Last Friday night when I was in Rome I received e-mails in quick succession from two science journalists asking what I thought about a new mathematical result, the “mapping of E8″ that was going to be announced at a press conference on Monday. Information sent to journalists was embargoed until Sunday night at 11pm, but the first journalist sent me a copy of the brief press release and told me that there was a longer one available. Reading the press release left me still baffled about what this could be about: what was the “century old problem” that this group of 18 mathematicians had solved? The obvious interpretation of “mapping of E8″, mapping it as a geometrical object, didn’t make sense since that’s a well-understood problem. The group E8 is a 248 dimensional space, but its local geometry is the same everywhere and completely understood in terms of its Lie algebra. The global topology is interesting, but also well understood.

I wrote back to both journalists that the best person I knew to comment on this and its possible relation to physics would be John Baez, and asked to see the longer press release. It wasn’t much more enlightening, but it did have a link to a web-site with details. After spending a little time reading this I understood that “mapping of E8″ was a calculation of the structure of representations of the split real form of E8, and decided that I was on vacation and not about to try and quickly write a blog posting about this.

Well, here are the press releases from MIT and AIM, and David Vogan did give a public talk about this yesterday at MIT. The media blitz was quite effective, getting the story into not just the usual suspects (there’s a good version of the story by JR Minkel at Scientific American), but also achieving a wide distribution in much less usual places such as today’s New York Times, the BBC, le Monde, and many, many others. I think this may be getting about as much attention as the proofs of Fermat’s Last Theorem and the Poincare Conjecture. There are also a huge number of blog postings, and I’m very pleased with myself to note that by far the best is the one by John Baez (crucially supplemented by the first comment there, from David Ben-Zvi), so I at least sent the journalists to the right place.

For mathematical details, John’s posting and the comments there are the best place to go besides the technical papers linked to from the AIM site.

While the calculation is a computational *tour de force*, and the computational methods may be useful elsewhere, the level of hype in the press releases, especially about the possible relations to physics, is somewhat disturbing. The AIM page on E8 and Physics contains statements such as

*…once one adopts the basic principles of string theory, it can be argued that we live in the universe we live in because it is the only one that is possible.*

as well as making the highly misleading claim that the new calculation has something to do with heterotic string theory.

What initially confused me about the press release is that, with the standard interpretation of what one means by “E8″, the “E8″ that appears in heterotic string theory, there is no open problem to be solved. The group is well-understood, and so is its representation theory. As a compact Lie group, the representation theory of E8 is part of the standard Cartan-Weyl highest weight theory, and was worked out long ago. To read about this, there’s an excellent book by Frank Adams about the representation theory of E8 and other exceptional Lie groups, called Lectures on Exceptional Lie Groups. It is this representation theory that appears in the heterotic string story. For more about E8, and one of the stranger things I’ve seen in a math paper, you might want to look up a 1980 paper by Frank Adams called “Finite H-spaces and Lie Groups”, in the Journal of Pure and Applied Algebra.

What the new result is about is something quite different, the “split real form” of E8. The classification of compact Lie groups proceeds by classifying their Lie algebras, giving a well-known list, with E8 the largest of the exceptional cases. In doing this, one complexifies (works over the complex numbers), studying the complex semi-simple Lie algebras, which are the Lie algebras of the complexifications of the compact Lie groups. In the simplest example, one studies SU(2) by complexifying its 3d Lie algebra (R^3 with vector product), i.e. studying the Lie algebra of SL(2,C) instead. Finite dimensional unitary representations of SU(2) correspond to holomorphic representations of SL(2,C), and the same correspondence works in general between finite dimensional unitary reps of compact Lie groups and holomorphic representations of their complexifications.

Given the complexified group, one can ask if it has other “real forms”, i.e. subgroups other than the compact one which would have the same complexification. In the case of SL(2,C), there is another real form: SL(2,R). The representation theory of SL(2,R) is a vastly more complicated subject than the case of SU(2). One reason is that the group is non-compact. Geometrical constructions of representations like the Borel-Weil construction give infinite-dimensional irreducible unitary representations. The case of SL(2,R) is difficult enough (and a central topic in number theory), but the case of representations of general real forms of semi-simple Lie groups is extremely difficult and complicated. Representations are infinite-dimensional and labeled by “Langlands parameters” instead of highest weights. This theory has been pretty well worked out over the last 30-40 years or so, with the case of E8 one where it was known how to do calculations in principle, but they had so far been computationally intractable. Dealing with this is the new advance.

What actually is calculated are things called “Kazhdan-Lusztig” polynomials; for an explanation, see John’s blog. These tell one how to build arbitrary irreducible representations out of something simpler which one does understand, certain induced representations called “standard” representations. The numbers involved here also have a beautiful geometrical and topological interpretation. This is a generalization of what happens in the compact case, where the cell decomposition of the flag variety governs how irreducibles are built out of Verma modules.

