing that it will be needed to explain what is seen at the LHC:

“Soon the LHC will start giving new information on particle physics,” she says. “But we know it will be difficult to interpret this data unless you also can digest all the data from the sky—all the observations from astrophysics and cosmology.”

The article ends with a large picture of one of the LHC detectors, captioned “Cradle of Collaboration: Will the LHC provide evidence for string theory?”

FQXI is funded by the Templeton Foundation, the goal of which is to bring science and religion together. Cormac O’Raifertaigh is at another Templeton funded event, a conference in Cambridge on From the Big Bang to the Brain: Current Issues in Science and Religion. This Wednesday will be devoted to cosmology, featuring talks on the anthropic principle, fine-tuning, God and time, and God and the Big Bang.

For another take on cosmology, this October the ENS in Paris will host a conference on Evolution and Development of the Universe. For more about this parallel universe of cosmologists who also study anthropics and the multiverse , see EvoDevoUniverse.

Posted in Multiverse Mania | 25 Comments

Awaiting a Messenger From the Multiverse

Posted on July 17, 2008 by woit

As particle physicists eagerly await results from the LHC, many theorists are already promoting interpretations of what they hope it will find. This week’s Chronicle of Higher Education has a cover story on the LHC entitled The Machine at the End of the Universe (see associated articles here and here). In it, Gordon Kane enthusiastically describes the LHC as “It is certainly the most important experiment of any kind in the past century, without qualification” and “the most important thing ever in our quest to understand the fundamental laws of nature and the universe.”

The question that the LHC will actually address, that of electroweak symmetry breaking, doesn’t get much attention. Instead, the focus is on supersymmetry, extra dimensions and string theory. While noting that there’s no evidence for string theory, the article reports:

The new collider could change that, says Joseph D. Lykken, a physicist at Fermi National Accelerator Laboratory. “Either the discovery of supersymmetry or extra dimensions is a triumph of string theory.” While such a finding would not conclusively show that string theory is correct, it would provide a first crucial experimental test, he says.

At New Scientist this week, there’s another article about the LHC, entitled Awaiting a messenger from the multiverse. In it, Savas Dimopoulos explains that there’s already quite a lot of experimental evidence against weak-scale supersymmetry:

“After a lot of experiments there has not been any hint of SUSY,” says Dimopoulos. For each individual one of those predictions, he says, you can find plausible explanations for why they are not seen. “But by the time you look at the whole package of ‘things that should have happened but didn’t’, you start getting a somewhat baroque structure.”

Dimopoulos instead promotes his joint work with Nima Arkani-Hamed on the idea of “split-supersymmetry”, where the supersymmetry breaking scale is very high, so can’t explain the hierarchy problem. One possible experimental signature of such models would be a long-lived gluino. They promote the idea that such a thing would be a “Messenger From the Multiverse”, the idea being that if supersymmetry doesn’t explain the hierarchy problem, the explanation must be the anthropic landscape:

That powerful piece of evidence would have dizzying implications. “It would be a strong indication that there is a string landscape or a multiverse,” says Dimopoulos. “I think the majority of opinion would come around to that point of view.”

One aspect of this argument is that it also works if no gluino is seen. If no superpartners at all are found at the LHC, and thus supersymmetry can’t explain the hierarchy problem, by the Arkani-Hamed/Dimopoulos logic this is strong evidence for the anthropic string theory landscape. Putting this together with Lykken’s argument, the LHC is guaranteed to provide evidence for string theory no matter what, since it will either see or not see weak-scale supersymmetry.

The New Scientist article does explain that this kind of argument for anthropics has its critics:

However, anthropic arguments remain controversial, and despite the authors’ heavyweight reputations, split supersymmetry is no exception. “I’m not a big fan of it,” says John Ellis, a theoretical particle physicist at the CERN laboratory, where the LHC is being prepared for its first run later this year. His criticism is that the anthropic approach gives you too much freedom to answer any troublesome question in physics. “At some point you might just as well say ‘let’s fine-tune everything’ and go home,” Ellis says.

Theorist Frank Wilczek at the Massachusetts Institute of Technology is also unhappy with the idea of split SUSY. “I think it’s a logical possibility, but if we really have to appeal to anthropic considerations I think that’s a big retreat. It would mean the explanatory power of theoretical physics would be limited.”
…
And even if the LHC does find Dimopoulos’s stopped gluinos, not everyone will be persuaded that arguments based on the multiverse are good science. “My opinion of anthropic reasoning is likely to remain unprintable,” says Ellis.


