Not Even Wrong

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.

I’ve been known to claim that string theory makes no experimental predictions, so this evening thought I better take a look at a preprint that just appeared entitled GUTs and Exceptional Branes in F-theory – II: Experimental Predictions. The abstract claims that to have found “a surprisingly predictive framework”.

This paper is 200 pages long, and a companion to part I, which was 121 pages. For part I, there’s a posting by Jacques Distler that explains a bit of the very complicated algebraic geometry going on. Making one’s way carefully through the entire 200 pages of the new paper looks like a very time-consuming project, so I thought I better start by identifying what the experimental predictions are. These days, one expects experimental predictions to say something about LHC physics, but I don’t see anything about that in the paper. Perhaps this is because, except for some comments in section 16, it appears that the authors are studying a model with exact supersymmetry.

Looking at the introduction and conclusion sections of the paper, the only predictions I can see are for neutrino masses, and there are two of them. Either .5 x10^{-2 +/-.5} eV or 2 x 10^{-1 +/- 1.5} eV is given for the neutrino mass, with the error bars just those due to an unknown value of one of the geometrical parameters involved. There’s no mention of which neutrino is being discussed, and as far as I can tell this is just an order of magnitude estimate of the neutrino mass scale, one which the author’s describe as “somewhat naive”, noting that “factors of 2 and π are typically beyond the level of precision which we can reliably estimate”. It’s unclear to me whether or not other mechanisms giving quite different neutrino masses would also fit into the author’s model.

Maybe I’m missing something and an expert can help me out, but I’m not seeing anything here of the sort one would normally describe as an experimental prediction. There’s certainly nothing falsifiable at all about the model, since one knows from limits on neutrino masses and measurements of oscillations that the neutrino mass scale has to very roughly be in this kind of range. Furthermore, again maybe I’m missing something, but I don’t see any way in which more detailed calculation in this framework can make it any more predictive.

Update: Lubos has a detailed posting about this, and from reading it, it doesn’t appear that the paper has experimental predictions that I missed. I do wonder what a “musculus maximus” is…

Update: For more about this, see presentations at PASCOS 08 here and here. The first describes this as “a modest step” in the direction of predictions, the second doesn’t mention predictions at all.

Most of the time the attention paid here to efforts to popularize physics is restricted to grumpy complaints about the hype surrounding string theory as well as the more general dubious phenomenon of scientists promoting things that are more science fiction than science. Today I’m in a much more positive mood, and thought I’d take the opportunity to make some unusually sunny comments for a change.

One reason for this is that I attended the opening party for my colleague Brian Greene’s World Science Festival Wednesday night at the American Museum of Natural History, and several people have told me about the enthusiastic reception the festival events have been getting. I was hoping to attend one of the events, but it was already sold out.

Things started off during the day Thursday with a World Science Summit here at Columbia featuring a speech by Mayor Bloomberg, and award of the new Kavli prizes. That evening, at the museum event, Brian, Fred Kavli and Senator Schumer all spoke, and the crowd was entertained by the choir of the Abyssinian Baptist Church.

Among the people I got a chance to talk to at the event were several string theorists. One of these was Jim Gates, who I had the pleasure of first meeting last year down in Orlando. He was there with his wife, just back from South Africa where he is involved with the African Institute for Mathematical Sciences.

Gates told me that his collection of video lectures with impressive graphics explaining quantum mechanics, general relativity and superstring theory called Superstring Theory: The DNA of Reality has been selling well, generating over a million dollars in sales, despite not being able to get it reviewed in major publications. Strong evidence of its popularity comes from the fact that if you google “The DNA of Reality” you get an impressive variety of sources for pirated versions. Evidently he has done an excellent job of reaching a wide audience with this material. From conversations with him I know that we’re in closer agreement than you would guess, sharing an interest in the mathematics behind supersymmetry and a skepticism about extra dimensions.

I have mixed feelings about the highly enthusiastic promotion of certain speculative ideas about physics involved both here and in some of the World Science Festival events, but it’s undeniable that these are reaching a lot of people and getting them excited about science. Perhaps I can convince Jim to market what could be the ideal package: his videos to get people excited and enthusiastic about the open problems in physics, and my book to give them some skepticism about the solutions now being promoted…

Update: For an article describing what happened at the World Science Festival program about unification and string theory, see here.

Since 1968 SLAC has been maintaining a database of HEP documents called SPIRES, and this has become one of the main tools used by anybody searching the HEP literature. In recent years CERN has developed a much more modern document management system known as CDS Invenio. The two projects are now being brought together into something to be called INSPIRE, which will combine the best of both, in particular making the SPIRES data available through the more modern Invenio software.

There’s a press release from DESY about this here, and an alpha version is up and running here. The current state of the project is that most of the SPIRES functionality has been reproduced, and they are working on getting a beta version ready of a complete replacement of SPIRES.

Last week at DESY a workshop was held about this, announced as an HEP Information Resource Summit, talks are available here. There were presentations from other HEP information providers, including the APS, commercial publishers and the arXiv. The arXiv presentation discussed their desire to better support blogging, and the role of the blogosphere, including the fact that Garrett Lisi’s paper was the most downloaded article on the arXiv. The current trackback system provides links to 21 discussions of the paper, but due to the Distler/arXiv policy of censoring links to this blog, one that is missing is the discussion here. More and more very worthwhile content is appearing on blogs, so the question of how to make this readily available in a useful form will become an increasingly important one.

Unfortunately, while the arXiv does a good job of bringing together mathematics and physics, there seems to be no discussion of the role of the mathematics literature in the new INSPIRE system. Besides the arXiv, the main database used by mathematicians is the excellent MathSciNet developed by the AMS.

Update: Travis Brooks of SPIRES has a posting about this here.

For the last 15 years the New York City subway has featured “Poetry in Motion”, which places extracts of poetry in subway cars. Starting next month this program will be expanded, joined by Train of Thought, which will add “short quotations in history, philosophy, literature, and science chosen by Columbia University’s Graduate School of Arts and Sciences.” I gather that my colleague Henry Pinkham, a mathematician now dean of the Graduate School, is responsible for this. Of the first two quotations to go up next month, one is dear to my heart, from Galileo:

The book of nature is written in the language of mathematics. Its symbols are triangles, circles, and other geometric figures, without which it is impossible to understand a single word; without which there is only a vain wandering through a dark labyrinth.