Number Theory News

  • Andrew Wiles is the recipient of this year’s Abel Prize. I have to confess that I found this surprising, since I assumed he’d already won this. His work in general and specifically the work that led to the proof of Fermat’s Last Theorem is on any reasonable list of the top few achievements in mathematics in recent decades.

    If you haven’t seen the documentary about the FLT proof, you really should, it was a BBC Horizon show in the UK, Nova here in the US, transcript here.

    I’d heard and Nature confirms that Wiles has for quite a while now been working quietly on the BSD Conjecture, maybe some day there will be another very dramatic moment in the subject, and another documentary.

  • Erica Klarreich at Quanta has the story of a surprising new result about prime numbers from Kannan Soundararajan and Rober Lemke Oliver. They have found that, given a prime number with a certain last digit, there are different probability for the last digit of the next one (among the various possibilities). This violates usual assumptions that such things are in some sense “random”, indicating just how subtle this “randomness” is.

    For more details, there’s an excellent blog post from Terry Tao. This might be a good time to point out that people sometimes complain about the quality of coverage of scientific advances aimed at non-experts. From what I’ve seen in recent years, the coverage of mathematics advances has been of extremely high quality, with this story a good example.

  • April 29 is the release date for The Man Who Knew Infinity, a film about the life of Ramanujan. It’s based on a great biography and a fascinating story. I hope this turns out better than the similar situation with the film about Turing.

Update: It turns out that an astronomer, Chung-Ming Ko, had already a while ago done some calculations showing non-randomness in the last digits of primes, see here. The new paper has been updated to refer to that.

Reports about the Ramanujan film are that they took great pains to get the mathematics right, with Ken Ono and Manjul Bhargava working extensively on the film.

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The Ultimate Simplicity of Everything

There’s a wonderful interview with Perimeter Institute director Neil Turok here, entitled The Ultimate Simplicity of Everything, and done for a Canadian radio program.

Turok discusses his point of view on whether we’re at “the end of physics”, and I’m very much in agreement with what he has to say:

I think what people are sort of expressing is that we haven’t had a big revolution in physics. String theory was hoped for to be that revolution in the 1980s but it hasn’t really panned out in the sense that it hasn’t given a single prediction. Instead it’s given us a huge collection of theories where, if you like, there’s no overarching theory to tell which particular version of string theory is the one that describes the world. It’s almost self-destructed, I would say because it turned out to be not just one theory but this vast collection of theories which could all give different descriptions of the world.

So I think that sort of theoretical catastrophe, as I view it — meaning the logical pursuit of quantum mechanics and relativity over a hundred years was tremendously successful at some level but finding its own successor theory, it hasn’t been successful. I think that is also laying the ground for some sort of revolutionary change in the sense that we basically will have to go back to the founding principles. It looks like the founding principles of modern physics — quantum theory and relativity — have played out and they have not given us the answers we need. And so we have to go back and question those founding principles and find whatever it is, whatever new principle will replace them. So matching these great puzzles posed by the observations are equally great puzzles in our fundamental theories. And so that is just a wonderful thing to contemplate in itself. I mean, partly people become very pessimistic and say, oh my god, I’ve devoted 50 years of my life to studying this incredibly technical and difficult theory and now I find it’s blown up in my face, it’s not giving any predictions at all…and so some people talk about the multiverse where the universe would be wild and chaotic on large scales and almost anything you could imagine would actually exist somewhere in the universe. I mean, this is literally a scenario which became very popular among a category of physicists, that there is a multiverse out there. Yet the evidence is exactly the opposite. That, as we look around us, things could not be simpler. There’s no evidence for chaos on large scales in the universe. It’s totally the opposite. It’s pristine, elegance, minimalism is all we see. So, I think this is a very, very exciting time to be doing theory. The challenge is enormous. The clues are enormous. We’re waiting and we’re preparing and we’re encouraging people to take radical leaps.

