Not Even Wrong

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.

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”.

This week there’s a Nobel Laureate Meeting in Lindau, devoted to physics. Many of the talks can be viewed on-line. From 3-5pm today (Lindau time) there will be a session devoted to a panel discussion of expectations for the LHC. Besides the Lindau web-site, a webcast will also be available here.

Blogging (in German) is going on here, including accounts of the late night activities there, featuring pictures of physicists dancing to the tune “Sex Bomb”.

Update: The panel discussion included two questions from the audience about the multiverse. At first Gross refused to address them leaving cosmologist Smoot to try and say something. Finally ‘t Hooft broke in to say that there were a lot of misconceptions being spread about the multiverse, but that the truth was that the LHC will never have anything to say about either the multiverse or string theory, and Gross did not disagree with him. ‘t Hooft explained that while in principle there could be indirect evidence for a multiverse (from direct evidence for aspects of a theory that implied multiple universes), at the moment the idea was completely untestable and the LHC would have nothing to say about it. Gross agreed, describing multiverse models and research as “very ill-defined”.

At the end, an argument between Veltman and others broke out over the selling of particle physics using astrophysics. He described claims that the LHC will “recreate the Big Bang” as “idiotic”, and as “crap”. He said that this is “not science”, but “blather”, and that the field would come to regret this, arguing that if you start selling the LHC with pseudo-science, you will end up paying for it. Gross and Smoot politely disagreed.

Update: See here for Gross’s talk on expectations of what will be seen at the LHC. He predicts definite observation of a Higgs particle, and says he has taken bets that supersymmetry will be seen, at 50-50 odds. Nothing about string theory at the LHC.

A workshop in Paris/Saclay is taking place this week entitled Wonders of Gauge Theory and Supergravity and the talks are now online. They show that some exciting new things have been happening in the study of gauge theory and supergravity amplitudes, and I’ll make the prediction that this field will attract a huge amount of attention in the coming years (at least until the LHC experiments announce results incompatible with the Standard Model…).

Perhaps the most remarkable part of this whole story is the mounting evidence that N=8 supergravity amplitudes are finite in perturbation theory. Remember the standard story about how quantum theory and general relativity are incompatible that has dominated discussion of fundamental physics for years now? Well, it turns out that this quite possibly is just simply wrong. See Zvi Bern’s talk on UV properties of N=8 supergravity at 3 loops and beyond for the latest about this. Bern shows that divergences everyone had been expecting to occur at 3 loops aren’t there, and gives evidence that they might also be absent at higher loops. He even sees this as a phenomenon not special to N=8 supergravity, but also occurring in theories with less supersymmetry, e.g. the N=5 and N=6 theories. Among the other talks, Nima Arkani-Hamed’s is also about this, advertising the idea that N=8 supergravity is the Simplest QFT.

Much of this story is about the N=4 SYM amplitudes and new insights into them and their relations to supergravity amplitudes, with some of this research growing out of and motivated by the AdS/CFT conjecture of the existence of a string dual to N=4 SYM. Quite a few of the talks are interesting and worth trying to follow, with a much higher proportion of new ideas than is usual at particle theory workshops in recent years.

To go out on a limb and make an absurdly bold guess about where this is all going, I’ll predict that sooner or later some variant (“twisted”?) version of N=8 supergravity will be found, which will provide a finite theory of quantum gravity, unified together with the standard model gauge theory. Stephen Hawking’s 1980 inaugural lecture will be seen to be not so far off the truth. The problems with trying to fit the standard model into N=8 supergravity are well known, and in any case conventional supersymmetric extensions of the standard model have not been very successful (and I’m guessing that the LHC will kill them off for good). So, some so-far-unknown variant will be needed. String theory will turn out to play a useful role in providing a dual picture of the theory, useful at strong coupling, but for most of what we still don’t understand about the SM, it is getting the weak coupling story right that matters, and for this quantum fields are the right objects. The dominance of the subject for more than 20 years by complicated and unsuccessful schemes to somehow extract the SM out of the extra 6 or 7 dimensions of critical string/M-theory will come to be seen as a hard-to-understand embarassment, and the multiverse will revert to the philosophers.