Update: Forgot to add one more piece of LHC-related news. France is now joining the US with its own national LHC web-site: LHC-France. Because of the Gallic fondness for cartoons, it includes a section Le LHC en BD.

Posted in Multiverse Mania | 28 Comments

Hawking to Perimeter?

Posted on July 16, 2008 by woit

The Canadian press today is putting out the story that Stephen Hawking may be abandoning Cambridge to move permanently to the Perimeter Institute, where he would join recently appointed director Neil Turok, as well as Lenny Susskind.

From the stories it appears that Hawking plans to visit Perimeter next year for a month or so, and that he hasn’t actually made any decision about a permanent move.

For more, see here, here, here, and here.

According to Sam Blackburn, an assistant of Hawking’s, Hawking is just mulling the idea over with no move imminent, but he is “obviously a man of few words so the first we would probably know of it is when he packs his bags.”

Posted in Uncategorized | 15 Comments

Surfing the Universe

Posted on July 14, 2008 by woit

This week’s New Yorker has a quite good article by Benjamin Wallace-Wells entitled “Surfing the Universe” about Garrett Lisi and the controversy generated last year by his paper An Exceptionally Simple Theory of Everything (which I wrote about here). Unfortunately the article is not available on-line as far as I know.

One of the main themes of the piece is how Garrett ended up getting enmeshed in the controversy over string theory. I’m quoted as agreeing with the writer’s impression that one thing that got Garrett “enlisted in the string wars” was having my name appear first in his acknowledgements:

“It was probably not the most politic thing to do,” Woit said.

The description of the state of the string theory controversy is pretty accurate. Wallace-Wells got the following from Steven Weinberg

String theory still has great attractions, and there aren’t any alternatives… Well, there are alternatives and they’re worse.

and describes the situation as follows:

In physics, as in politics, the competition is crueller in lean times. “In terms of development of new theories, it has been maybe the slowest period in two centuries,” the science historian Spencer Weart said. By 2006, the fight over string theory had begun to leak out of the scientific community. Smolin and Woit published widely reviewed books criticizing string theory, and USA Today published an account of the assault headlined “HANGING ON BY A THREAD?”

The article explains the role of the arXiv and the blogs:

In recent years, as science reporters and interested amateurs have turned to the arXiv – and as some physics personalities have started blogs – the audience for physics has both expanded and fragmented. “I know for a fact that many of the leading figures in the field read the blogs, but so do high-school science students,” Woit, the Columbia mathematician and string-theory critic, said. “The scary thing is that frequently you can’t tell which is which.” The leading blogs have readerships that, while including some loud dissenters, tend to align with the perspectives of their authors – Distler, of the University of Texas, has a blog that attracts many string theorists and enthusiasts, while Woit’s blog draws more skeptics. It works somewhat in the way the blogosphere operates in politics. Andreas Albrecht, a physics professor at the University of California at Davis, said that the blogs had opened physics to a new sort of populism, one that the academic establishment had to figure out how to manage. “It just pushes thoses buttons,” Albrecht said. “There’s some really good stuff, but a lot of really sloppy stuff.” What you have, in other words, is the erosion of the referee and the rise of a scientific underclass.

The above quotes are from passages about the string theory controversy and ones where I had some involvement, but that’s only one aspect of the piece. There’s quite a lot more about Garrett, his story, and the physics/math context he is working in, together with a reasonable take on its significance, with everyone acknowledging that the ideas he is pursuing have problems and are still not such that success can be claimed. At the end of the article, Garrett explains that he’s still at work, now trying to see if an alternate form of E8 will work better.

Update: Lubos has the usual sort of rant about this. He doesn’t seem to have access to the article itself, so is basing his rant on my extracts. As a result, it includes an extensive personal attack on Spencer Weart….

Update: I hear that Bert Kostant has posted on his office door at MIT a copy of his e-mail exchange about E8 with the author of the New Yorker piece. So I guess that it’s all right to point to these files. Also, there’s an on-going Distler/Lisi exchange going on here. I haven’t followed the technicalities of this particular discussion, but the trademark Distlerian argumentation tactics are in operation, ensuring vastly more heat than light.

Update: It turns out that not everyone involved in that e-mail exchange had given their permission to make it public, so the files linked to above have been removed. Kostant has made his comments public, posted here.

Update: The New Yorker article is now available on-line.