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This and That

A few short items:

  • Nature has an editorial this week summarizing the situation with the 750 GeV possible diphoton bump. It mentions a new paper analyzing related data (the number of theory papers on this as a function of time). The paper is called A Theory of Ambulance Chasing, and claims that looking at a large collection of similar fads producing theory papers, the high-level behavior of the HEP theory community can be well summarized by a model that requires only two parameters to fit the data.
  • If you’re at Stanford tomorrow and a fan of multiverse mania, you can go hear Alexander Vilenkin talk about The Universes Beyond the Horizon. According to the Stanford PR for this

    Despite the similarities between Vilenkin’s theory and the Wikipedia summary of the film Interstellar, many scientists have hope for the multiverse theory.

    Stanford physics faculty members seem to have innovative ideas about the scientific method, with one of them quoted as claiming

    Once a reasonable idea comes, you can never say it’s wrong.

    which I guess could be taken as some sort of motto for research into string theory and the multiverse.

  • CERN is running a series of articles about the Theory group there, first one is here.
  • Norbert Bodendorfer has a nice new blog about loop quantum gravity and related topics.
  • The Templeton Foundation has mercifully stopped giving huge financial prizes to people for dubious attempts to bring religion and science together. I hadn’t even realized they had already given out this year’s Templeton Prize, which went to a British Rabbi.

    Among the many things they fund is a recent $1.1 million grant for this project on the philosophical implications of quantum gravity. They will hold a summer school this year (with Amanda Peet and Carlo Rovelli, that should be fun), and there’s a Youtube channel.

  • Chris Quigg has an interesting overview of the future of HEP physics. I particularly like his emphasis on the questions

    How are we prisoners of conventional thinking?


    Might we have misunderstood the hierarchy problem, and so need to reframe it? Perhaps it is time to ask whether the unreasonable effectiveness of the standard model (to borrow a turn of phrase from Eugene Wigner) is itself a deep clue to what lies beyond.

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Multiverse Observed, South of Glasgow

It turns out the multiverse does exist, just off the A76, 25 miles north of Dumfries in Scotland. It’s called the Crawick Multiverse and is now open 10 am to 4 pm. Admission is 5 pounds, but parking is free.

For more details, see this story, which explains about the huge mounds, and that

There’s also a mound where mudstone slabs trace a spiral path up to the top that represents the multiverse. Along the way, some of the slabs are carved out to symbolise other potential universes where different physical laws apply.

The idea seems to be that

the park is a modern take on Neolithic monuments such as Stonehenge, which paid tribute to the movements of the Solar System – but this time the focus is on the latest advances in physics, such as chaos theory and the idea of parallel universes.

so while Stonehenge was designed to last forever and show that a primitive people understood about the solar system, this is going to show our descendants that 21st century humans knew about the multiverse.

Since it is well known that the LHC is our best chance of figuring out if multiverses exist, it seems that the people at CERN have agreed to have the same person build something like this there. This should ensure that when any future generations excavate the LHC site, they will get some idea of its purpose, and marvel at how much progress the human race made after the Neolithic era.

Update: Another observation of a multiverse, this one in Trieste.

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Michael Atiyah’s Imaginative State of Mind

Quanta magazine has an intriguing article by Siobhan Roberts out about Michael Atiyah, and what he’s up to these days. It mentions some new ideas about twistor theory I hadn’t heard about, that emerged from a conversation with Penrose, which Penrose wrote up as Palatial twistor theory and the twistor googly problem. Penrose explains the name:

The majestic ambiance of the unusual location (Buckingham Palace) of a brief discussion with Atiyah, no doubt provided inspiration for the initial thought that non-commutative twistor algebra should be the key to those subsequent developments described in this paper.

The Quanta article explains:

One day in the spring of 2013, for instance, as he sat in the Queen’s Gallery at Buckingham Palace awaiting the annual Order of Merit luncheon with Elizabeth II, Sir Michael made a match for his lifelong friend and colleague, Sir Roger Penrose, the great mathematical physicist.

Penrose had been trying to develop his “twistor” theory, a path toward quantum gravity that’s been in the works for nearly 50 years. “I had a way of doing it which meant going out to infinity,” Penrose said, “and trying to solve a problem out there, and then coming back again.” He thought there must be a simpler way. And right then and there Atiyah put his finger on it, suggesting Penrose make use of a type of “noncommutative algebra.”