Many of the titles of the talks at Strings 2008 have recently been announced. The plenary talks will include several talks mostly not about string theory, including 3 about the LHC and one by Lance Dixon on the amplitudes story. It seems that the string theory anthropic landscape is a topic the conference organizers don’t want anything to do with, since the only person from the Stanford contingent speaking will be Kallosh on prospects for getting something observable out of string cosmology models of inflation. As for what is popular, it clearly helps a lot to be from one of my alma maters, with Princeton (7 speakers), and Harvard (3 speakers) the best-represented institutions.

Update: For an extensive rant about this, see here.

Update: Last week was Paris, this week it’s Zurich. Amplitudes are all the rage this summer.

Today’s Bloggingheads.tv diavlog features Sean Carroll and philosopher of science David Albert, discussing a variety of issues. Albert tells about his unfortunate experience with the What the Bleep? film, a good example of why it’s not always a good idea to get involved with people doing a supposedly science-related media project. He also discusses the hostility towards study of the quantum measurement problem from within the physics community over the years, a situation that has changed recently. The two also had a long discussion concerning Carroll’s claims about the arrow of time, about some of which Albert seemed to be rather skeptical.

The discussion of criticisms of string theory were on the whole ill-informed, misleading, and devoted to ferocious attacks on straw men. For some reason, only John Horgan was mentioned, with the existence of trained theoretical physicist critics and two recent books on the topic completely ignored. Albert insisted repeatedly on the idea that Horgan and other critics were not acknowledging the “spectacular predictive success” of string theory. He was referring to claims that string theory “predicts gravity”, since it contains a massless spin-two particle (ignoring the fact that it is in the wrong dimension; to quote Lisa Randall “string theory predicts gravity: 10d gravity”). Later on Carroll did explain the problem with this, that string theory seems to allow an infinite variety of ground states with different physics, many of which don’t have 4d gravity. Carroll told about having asked various string theorists if they could imagine any kind of experimental result at any energy that would be incompatible with string theory, and getting the answer “No” from at least some of them. This seemed to rather shock Albert.

There was no real discussion of the multiverse, a topic where philosophers of science might be able to perform a public service by taking a serious look at what physicists are up to and analyzing what they learn. Carroll launched the standard attack on string theory critics as having a “sophomore-level” understanding of the philosophy of science, unaware that there is anything to the problem of what is science and what isn’t other than Popper’s falsifiability criterion. He also claimed that string theory critics have created a “20 year statute of limitations” criterion, that theoretical work must lead to a falsifiable prediction within 20 years or cease to be science, chuckling with Albert about how ignorant people must be who think such a thing. This kind of willful misrepresentation of the views of people you disagree with seems to me to be less than honest. From what I remember of Lee Smolin’s book, there’s a long section about his engagement with the philosophy of science, and his sympathies are not with Popper, but with Feyerabend’s “anarchistic” views on the subject, which are very different. In my book there’s an entire chapter devoted to explaining what is wrong with just invoking falsifiability. I assume Carroll has read at least one of the two books, so it’s unclear why he thinks it’s acceptable to go on like this. He does make one more accurate accusation, that critics of the multiverse are stuck in an out-dated 1960s particle physics paradigm of what it means to test a theory. I suppose this is true enough. Not the first time I’ve been accused of being stuck in the 60s, which, if one has to be stuck somewhere, doesn’t seem like that bad a choice…

Update: Evolving Thoughts has a link to The Ideas of Quine on Youtube, an interview of the philosopher Willard Van Ormond Quine by Brian Magee. The relation of physics and philosophy was one of Quine’s main concerns, and one of the main topics of the interview. I suppose that to the extent my own philosophical views could be characterized by picking one philosopher I find most sympathetic, Quine would be a good choice. Perhaps he’s responsible for my “sophomore-level” philosophy of science, since I took a course from him as a sophomore.

The relationship between mathematics and physics is a topic that has always fascinated me, and today I noticed two interesting blog postings related to the topic. The first was Ben Webster’s posting inspired by a recent XKCD comic. The discussion in the comment section is well worth reading, especially the contributions from Terry Tao.

