Langlands and Twistors?

I just heard about this from George Sparling, who is giving a talk this afternoon with the title “From Roger Penrose to Robert Langlands and back” at a symposium in Pittsburgh. I don’t at all know what this is about, but am posting this quickly because he tells me there may be a live stream of his talk at 5pm today, available at

http://live.twistor.org

Evidently yesterday and today at Pittsburgh there’s a symposium on “Towards the Unification of Mathematics and Physics”, with the following schedule:

4pm Wednesday: Mellon Professor Thomas Hales, “The present status of the Langlands Conjecture”.
5pm Wednesday: Jonathan Holland, “Parabolic geometry”.

4pm Thursday: Tim Adamo, University of Oxford, “Twistor-String theory and gravity”
5pm Thursday: George Sparling, “From Roger Penrose to Robert Langlands and back.”

Last week there was a workshop at Banff on The Geometry of Scattering Amplitudes, with videos now available. I’ll try soon to take a look at some of them, Sparling’s may give an idea of what he is up to.

And, for something completely different, see this interview with Edward Frenkel about the Langlands Program at the Fields Institute blog for this fall’s Fields Medal symposium.

Update: There’s a video of the Sparling talk available here.

Posted in Langlands | 5 Comments

This Week’s Hype

BBC Horizon this week is running an episode How Small is the Universe? with a description that features the usual sort of hype about modern physics:

It is a journey where things don’t just become smaller but also a whole lot weirder. Scientists hope to catch a glimpse of miniature black holes, multiple dimensions and even parallel Universes. As they start to explore this wonderland, where nothing is quite what it seems, they may have to rewrite the fundamental laws of time and space.

Access to the video is restricted to IPs in the UK, so I can’t watch the thing, and should avoid being too critical. One of the two clips though advertises The landscape of String Theory and somehow I doubt that the clip explains why this is pseudo-science. Associated with the show is this article by Andy Parker of ATLAS, which gives the idea that ATLAS is looking for strings:

Strings can vibrate, and this allows us to explain all of the strange fundamental particles which we see as different vibrations of the strings – different notes from a cosmic violin.

So far, so simple – but to explain the particles we know about, the strings have to vibrate in lots of different ways.

Superstring Theory allows them to vibrate in a bizarre space with 11 dimensions – up, down, sideways, “crossways” and 7 other ways!

Experiments at the LHC are looking for evidence that you can move “crossways”. If we can, there could be whole universes, as big and marvellous as our own, sitting just down the road “crossways”.

No mention is made of the fact that the LHC has seen zero evidence for any such thing, or that few if any physicists ever thought there was any real chance it would.

The other experiment invoked is the MAGIC gamma ray telescope, presumably in the context of the search for Lorentz-violating dispersion of gamma rays from gamma ray bursters. This was discussed in an edition of This Week’s Hype from five year’s ago, which featured a Slashdot report that Gamma Ray Anomaly Could Test String Theory. At Scientific American, the story was Hints of a breakdown of relativity theory?, which was about this paper, and contained the news:

Another co-author, string theorist Dimitri Nanopoulos of Texas A&M, writes to me: “I am very excited about this, because as you know we suggested this effect about ten years ago and we have follow through with several analyses and/or improvement on theory. Notice that the 0.4 x 1018 GeV is the typical string scale!!!!”

Since 2007 there have been a series of much more sensitive results from Fermi ruling out the quantum gravity interpretation of the MAGIC observations (see e.g. here, here, here and here.)

Since I can’t watch the video, I don’t know what the BBC has to say about MAGIC’s results, in particular whether the show explains the story of the 2007 claims and how they were later shown not to have anything to do with space-time structure by the newer Fermi observations.

Update: I did just get a chance to watch the program. It was very well made, with the first half quite interesting, featuring the LHC and some atomic-physics scale experiments I would have loved to hear more about. About half-way through though, it started to go off the rails, with the usual kinds of problems. The extra dimensions at the LHC stuff made no mention of the fact that even string theorists see no good reason for them to show up at this scale, and the results to date confirm this. The Mike Green segment was pretty much pure string theory/multiverse hype. Reference to the “mind-boggling predictions” of string theory misses the main problem, that there are no predictions. In particular, no predictions about the gamma-ray dispersion MAGIC is looking for, which ended the show. The 5 second discrepancy described at the end in MAGIC 2005 observations I suspect has been shown to not be plausibly due to such dispersion by later Fermi results which went unmentioned.