So, this is a result about the structure of the irreducible representations of one of the real forms of E8 called the “split” real form. As far as I know it has nothing to do with heterotic string theory. The only thing I can think of that physicists have worked on that might make contact with this result is the work of people like Hermann Nicolai and Peter West trying to get physics out of Kac-Moody algebras like E10 and E11. I have no idea whether they have run into the split real form of E8 subalgebras and the representation theory of these in their work. In Pisa I had the pleasure of meeting blogger Paul Cook, a student of Peter West’s who is now a postdoc in Pisa and has worked on this kind of thing. Perhaps he would know about this.

**Update**: I hear from Jeffrey Adams that he has put together a web-page about this, aimed at mathematicians, and designed to explain the nature and significance of this result. It’s quite clear and does a good job of this, accessible if you have a bit of background in representation theory. If not, you may at least enjoy his comment on the media attention:

*This leaves the question of why this story took off in the press. For us, that is harder to understand than the Kazhdan-Lusztig-Vogan Polynomials for E8.*

Dear Peter,

I think that AIM link is to an older posting of John Baez which is not

about the recent calculation but about compact form of E_8.

Matti,

Not sure which link you mean. The “E8 and Physics” link is on the main page announcing the new result. It does just refer to the compact case and links to an old Baez piece about this. I think it would be better if they made clear that this is rather different than the case they’re issuing a press release about.

The analogy in SL(2,C) would be trying to promote a new result about SL(2,R) representation theory by claiming that it had do with the physics of spin (SU(2)). They’re two very different things….

The split real form of E8 shows up when you compactify 11D SUGRA on an 8-torus. In particular, all the fields organize them into representations of the group (except for the scalars which organize themselves into a sigma model into the homogeneous space E8/K where K is the maximal compact subgroup). This is the inspiration for the ideas of Nicolai, West, Ganor and others about exceptional symmetries in M-theory.

Is it common for mathematical results to be announced via press release? What kind of coverage could anything less than a result like FLT hope to get?

Thanks Aaron,

Very interesting. Do you know if there’s any possible application of this calculation to that story?

Cody,

No, it’s not common to announce math results this way. I’m rather surprised at how successful it was in getting attention. Lots of universities and other organizations issue press releases about the work of their people, but normally these are mostly ignored by the general media. Sounds like AIM has a really, really good person handling this.

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Thanks for the plug!

One of the people on the Atlas team, Jeffrey Adams, just let me post an email explaining more about what they calculated.

Peter,

Yes,I meant “E8 and physics”. By the way, the flag manifold discussed in the page of John Baez is similar to that appearing in the construction of modular representations in Langlands program.

I just tried to read that about 5 times and now I want to hit my head on my desk because I am lost after about the first 8 words.

The tone of your post is unfortunately negative. You have allowed your antipathy to all things stringy to mislead you into misreading what has been done and what has been claimed to have been done. Look at the Atlas webpage and you will find nothing about string theory, only a coherent project to describe representations of real reductive groups. The split real form of E8 being the largest exceptional simple real Lie group, the calculation of things like its character table is impressive both theoretically and computationally. The goal of representation theory is not to know in the abstract how to find representations, rather to know the representations and how to decompose their tensor products. When G has many, and their structure is complicated, it is at first bewildering how one might even organize a map (a page in an atlas) of the representations of G. This is what the Atlas project is after.

There is a certain tendency to wish to find the exceptional simple real Lie groups in nature, but this tendency has really very little to do with a mathematician’s interest in these groups.

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To assist readers in better understanding the E8 relation to the Leech Lattice, the Monster and [string] physics, consider:

1 – The Terry Gannon arXiv paper ‘Monstrous Moonshine: the first 25 years’ [33 pages with 124 references].

http://www.arxiv.org/PS_cache/math/pdf/0402/0402345.pdf

2 – This appears to form the framework of the Gannon book, ‘Moonshine Beyond the Monster’ [477 pages, 575 references].

Lieven le Bruyn refers to [1] as a survey paper at ‘Never Ending Books’.

http://www.neverendingbooks.org/?p=133

Peter,

This is a worthy but quite modest result. But the hype surrounding it that has been confected by the people involved is just embarrassing. That the media have fallen so badly for that hype is just one more testament to how intellectially impoverished reporters and media outlets have become.

The classification of irreducible unitary representations of reductive Lie groups has been a central focus of David Vogan’s work for over 30 years. But no general picture has emerged, not even a conjectural one. For the last few years he has been making a computational assault. But for this kind of question a computational approach all but concedes defeat. The original goal was a computer program that would tell you whether or not a parametrically inputted potentially unitary representation is actually unitary. Hypothetically, if there were a conceptual description of the irreducible unitary representations for all representations other than the split real form of E8 (rather than computational description), the omission would be of much concern to no one. Conversely for a computational approach, a general answer would not add all that much to an answer for all cases other than E8.