Over at Backreaction, Sabine Hossenfelder discusses an interview with Max Tegmark from the latest issue of Discover magazine entitled Is the Universe Actually Made of Math?. Much of the discussion is about Tegmark’s comments on how he dealt with the potential danger to his career caused by his unconventional publications on the “Mathematical Universe Hypothesis”. This says that

Our external physical reality is a mathematical structure

I’ve always had an extreme case of mixed feelings about this, thinking that Tegmark manages to bring together the extremely deep and the extremely dumb. He embeds this as “Level IV”, the highest level, of the multiverse, and multiverse mania is one reason he has gotten attention for this and not had it dismissed as crackpotism. The idea he is pursuing is that any mathematical structure can be thought of as a “universe”, and we just happen to be in some random one of these. This seems to me to be pretty much content-free, and the attempts to fit it into more conventionally popular multiverse studies don’t help.

At the same time, this does get at an incredibly deep problem, that of the relationship between mathematical structures and physical reality. Some of the central mathematical structures that mathematicians have discovered have turned out to be identical to those found by physicists pursuing models of fundamental physics. This has happened in several very striking ways over the years. Thinking of the universe as a mathematical structure has turned out to be extremely fruitful, both for mathematics and for physics.

What is important though is that not all mathematical structures are equally important, central, or interesting, and this is the crucial point that Tegmark seems to me to be missing. Once you learn enough mathematics, you find certain recurring themes and deep structures throughout the subject. What fascinates me is that these often also turn out to be central in theoretical physics. Tegmark just accepts every mathematical structure as equally important, creating a huge undifferentiated multiverse where we occupy some random anthropically acceptable point. But the evidence is that the mathematical structure we inhabit is a very special one, sharing features of the very special structures that mathematicians have found to be at the core of modern mathematics. Why this is remains a great mystery, one well worth pursuing from both the mathematician’s and physicist’s points of view.

This week I’m in Montreal attending a conference in honor of Raoul Bott (who I wrote about in some detail here a couple years ago).

On the first day of the conference, a documentary film about Bott made by his grandaughter Vanessa Scott was shown, and there was a panel discussion about the man and his influence on students and collaborators. Bott was both a wonderful mathematician and human being, and many people at the conference paid tribute to him as someone who encouraged them and taught them beautiful and deep mathematics. Quite a few commented on visits to him and his family at their summer place on Martha’s Vineyard. It adjoined a clothing-optional beach frequented by Alan Dershowitz among others, a beach where Bott was supposedly known as the “Mayor”.

Bott was very much involved with physics later in his career, and two physicists spoke at the conference: Cumrun Vafa on topological strings and Edward Witten on the 6d QFT point of view on geometric Langlands. I greatly enjoyed both talks, but I suspect they were pretty difficult for the mathematicians in the audience to follow, covering a great sweep of material bringing together new mathematical developments not understood by most mathematicians with QFT and string theory techniques far from their background.

Michael Atiyah gave a truly wonderful talk to open the conference. He described how his friendship with Bott had covered a period of 50 years, from 1955 until Bott’s death in 2005. From 1964-1984 they did some of the best work of their careers together, and Atiyah tried to summarize the high points of this, as well as point out problems their work raised that he sees as still open and worthy of investigation by a new generation. Editions of the collected works of both of them are available that include their commentaries on the papers, and these are very much worth reading. Each of them is a masterful expositor, so their joint papers are uniformly models of clarity.

Atiyah broke things up into the following main topics:

In work with Hirzebruch, Atiyah realized that Grothendieck’s construction of K-theory in algebraic geometry could be turned into a generalized cohomology theory in the topological category. To make this work uses Bott periodicity in a crucial way, to show that K(MxS²) is the tensor product of K(M) and K(S²) for any compact manifold M. Atiyah realized he didn’t know how to prove this, and got Bott to produce an appropriate proof, which appeared in a paper of Bott’s in 1959. The paper is written in French, clearly not by Bott, and Atiyah says he still doesn’t know who translated it into French for Bott.