Posted in Uncategorized | 71 Comments

This and That

Posted on July 8, 2008 by woit

Some quick links:

The Clay Mathematics Institute is now making available for free online the books whose publication it has sponsored. These include the Morgan-Tian exposition of the proof of the Poincare conjecture. Surveys in Non-commutative geometry, which contains two excellent articles by Jeffrey Lagarias and Paula Tretkoff explaining ideas about the Riemann Hypothesis that have been motivating Connes and others recently. Also the excellent huge group-effort expository volume on Mirror Symmetry, and a more recent volume on the topic, which includes a good review article by Michael Douglas about the string theory motivations for this work.

There’s an interview with Susskind here about his latest book. About anthropics and the multiverse he claims

…since I wrote “The Cosmic Landscape,” it has practically become the conventional view.

A couple of relatively new physics bloggers are Sunil Mukhi and Marco Frasca. P.P. Cook has revived his blog and is reporting from Eurostrings 2008 here and here.

Among the posts worth reading over at Secret Blogging Seminar, there’s a nice posting by A. J. Tolland explaining what a “stack” is. The comments contain a valuable discussion about the different versions of a “classifying space” that show up in this story.

Posted in Uncategorized | 12 Comments

Back to the Future

Posted on July 7, 2008 by woit

Several things have come up recently that brought up the year 1985, the year the film “Back to the Future” came out.

This summer the IAS will be running a two-week program at the IAS on Strings and Phenomenology, designed to train a new generation of graduate students and postdocs in the details of compactification methods which mostly go back to 1985, quite possibly before some of the attendees were even born. No evidence that there will be any mention of the fact that 23 years of work on these topics has led simply to a dead-end: the landscape.

At SUSYO8, the first two speakers harkened back to the 1985 period, with Hans Peter Nilles (who also will be lecturing at Princeton) quoting his own words from 1984 (Physics Reports 110):

Experiments within the next five to ten years will enable us to decide whether supersymmetry at the weak interaction scale is a myth or reality

He notes that “This statement is still true today!”

Andrei Linde in his talk on cosmology crows about what he sees as Witten’s recent capitulation to the anthropic landscape point of view about string theory that Linde was pushing back around 1985 (actually 1986) when he wrote:

An enormously large number of possible types of compactification which exist e.g. in the theories of superstrings should be considered not as a difficulty but as a virtue of these theories, since it increases the probability of mini-universes in which life of our type may appear.

which he compares to this from the New York Times

Now, Dr. Witten allowed, dark energy might have transformed this fecundity from a vice into a virtue, a way to generate universes where you can find any cosmological constant you want. We just live in one where life is possible, just as fish only live in water.

At the same time, I’ve been reading and thinking about some papers written back in 1985 which deal with the mathematics of gauge theory and anomalies. At least some of these were never published, including one that I’ve seen references to (by Igor Frenkel and Iz Singer), but never a copy of (does anyone have a copy?). Looking at the history of this subject, it is clear that some very good people were working on this until 1985, at which point quite a few of them dropped it to take up the new fashion of string theory.

Perhaps the LHC will revive the subject of particle theory, by producing a wormhole that will take the world back to its other end, opened up in 1985 by a DeLorean in the movie, from there setting us off into a more promising part of the multiverse.

Posted in Uncategorized | 27 Comments

LHC Update

Posted on July 7, 2008 by woit

Latest press release from CERN about the LHC says first beams “currently scheduled for August”. According to a presentation at the July 2 meeting of the LHC Technical Committee, the latest news is that “circulating beam not before September” (the presentation includes a detailed version of the schedule of what has to take place between now and the end of August). At this point the second to last sector is just about cool, the final one will take another two weeks. The last of 470 trucks of liquid nitrogen has arrived. Assuming it will take 1-2 months from first circulating beam until physics collisions, it looks like time for data-taking will be rather short before the shutdown for the winter.

Posted in Experimental HEP News | 1 Comment

Interview With Atle Selberg

Posted on July 4, 2008 by woit

Sticking with the theme of the Riemann Hypothesis, the AMS has recently posted some articles to appear in an upcoming issue of the AMS Bulletin, one of which contains a long interview with Atle Selberg, who died last summer at the age of 90. Selberg had been a professor at the IAS and an expert in analytic number theory, responsible for some of the most important developments in the subject during the 20th century. A large part of the interview concerns in one way or another the Riemann Hypothesis, which is a central concern of Selberg’s mathematical research, with his work on it beginning during the German occupation of Norway when he was still a student, Some thoughts from Selberg on the subject:

“If anything at all in our universe is correct, it has to be the Riemann Hypothesis, if for no other reasons, so for purely esthetical reasons.” He always emphasized the importance of simplicity in mathematics and that “the simple ideas are the ones that will survive.”