Posted in This Week's Hype | 15 Comments

Proof of the abc Conjecture?

Jordan Ellenberg at Quomodocumque reports here on a potential breakthrough in number theory, a claimed proof of the abc conjecture by Shin Mochizuki. More than five years ago I wrote a posting with the same title, reporting on a talk by Lucien Szpiro claiming a proof of this conjecture (the proof soon was found to have a flaw). One change over the last five years is that now there are excellent Wikipedia articles about mathematically important questions like this conjecture, so you should consult the Wikipedia article for more details on the mathematics of the conjecture. To get some idea of the significance of this, that article quotes my colleague and next-door office neighbor Dorian Goldfeld describing the conjecture as “the most important unsolved problem in Diophantine analysis”, i.e. for a very significant part of number theory.

Jordan is an expert of this kind of thing, and he has some of the best mathematicians in the world (Terry Tao, Brian Conrad and Noam Elkies) commenting, so his blog is the place to get the best possible idea of what is going on here. After consulting a couple experts, it looks like this is a very interesting and possibly earth-shattering moment for this field of mathematics. In the case of the Szpiro proof, the techniques he was using were relatively straightforward and well-understood, so experts very quickly could read through his proof and identify places there might be a problem. This is a very different situation. What Mochizuki is claiming is that he has a new set of techniques, which he calls “inter-universal geometry”, generalizing the foundations of algebraic geometry in terms of schemes first envisioned by Grothendieck. In essence, he has created a new world of mathematical objects, and now claims that he understands them well enough to work with them consistently and show that their properties imply the abc conjecture.

What experts tell me is that, very much unlike the case of Szpiro’s proof, here it may take a very long time to see if this is really a proof. They can’t just rely on their familiarity with the usual scheme-theoretic world, but need to invest some serious time and effort into becoming familiar with Mochizuki’s new world. Only then can they hope to see how his proof is supposed to work, and be able to check carefully that a proof is really there, not just a mirage. It’s important to realize that this is being taken seriously because such experts have a high opinion of Mochizuki and his past work. If someone unknown were to write a similar paper, claiming to have solved one of the major open questions in mathematics, with an invention of a strange-sounding new world of mathematical objects, few if any experts would think it worth their time to figure out exactly what was going on, figuring instead this had to be a fantasy. Even with Mochizuki’s high reputation, few were willing in the past to try and understand what he was doing, but the abc conjecture proof will now provide a major motivation.

Mochizuki has been at this for quite a while. See this page for some notes from him about how he has been pursuing this project in recent years. This page has notes from lectures he has given on the topic, starting in 2004 with A Brief Introduction to Inter-universal Geometry. For the proof itself, see here, but this is the fourth in a sequence of papers, so one probably needs to understand parts of the other three too.

Update: Barry Mazur has recently made available his 1995 expository article on the abc conjecture, entitled Questions about Number.

Posted in abc Conjecture | 17 Comments

Fall Course: Quantum Mechanics for Mathematicians

This fall I’m teaching on quantum mechanics for mathematicians, at the undergraduate level. There’s a web-page with more information here. I’ll be writing up lecture notes, which should appear on that web-page as the course goes on, starting Wednesday.

We’ll see how this works, but the plan is to teach many of the standard topics, although starting from a different point. Most quantum mechanics classes start out with classical mechanics, then somehow try and motivate quantum mechanics from there, following the historical logic of the subject. I’ll instead start with the simplest purely quantum systems, especially the two-state, spin-1/2 system, now famous as the “qubit” of quantum computation. This is also a central example for the theory of Lie groups, Lie algebras and representations, so something that every mathematician should become familiar with. Another advantage of starting here is that there’s no analysis, just linear algebra, and one can easily do everything rigorously.

Later on in the course I’ll get to the standard material about wave-functions and quantum particles in potentials. The emphasis will be though not on the analytical machinery needed as a rigorous foundation for this subject in general, but on specific problems and their symmetries, and the use of these symmetries to do real calculations, ending up with the spectrum of the hydrogen atom.

We’ll see how this goes, and what the students think of it. As lecture notes appear, corrections and suggestions of how to improve them would be appreciated.