Atiyah and Bott worked together on extending the Atiyah-Singer index theorem to the case of manifolds with boundary, where the issue of how to handle the boundary conditions so as to get a good index problem is a subtle one. As part of this, they needed a new, more “elementary” proof of Bott periodicity, finally finding one that “even MIT faculty could understand”. This is periodicity for the unitary group and crucially uses Fourier analysis. Atiyah gave as a problem deserving attention that of extending the proof to the case of the orthogonal group, where the use of Fourier series would have to be replaced by the representation theory of O(N). Multiplying Fourier series becomes taking the tensor product of representations, which is much more non-trivial to deal with.

Using the McKean-Singer formula, one can relate the computation of the index of a differential operator to the asymptotics of a related heat equation. Patodi and Gilkey had carried this through using complex and skillful algebraic calculation, but Atiyah and Bott felt they couldn’t understand these proofs, so with Patodi came up with a new proof, one that just used Weyl’s invariant theory for the orthogonal group and the Bianchi identities of Riemannian geometry.

As a problem for the future, Atiyah listed finding a better understanding of the relation of the Atiyah-Bott-Patodi argument with the supersymmetric quantum mechanics proof. More explicitly, he conjectures that one should be able to just use invariant theory, but perhaps invariant theory of the infinite dimensional group Diff(M), of a sort advocated by Gelfand.

The Atiyah-Bott fixed point formula computes the Lefschetz number one gets in the context of index theory and a mapping of the manifold to itself (coming for instance from a group action) in terms of data at the fixed points of the mapping. This has many beautiful applications, including a new proof of the Weyl character formula. The formula is sometimes known as the “Woods Hole formula”, since Atiyah and Bott conjectured it at a conference at Woods Hole, where certain experts told them it couldn’t be true, since they had computed counter-examples. Atiyah didn’t name names, but described the experts involved as now claiming to not remember this. “They deny it to a man” he said, but he remembers it distinctly while being in the frustrating position of having nothing written down to provide incontrovertible documentary evidence.

One generalization of the fixed point formula shows that for manifolds with circle action the index is zero, and Atiyah mentioned Witten’s extension of this to the case of a loop space, leading to a relation to modular forms and the subject of “elliptic cohomology”. This continues to be an active subject, with Mike Hopkins and others developing the theory of “topological modular forms”, something that has shows interesting relations to number theory. Atiyah described a “moral bet” between him and Andrew Wiles about whether QFT will ultimately influence number theory. Wiles thinks not, but Atiyah believes it will happen, and hopes to be around long enough to find out if he is right.

Morse theory and equivariant cohomology were two of Bott’s favorite tools, and he and Atiyah did some wonderful work applying these to the case of Yang-Mills theory in two dimensions. Here the Yang-Mills functional is a Morse function, the space of connections is a symplectic manifold, and the reduced space for the gauge group action is an important mathematical object that can be thought of as the moduli space of flat connections, or of stable holomorphic bundles on the 2d surface. Their main result, a calculation of the Betti numbers of this space, reproduced earlier results coming from a very different approach, that of using algebraic curves over finite fields and the Weil conjectures.

As a problem for the future, Atiyah asks if there is some infinite-dimensional version of the Weil-conjectures, and some QFT where the Feynman integral is analogous to Tamagawa measure. He says he has been thinking about this off and on for 30 years, hasn’t found anything satisfactory, but offers the problem as a gift to younger mathematicians, as long as they let him know if they solve it. For some recent work related to this, see here.

Atiyah described work of his with Bott on this topic as “performed under a subcontract” with Garding.

Finally, Atiyah commented on how his mathematical style was quite different than Bott’s with Bott always advocating an “old-fashioned” way of proceeding, involving concrete formulae, where Atiyah favored “new-fangled” abstraction, only writing down formulae when forced to by Bott (and then discovering that this gave important insight). Later he found himself in the opposite position when working with Graeme Segal, with respect to whom he was the “old-fashioned” one, resisting abstraction. He commented that Bott and Segal had written a paper together, and he was shocked to see that such a thing was possible.

He noted that he had met many fascinating people through Bott, including one of the world’s best known mathematicians: Tom Lehrer. Finally, he ended with the comment that, while Bott was no longer here, the great thing about doing math the way he did it is that you become immortal.