About whether there is a spectral problem that gives the zeros of the zeta-function, useful for proving the RH:

That is certainly a thought that several people have had. In fact, there have been some people that have been able to construct such a space, if they assume that the Riemann hypothesis is correct, and where they can define an operator that is relevant. Well and good, but it gives us basically nothing, of course. It does not help much if one has to postulate the results beforehand—there is not much worth in that.

About his own attempts to find a proof:

Once I had an idea that I thought perhaps could lead to a proof….

[gives some details]

After a while I became more and more convinced that it would not work as I had thought initially. It just seemed unlikely to me. However, I have now and then seen that people have attacked a problem in a way that seemed “hare-brained”, to use an English term, but then it turned out that they could make it work. They have proven something that would not be easy to prove in another way. On the other hand, I have seen people have ideas that seemed absolutely brilliant, but the only problem is that if one follows these to the end one is not able to get anything out of it after all. So it works both ways: sometimes a good idea does not work, and what seems like a bad, even idiotic idea, may actually work.

About Connes’s work on the RH:

Yes, that is a new way to arrive at the explicit formulas—a new access, so to say—but it basically does not give more than what one already had. Connes undoubtedly believed to begin with that what he was doing should lead towards a proof, but it turned out that it does not lead further than other attempts. When I last talked with him he had realized this. This often happens with types of work that are rather formal. There was, for example, a Japanese mathematician, Matsumoto, who gave several lectures that made quite a few people believe that he had the proof.

and finally:

I think it is a good possibility that it will take a long time before it is decided. From time to time people have been optimistic. Hilbert, when he presented his problems in 1900, thought that the Riemann hypothesis was one of the problems that one would see the solution of before too long a time had elapsed. Today it is a little more than one hundred years since he gave his famous lecture on these problems. So one must say that his opinion was wrong. Many of the problems that he considered to be more difficult turned out to be considerably simpler to solve.

Posted in Uncategorized | 8 Comments

Proof of the Riemann Hypothesis?

Posted on July 2, 2008 by woit

Last night a preprint by Xian-Jin Li appeared on the arXiv, claiming a proof of the Riemann Hypothesis. Preprints claiming such a proof have been pretty common, and always wrong. Most of them are obviously implausible, invoking a few pages of elementary mathematics and authored by people with no track record of doing serious mathematics research. This one is somewhat different, with the author a specialist in analytic number theory who does have a respectable publication record. Wikipedia has a listing for Li’s criterion, a positivity condition equivalent to the Riemann Hypothesis.

Li was a student of Louis de Branges, who also had made claims to have a proof, although as far as I know de Branges has not had a paper on the subject refereed and accepted by a journal. He describes his approach as using a trace formula and “in the spirit of A. Connes’s approach”. Li thanks

J.-P. Gabardo, L. de Branges, J. Vaaler, B. Conrey, and D. Cardon who have obtained academic positions in that order for him during his difficult times of finding a job.

but it is a little worrisome that he doesn’t explicitly thank any experts for consultations about this proof. If the arXiv submission of the preprint is the first time he has shown it to anyone, that dramatically increases the already high odds that there’s most likely a problem somewhere that he has missed.

I’m no expert in this subject, so in no position to check the proof or to have an intelligent opinion about whether his method of proof contains a new, promising idea. I suspect though that experts are already looking at this proof, and it appears to be written up in a way that should allow them to relatively quickly see whether it works. Given the history of this subject, I think the odds are against Li, but I’m curious to know what experts think of this.

This also has appeared on Slashdot. If your comment is like any of the ones there, please don’t submit it, but comments from the well-informed are strongly encouraged.

Update: It looks like a problem with the proof has been found. Terry Tao comments on his blog

It unfortunately seems that the decomposition claimed in equation (6.9) on page 20 of that paper is, in fact, impossible; it would endow the function h (which is holding the arithmetical information about the primes) with an extremely strong dilation symmetry which it does not actually obey. It seems that the author was relying on this symmetry to make the adelic Fourier transform far more powerful than it really ought to be for this problem.

Update: Another Fields medalist heard from: Alain Connes comments as follows on his blog:

I dont like to be too negative in my comments. Li’s paper is an attempt to prove a variant of the global trace formula of my paper in Selecta. The “proof” is that of Theorem 7.3 page 29 in Li’s paper, but I stopped reading it when I saw that he is extending the test function h from ideles to adeles by 0 outside ideles and then using Fourier transform (see page 31). This cannot work and ideles form a set of measure 0 inside adeles (unlike what happens when one only deals with finitely many places).

Update: The paper has now been withdrawn by the author, “due to a mistake on pg. 29”.

Posted in Uncategorized | 25 Comments