“I thought, ‘Oh, my God,’” Penrose said. “Because I knew there was this noncommutative algebra which had been sitting there all this time in twistor theory. But I hadn’t thought of using it in this particular way. Some people might have just said, ‘That won’t work.’ But Michael could immediately see that there was a way in which you could make it work, and exactly the right thing to do.” Given the venue where Atiyah made the suggestion, Penrose dubbed his improved idea “palatial twistor theory.”

The article links to this recent talk about the role of beauty in mathematics, and describes some very speculative ideas he’s been working on, which I guess correspond to for instance this paper.

About this kind of work he has this to say:

If you try to direct science, you only get people going in the direction you told them to go. All of science comes from people noticing interesting side paths. You’ve got to have a very flexible approach to exploration and allow different people to try different things. Which is difficult, because unless you jump on the bandwagon, you don’t get a job.

Worrying about your future, you have to stay in line. That’s the worst thing about modern science. Fortunately, when you get to my age, you don’t need to bother about that. I can say what I like.

When asked if he’s risking his reputation this way, he has this sensible response:

My reputation is established as a mathematician. If I make a mess of it now, people will say, “All right, he was a good mathematician, but at the end of his life he lost his marbles.”

A friend of mine, John Polkinghorne, left physics just as I was going in; he went into the church and became a theologian. We had a discussion on my 80th birthday and he said to me, “You’ve got nothing to lose; you just go ahead and think what you think.” And that’s what I’ve been doing. I’ve got all the medals I need. What could I lose? So that’s why I’m prepared to take a gamble that a young researcher wouldn’t be prepared to take.

Update: For an alternate source of information about “palatial twistor theory”, see slides here, video here.

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Who’s Winning the String Wars and Why Should You Care?

Part two of Gerald Alper’s piece at Smashpipe is now available there, with the title Who’s Winning the String Wars and Why Should You Care?, and some more substantive material than in part one. One of the great things about having a blog is that whenever anyone writes anything about you that you think might not be 100% correct, you can blog about it, and explain yourself ad nauseam. So, here are a few clarifications for readers of that article:

  • About the “horrible sentence”

    The Hilbert space of the Wess-Zumino-Witten model is a representation not only of the Kac-Moody group, but the group of conformal representations [transformations] as well.

    I don’t think it’s a bad sentence, it succinctly conveys the main point about the close relationship of the WZW QFT to representation theory. Like a certain number of things in the book though, it’s not intended for everyone. There were certain things I wanted to explain, and the way I went about this was to try to as clearly write them down as possible, in a way accessible to as many people as possible, but well aware that not everyone would understand everything. Unlike writing “the WZW model is related to mathematics like X is to Y”, where X and Y are things most people would recognize, you’re not going to get fooled into believing you understand something you don’t by what I was writing. Those who do understand the sentence will understand a real idea.

  • I AM VERY CONFIDENT that I AM RIGHT. No one has ever critiqued string theory with the level of detail that I have.

    Not sure exactly how I said this, but I suspect the “no one has ever” wasn’t intended to convey that this was a good thing. I’ve clearly spent too much of my life thinking about this. I also should specify that what I’m confident about is that current “string vacua” models don’t correspond to reality. They’re complicated, ugly, and don’t explain anything. My suspicion is that even Witten might not completely disagree with this, acknowledging that at our current understanding of string theory, there is no convincing model. I think a more accurate way of characterizing where Witten and I disagree here is with how promising it is to pursue this particular vision of unification. I am not at all confident that Witten or someone else pursuing it might not come up with something really new and successful some day. I just think it’s a relatively unpromising route to keep heading down, although I acknowledge it’s possible it might lead to finding a more interesting path. Doubtless Witten feels the same way about things I find more promising.

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Beyond Experiment: Why the scientific method may be old hat

This week’s New Scientist has an article by Jim Baggott and Daniel Cossins entitled Beyond Experiment: Why the scientific method may be old hat, which deals with the recent controversy over attempts to excuse the failure of string theory by invoking the multiverse. The article (unfortunately behind a paywall) does a good job of describing the nature of the controversy: what do you do when it becomes clear your theory can’t be tested? Do you follow the conventional scientific norms, give up on it and work on something else, or do you try and find some kind of excuse, even if it means abandoning those norms?