Posted in Uncategorized | 35 Comments

The Templeton Effect

The Chronicle of Higher Education has a long story about the Templeton Foundation, entitled The Templeton Effect. Much of it is about various subfields of philosphy where Templeton money has been successful at bringing religion, theological concerns and religious philosophers to greater prominence. One section however describes the Templeton funding promoting a new field of Philosophy of Cosmology. Religion doesn’t explicitly appear here, but the story of how this “Philosophy of Cosmology” got underway gives a good example of how money influences intellectual pursuits:

Barry Loewer, a philosopher at Rutgers University at New Brunswick, isn’t likely to turn up at a Society of Christian Philosophers meeting with Newlands and Miller. “I myself have no interest in philosophy of religion and am not a religious person,” he says. For years, Loewer has been working with a group of philosophers, mathematicians, and physicists in the New York area, meeting and collaborating on papers—nothing very expensive. But about five years ago a colleague at Rutgers, Dean W. Zimmerman, told the group about the Templeton Foundation and suggested that they apply for a grant. Zimmerman, a top Christian philosopher, had already served on Templeton’s advisory board and participated in many foundation-sponsored activities.

The idea at first was to do a project about quantum mechanics and the foundations of physics, which was an interest of Loewer’s group. Templeton had other ideas. The foundation pointed the group in the direction of cosmology, with the prospect of a much bigger grant, and the researchers jumped at the idea. They realized that cosmology encompassed the questions of time and physical laws that had concerned them all along.

“You know that story of Molière’s where someone discovers that he has been speaking prose his whole life?” says Loewer. “It was a little bit like that.”

The nearly $1-million grant his team received from Templeton last year coincided with another, slightly larger one called “Establishing the Philosophy of Cosmology,” which was awarded to scholars at the University of Oxford. Despite the change of plans at Templeton’s behest, Loewer stresses, “They’ve been really helpful, and totally noncoercive in terms of any agenda that they might have. I had my eyes open for it.”

Not that philosophers are especially well practiced in negotiating the terms of million-dollar grants, much less in thinking about how such money might sway them. Neither Loewer nor Mele nor Miller nor Newlands could have anticipated back when they were in graduate school that they’d be administering projects like this; their training was for armchairs, libraries, and conferences. But now that the money is coming into the field, it is being welcomed even by those who lack the foundation’s spiritual proclivities. “Templeton picks some people whose Christian epistemology I might not share,” Brian Leiter says, “but there’s no quarreling that they’re serious philosophers.” Suspicions about some secret religious agenda tend to lessen the more widely the foundation’s substantial sums begin to spread.

Posted in Uncategorized | 7 Comments

Short Math Links

Some links of mathematical interest that I’ve recently run across:

  • The life and work of Alexander Grothendieck is one of the great stories of modern mathematics. Winfried Scharlau’s first volume of a biography of Grothendieck, covering the years up to 1948 is now available in English, see here. The third volume, covering Grothendieck’s life after 1970 is only available in German, see here. Leila Schneps is writing a second volume, covering his life and mathematics during the height of his career, from 1948-1970, with chapters appearing as they are written on this page. She is now up to 1952.

    The same page contains links to various wonderful articles about Grothendieck’s mathematics, many by mathematicians who interacted with Grothendieck during his period of greatest mathematical activity.

  • Cédric Villani joins other Fields Medalists with blogs, see here. Villani has just published in France a memoir called Théorème Vivant, a mix of autobiography and description of a collaboration on a mathematics problem. More about the book here, here and here, with a video here.
  • Many of the talks given at this summer’s String-Math 2012 conference are now available as slides or video, see here.
Posted in Uncategorized | 3 Comments

Simons Foundation and the arXiv

Via the Quantum Pontiff, news that the Simons Foundation will be providing up to \$300,000 in financial support to the arXiv for each of the next five year. Last year, the arXiv announced a \$60K planning grant from Simons. Now the Foundation is stepping in with a much bigger contributions, for details see here.

This kind of support for open-access publication is an excellent way for Simons to use its resources. Perhaps this will be the beginning of a larger effort to buy back control of the math and physics literature from commercial publishers and set up a viable model for making this literature available to all going forward. This may be an expensive undertaking, but Simons (and other math/physics-friendly financiers) have resources on the scale necessary to do this.

Posted in Uncategorized | 22 Comments

Linde on Inflation and the Multiverse

Andrei Linde is one of Yuri Milner’s $3 million dollar men, best known for his “chaotic inflation” version of inflationary theory, as well as being one of the main proponents of anthropic multiverse mania. There’s a long piece based on a conversation with him up now at the Edge web-site.