The matrix that relates the irreducible (unitary or not) representation to the so-called standard representations is called the “Kazhdan-Lusztig matrix.” This effort computes the Kazhdan-Lusztig matrix for E8, as well as the inverse of that matrix. The algorithm to do this in all cases is well known. The only problem was that some of the matrix entries are very large integers, so computation for E8 required too much computer memory. Someone I will not name – someone who has not even been credited publicly in all this hype — suggested to the Adams-Vogan group that they perform the E8 computation modulo several large (but not too large) primes. That reduces the amount of memory enormously and the actual answer can then be reconstructed. This is exactly what the Adams-Vogan group now has done. The program can run on a laptop for groups other than E8. The hype (although not the result) is risible because just knowing the Kazhdan-Lustig matrix is parsecs from solving the unitarity problem.

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Peter, if I needed an opinion about a scientific breakthrough, you’d be my first choice as well. I’ll keep you on speed dial…

Turkey Royale,

To the extent my comments were negative, they referred not to the Atlas project or its web-pages, just to the AIM web-pages and press release. I think the material there about relations to physics is seriously misleading, and one of the main things I wanted to do in the posting was to clearly explain what the issues are, something I don’t see getting explained anywhere else on the web or in the media.

As for the significance of this as pure mathematics, sure, it’s an impressive piece of work, especially as a computational achievement. The fact that the AIM people managed to get it so much attention is quite a phenomenon. In one sense it’s great: if every mathematical advance of this magnitude was widely covered in the press, that would be wonderful, and there would be a lot more press stories about math. But, given how traditionally only the most dramatic millennium-prize sort of advances normally get covered, I do think it’s a good idea to offer some perspective on this one.

I particularly enjoyed digesting this hi-calorie forkfull of hyperblather (http://news.yahoo.com/s/afp/ussciencemathematicsfrancegermany):

“Today string theorists search for a theory of the universe by looking at E8 X E8. The scientists said the magnitude of the E8 calculation invited comparison with the Human Genome Project. While the human genome, which contains all the genetic information of a cell, is less than a gigabyte in size, the result of the E8 calculation, which contains all the information about E8, is 60 gigabytes in size, they said.”

But this implied claim that the mathematicians who “have successfully mapped E8″ have accomplished something about 60 times as difficult and important as the biologists “mapping” the human genome raises a difficult issue: How is it that the marketing geniuses who developed and sold string theory failed to call it “Strand Theory” instead? After all, if it were called Strand Theory the facile equation of (1) the importance of this branch of physics without a testable hypothesis or prediction and (2) the ultra-important and practical structure of DNA (which comes in the most famous double strands of all!) could have been enormously facilitated in the mind of the public. As it is, the Adams-Vogan group (or at least their publicists) have to labor mightily to insinuate that their result is as epoch making as the mapping of the human genome at the same time they are laboring to imply that E8 contains – through a completely preposterous and non-existent string theory connection – something like the “genetic code” of particle physics.

Gosh, they must be tired. And all of that extra labor could have been eased so much by just one simple change in nomenclature!

It just makes you wonder where their heads were at when they were poking them into all those extra dimensions.

Peter,

My post did not intend to defend the hype-seekers and fame-mongerers, just to say that the culpability for the allusion to string theory did not appear to belong to the representation theorists in the Atlas Project. That is not to say that their efforts merit or do not merit press coverage; the folks actively seeking press coverage are the same folks all the time scheming to dominate this or that aspect of academia, and they are very tiresome for the rest of us who just want to modestly understand a few modest things within our comprehension.

In this sense, I think you are basically right to criticize the publicity seeking efforts of AIM. Overstating one’s case undermines one’s credibility in the eyes of those who know something.

I think the unnamed suggestor mentioned in Musil’s post was Noam Elkies.

Now that the dust has settled I would like to make a comment about the attention this story has gotten in the press.

The goal of the Atlas of Lie Groups and Representations is to classify the unitary dual of a real Lie group G by computer. A step in this direction is to compute the admissible representations of G, including their Kazhdan-Lusztig-Vogan polynomials. The computation for E8 was an important test of the technology. While an impressive achievement, it is but a small step on the way towards the unitary dual, and not remotely as important as the original work of Kazhdan, Lusztig, Vogan, Beilinson, Bernstein et. al.

Nevertheless, because of the nature of the result, the Atlas team

and the American Institute of Mathematics decided this would

be an excellent opportunity to educate the public about research in pure mathematics. The intended audience of this campaign was the general public, and it was undertaken for the benefit of mathematics awareness as a whole, and not for the Atlas project itself. We are happy to have been successful in raising awareness of mathematics research worldwide.

For more information see some details about the Atlas project.