Much of the article deals with the issues raised at the recent Munich conference (discussed here). Two of those quoted (Dawid and Gross) are not multiverse partisans, instead argue that the motivations that got people interested in string unification more than 30 years ago are good enough to justify indefinitely pursuing the theory, no matter how bad things look for prospects of connection to experiment. On the other hand:

Their enthusiasm is far from universal, and some physicists are downright alarmed. Woit warns that the need for empirical vindication could be pushed so far into the background as to be invisible. Carlo Rovelli, a theorist at the University of Aix-Marseille in France, believes that this scenario has already come to pass. Rovelli … argues that the last thing we need is a system that legitimises failed theories. “A theory is interesting when it teaches us something new about the real world,” he says. “Not when it becomes a house of cards that delivers nothing but university positions.”

On the question of the string theory multiverse as science, those gathered at the Munich conference were pretty uniformly hostile. As a proponent of this, the article quotes only one person, who wasn’t there:

Sean Carroll, a theorist at the Caltech Institute of Technology at Pasadena and a leading advocate of the multiverse, insists that if anyone is being unscientific, it is those physicists who seek to enforce outmoded philosophical principles and impossibly high standards. “People support these theories because they offer the best chance of providing a useful account of the data we actually do collect here in our universe.”

I’m not sure how the string theory multiverse provides an account of data we have collected that is “useful”, except in the sense of “useful to those who don’t want to give up on string theory.”

Carroll has explained his views in more detail here, arguing that falsifiability is an idea that needs to be retired, to be replaced by “empiricism”. “Empiricism” seems to mean “ability to account for the data”, with “the multiverse did it” an acceptable way to account for data, even if not falsifiable. He’ll be giving a talk on this at the American Astronomical Society meeting in San Diego this summer, with abstract:

A number of theories in contemporary physics and cosmology place an emphasis on features that are hard, and arguably impossible, to test. These include the cosmological multiverse as well as some approaches to quantum gravity. Worries have been raised that these models attempt to sidestep the purportedly crucial principle of falsifiability. Proponents of these theories sometimes suggest that we are seeing a new approach to science, while opponents fear that we are abandoning science altogether. I will argue that in fact these theories are straightforwardly scientific and can be evaluated in absolutely conventional ways, based on empiricism, abduction (inference to the best explanation), and Bayesian reasoning. The integrity of science remains intact.

Carroll’s argument seems to be that the conventional understanding of how science works that we teach students and use to explain the power of science has always been wrong. Falsifiability by experiment isn’t necessary, instead, what is the “absolutely conventional” way to do science is “empiricism, abduction (inference to the best explanation), and Bayesian reasoning”. I’d never heard of abduction as a basis of science before. If you believe Wikipedia, this goes back to Charles Sanders Peirce, whose view in later years was:

Abduction is guessing. It is “very little hampered” by rules of logic. Even a well-prepared mind’s individual guesses are more frequently wrong than right. But the success of our guesses far exceeds that of random luck and seems born of attunement to nature by instinct (some speak of intuition in such contexts).

As for “Bayesian reasoning”, I would have thought that Polchinski’s Bayesian calculation of an “94% chance” of a multiverse would have conclusively shown the absurdity of that.

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Various and Sundry

  • The online magazine Smashpipe has the first part of a two-part article written by Gerald Alper, who recently came up here to Columbia to talk to me about string theory/etc. It was an interesting conversation, so I’m curious to see what he makes of the more substantive part, which is in part two, planned for next week.
  • If instead you’d like to read about a conversation with my colleague Brian Greene, there’s a piece at Cosmos Magazine. Brian is taking his World Science Festival to Australia next month and will be on tour there.
  • In other Columbia news, LHC experimentalist Emlyn Hughes has evidently
    baffled the students in Frontiers of Science again. Three years ago he undressed for the students, this year the performance somehow involved a student mistress (see here and here). No, I don’t understand any of this either.
  • As a last Columbia story, this semester in the physics department Bill Zajc is teaching a string theory course for undergraduates, Physics W4012, based on the Zwiebach book. While Zajc isn’t a string theorist, he is a frequent commenter at Lubos Motl’s blog.
  • There’s a new issue out of Inference, which has some interesting articles, including an essay by Pierre Schapira on category theory (French version here). Also there’s Jean-Pierre Luminet on holography.