Much of the piece is just a retread of the usual heavily-promoted ideology of the past 30 years of fundamental physics research: we must have SUSY, so must have supergravity, so must have string theory, so must have the landscape, so must have a multiverse where we can’t predict anything about anything, thus finally achieving success. Linde claims he pretty much had this picture 30 years ago back in 1982, with the string theory component in 1986, with others coming around to his point of view in the last 10 years, partly because of the KKLT work he was co-author of in 2003.

Besides the tired Stanford pseudo-scientific ideology, there’s also a wonderful history of the subject of inflation, from a Moscow point of view, which is rather different than the Western, Alan Guth-oriented, point of view from which the story is often told. Linde’s description of Hawking’s visit to Moscow is not to be missed:

The next morning after I gave a talk at this conference, I found myself at the talk… oh, my God, this is going to be a funny story… I found myself at the talk by Stephen Hawking at Sternberg Institute of Astronomy in Moscow University. I came there by chance because I have heard from somebody that Hawking was giving a talk there. And they asked me to translate. I was surprised. Okay, I will do it. Usually at that time Stephen would give his talk well prepared, which means his student would deliver the talk, and Stephen from time to time would say something, and then the student would stop, and change his presentation and do something else. So Stephen Hawking would correct and guide the student. But in this case they were completely unprepared; the talk was about inflation. The talk was about the impossibility to improve Alan Guth’s inflationary theory.

So they were unprepared, they just finished their own paper on it. As a result, Stephen would say one word, his student would say one word, and then they waited until Stephen would say another word, and I would translate this word. And all of these people in the auditorium, the best scientists in Russia, were waiting, and asking what is going on, what it is all about? So I decided let’s just do it, because I knew what it’s all about. So Stephen would say one word, the student would say one word, and then after that I would talk for five minutes, explaining what they were trying to say.

For about a half an hour we were talking this way and explained to everyone why it was impossible to improve Alan Guth’s inflationary model, what are the problems with it. And then Stephen said something, and his students said: “Andrei Linde recently proposed a way to overcome this difficulty.” I didn’t expect it, and I happily translated it into Russian. And then Stephen said: “But this suggestion is wrong.” And I translated it… For half an hour I was translating what Stephen said, explaining in great detail why what I’m doing is totally wrong. And it was all happening in front of the best physicists in Moscow, and my future in physics depended on them. I’ve never been in a more embarrassing situation in my life.

Then the talk was over, and I said: “I translated, but I disagree,” and I explained why. And then I told Stephen: “Would you like me to explain it to you in greater detail?” and he said, “Yeah.” And then he rode out from this place, and we found some room, and for about two or three hours all the people in Sternberg Institute were in panic because the famous British scientist just disappeared, nobody knew where to.

During that time, I was near the blackboard, explaining what was going on there. From time to time, Stephen would say something, and his student would translate: “But you did not say that before.” Then I would continue, and Stephen would again say something, and his student would say again the same words: “But you did not say that before.” And after we finished, I jumped into his car and they brought me to their hotel. We continued the discussion, which ended by him showing me photographs of his family, and we became friends. He later invited me to a conference in Cambridge, in England, which was specifically dedicated to inflationary theory. So that’s how it all started. It was pretty dramatic.

Addressing the question of “what evidence is there for any of this?”, here’s what Linde has to say:

Usually I answer in the following way: If we do not have this picture, then we cannot explain many strange coincidences, which occur around us. Like why vacuum energy is so immensely small, incredibly small. Well, that is because we have many different vacua, and in those vacua where vacuum energy is too large, galaxies cannot form. In those vacua, where energy density is negative, the universe rapidly collapses, and in our vacuum the energy density is just right, and that is why we live here. That’s the anthropic principle. But you cannot use anthropic principle if you do not have many possibilities to choose from. That’s why multiverse is so desirable, and that’s what I consider experimental evidence in favor of multiverse.

So, the main experimental evidence for the multiverse is that an anthropic argument works. Some might not find this completely convincing.

About 5 years ago, the field of “string cosmology” was quite active, with even a graduate-level textbook appearing. My impression (contrast the tone of this review, with those of 5 years earlier) is that there’s much less interest in this area during recent years, since it became obvious that no predictions about physics were going to emerge from it. Late this year or early next year the Planck experiment will finally report what it sees in the CMB. I’m curious to know whether Linde and other string cosmology proponents have any predictions for what Planck will tell us.