    Inference is a bit of a mystery, unclear who is editing it (some speculation here). Whoever it is though, it’s quite worth paying attention to.

  • The HEP Postdoc Project is collecting anonymously information aimed at helping potential postdocs (or even Ph.D students) find out more about what it’s like to work with various senior HEP theorists. No, like Inference, I have no idea who is behind this.
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Yet More About Grothendieck

Since Grothendieck’s death somewhat more than a year ago, quite a lot of new material about him and his mathematics has become available. Visit the Grothendieck Circle to find a lot of this, with just one example some new chapters of the English translation of the third volume of Scharlau’s biography.

This month’s AMS Notices has the first of two parts of a long article with contributions from many mathematicians discussing Grothendieck’s work and their memories of him and his influence on their careers. Colin McLarty has an excellent expository article, maybe the best of attempts I’ve seen to explain some of the themes of Grothendieck’s mathematics in a relatively accessible manner.

While you’re there, this latest issue of the Notices has quite a bit else worth reading, from my colleague Ivan Corwin on KPZ universality to Beilinson on Gelfand’s seminar, and an amusing attempt by Jeremy Gray to guess not the next Fields medalist, but who would have gotten one in 1866.

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This Week’s Hype

This week’s dramatic announcement of the discovery of gravitational waves was a major milestone for the fields of physics and astrophysics. The LIGO observation validates a lot of previously untested aspects of our understanding of general relativity, and promises the imminent opening up of a new field of observational astronomy, as LIGO sees other astrophysical sources of gravitational waves. Watching the announcement, the lead up to it, and the press stories that came out, many immediately as the embargo was lifted, I was struck by the general high quality of the stories in the press (I linked to a few of them in the last posting, but there are many more). Congratulations to whoever organized this, and to all the science writers who have done a great job producing enthusiastic but generally hype-free coverage of the story.

Unfortunately, those physicists brought in by major news organizations to tell the public what the significance of this is often can’t resist the temptation to indulge in the usual hype. At the Wall Street Journal today, Michio Kaku’s commentary is labeled Riding Gravity Waves to the Big Bang and Beyond, and subtitled “Once again, Einstein’s theory of relativity is confirmed by scientists. Next stop: Creation.”

There’s nothing in his piece about what else LIGO might observe and what we might learn from it about the universe. Instead, it’s all about the big bang, Creation, and before the big bang, things which as far as I can tell, LIGO data is highly unlikely to tell us anything about:

Now we are witnessing the third great revolution in telescopes, the use of gravity waves to open a new chapter in astronomy. For the first time, waves from the very instant of creation might be observed, giving us “baby pictures” of the universe as it was born. High-school textbooks may have to be rewritten to incorporate the new discoveries coming from this third generation of telescopes.

This may also have philosophical implications. Right now the big-bang theory doesn’t tell us what banged, why it banged, and what caused it to bang. It only tells us that there was a bang. But if space-based gravity-wave detectors similar to LIGO’s detectors can measure the radiation emitted an instant after the big bang, then, using mathematics, one can run the equations backward to determine what set off the big bang in the first place, in effect answering the biggest question of all: What banged and why?

When Einstein postulated gravity waves a century ago, he not only opened up an entirely new chapter in astronomy, he also opened the door to answering the most important philosophical questions of all time, including the creation of the universe.

Over at the New York Times, in the Sunday Review, Lawrence Krauss has a more sensible piece, entitled Finding Beauty in the Darkness. Multiverse mania seems though to be irresistible, as he ends up with this summary of the physics significance:

Ultimately, by exploring processes near the event horizon, or by observing gravitational waves from the early universe, we may learn more about the beginning of the universe itself, or even the possible existence of other universes.

Posted in Multiverse Mania, This Week's Hype | 30 Comments