Update: The Annenberg Foundation funds Annenberg Learner, a site designed to provide information to high school teachers. Their Physics course includes a unit from Stanford’s Shamit Kachru, which is pretty much pure hype, unadulterated by any skepticism that string theory might not be the way the world works. Physicists may have lost interest in string cosmology, seeing it as a failure, but that’s no reason not to teach it to high school students…

Update: Historian of science Helge Kragh has a new article Criteria of Science, Cosmology, and Lessons of History discussing the Multiverse, philosophy of science, and the dubious use of historical analogies. About the Multiverse: “it explains a lot but predicts almost nothing”.

Update: Tonight’s arXiv listings include The Top 10500 Reasons Not to Believe in the Landscape from Tom Banks, which starts off with:

The String Landscape is a fantasy.

He goes on to claim co-credit with Linde for the anthropic explanation for the value of the CC (in inflationary models), as well as to argue that it is wrong:

Linde and I were the first to suggest an anthropic explanation for the value of the c.c. based on inflationary models [2], but within the context of the string landscape, or most any contemporary view of global EI [i.e., Eternal Inflation], I don’t think anthropic reasoning leads to good phenomenology.

Posted in Multiverse Mania | 38 Comments

Bill Thurston, 1946-2012

Bill Thurston passed away yesterday, at the age of 65, after a battle with melanoma. Thurston was for many years the dominant figure in the study of 3 dimensional topology and geometry, winning a Fields medal for this work in 1982. His “Geometrization Conjecture” classifying the topology of 3 manifolds was finally proved by Perelman as part of his work on the Poincaré Conjecture.

For an exposition of some of his work, see The Geometry and Topology of Three-manifolds, which exists as a set of unpublished notes here, and a book covering the first few chapters of the notes here. Thurston was sometimes criticized for not writing up full proofs of his results, making it difficult for others entering the field (and sometimes students were advised not to enter the field since Thurston was so good the danger was he would just solve all open problems). He wrote a truly wonderful essay On Proof and Progress in Mathematics, responding to this and laying out part of his vision of how to do mathematics.

My first encounter with Thurston was in the early eighties, when I was a physics graduate student at Princeton. I was working on the problem of defining the topological charge of a lattice gauge field, and it became clear that one approach to do this would require computing the volumes of “spherical tetrahedra”, the 3d analog of the problem of computing the areas of spherical triangles. I’d had some experience trying to talk to mathematicians about the problem I was working on, with the usual result a baffling response about principal bundles, sections, characteristic classes, and all sorts of what seemed to be abstract nonsense (which later on of course I learned was the right way to think about the problem…). So, I was pretty convinced that mathematicians were uniformly experts in a lot of abstract, high-powered technology, surely no longer conversant with the kind of more concrete formulas of the mathematics of earlier centuries.

This was before the days of the internet, so the answer to my problem couldn’t be found via Google, and a bit of library research got me nowhere. So, I stopped by to see a friend who was a math grad student and asked him my volume question. He said that while he didn’t know, he knew someone who could surely help me, and took me over to the math lounge, where Thurston could often be found. After I asked my question, Thurston immediately knew the answer, explained it to me on the blackboard, and gave me the proper reference of where to read more (you break them up in a certain way and then get an answer in terms of things called Schläfli functions, see here). I realized that my views of how much the best research mathematicians knew about concrete calculations and lore from previous centuries had been rather naive.

Thurston’s death at such a relatively young age is a loss for us all. My condolences to his family, including his son Dylan, a very talented topologist in his own right, who has been my colleague here for the last several years.

Update: Terry Tao has more about Thurston and his work here.

Update: More here, here and here. Also worth the time is seeing what he had to say on MathOverflow.

Update: Jordan Ellenberg has something here.

Update: The New York Times has an obituary here.

Update: There’s a Cornell site here, Scientific American has a piece by Evelyn Lamb here, John Horgan here.

Update: Jonah Sinick has put together a memorial slideshow here.

Posted in Obituaries | 15 Comments

This Week’s Hype

The Higgs suggests that there could be more dimensions of space-time than we previously thought.

From a New Yorker piece this week (subscription required) about Joe Incandela of CMS and the Higgs discovery.

Even the famed New Yorker fact-checkers are no match for extra-dimensional hype. Will someone tell them they’ve been had?

Posted in This Week's Hype | 38 Comments