Math News

Various mathematics-related news:


Update
: This Friday and Saturday there will be a meeting in Cambridge on the topic of Ethics in Mathematics. Supposedly talks will be livestreamed on Youtube (perhaps here?).

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Are We In the Swampland?

Way back in 2005, soon after the emergence of the “String Landscape” and the ensuing debate over whether this made string theory untestable pseudo-science, Cumrun Vafa in response started writing about the “Swampland”. In contrast to the “Landscape” of effective field theories that are low energy limits of string theory, the “Swampland” is the space of effective field theories that are not low energy limits of string theory. One motivation here is to be able to claim that string theory is predictive, since if you can show a theory is in the Swampland, then string theory predicts that theory doesn’t describe our world.

I wrote a couple blog postings about this back then, see here and here. The situation was rather comical, with Jacques Distler unintentionally making clear one problem with the whole idea. He was enthusiastic about it, and gave as one example the suggestion that one and two generation versions of the Standard Model were effective theories that could not be derived from string theory, but Volcker Braun immediately wrote in to tell him about such a derivation. At this time I started having my problems with the arXiv (and its moderator, Distler) about trackbacks, but that’s another story.

I haven’t paid much attention to the Swampland business since then, but noticed last night a new preprint with the title What if string theory has no de Sitter vacua?. The authors summarize their argument:

From this analysis we conclude that string theory has not made much progress on the problem of the cosmological constant during the last 15 years. There is a general agreement that the presence of dark energy should be an important clue to new physics. So far, string theory has not been up to the challenge. Or to be more precise, string theorists have not been up to the challenge.

The well-motivated introduction of the anthropic principle and the multiverse, was a big relief. The mathematical standards were lowered, and unconstrained model building could set in exploring a wild and free landscape of infinite possibilities. But beyond this suggestive connection between a possible multiverse and the rich mathematical structures of string theory not much solid results have been achieved. We reviewed some fraction of the mounting evidence that most, if not all of this landscape, is a swampland and we refer to [14,16,149] for similar lines of thought. We believe it makes more sense to listen to what string theory is trying to tell us, then to try to get out of the theory what one would like to have. In recent years, especially with the program of the Swampland [14, 150–152], there is luckily a growing community that embraces this idea. Perhaps this program really already made its first prediction: no measurable tensor modes in the CMB.

From what we have seen so far, we believe that the most sensible attitude is to accept there are no dS vacua at all because string theory conspires against dS vacua.

The suggestion here is basically that effective field theories on a deSitter background are in the Swampland, so can’t be derived from string theory. Since we seem to live in a deSitter space, the obvious conclusion to draw from this is that string theory is falsified: it can’t be the fundamental theory we are looking for. The authors discuss various unconvincing ways to try and avoid this conclusion.

By the way, the authors make the usual flawed argument that “the string theory landscape is just like the Standard Model”:

This kind of criticism is, however, misguided [10, 11]. One might compare with quantum field theory, where there is an infinity of fully consistent theories. Experiments are needed to pick the right one, and parameters must be fitted. When this is done the theory still has enormous predictive power, and no one would claim that the Standard Model is useless. One could argue in a similar way concerning the string landscape.

They also claim:

Paradoxically the critics of string theory and the proponents of the string landscape all agree on one thing: the landscape exists and we more or less know its properties.

At least this string theory critic has never agreed on this. I don’t believe “string theory” is a well-defined enough framework to answer the question of what all its ground states might be, or to properly characterize them. If you accept conjectures about the theory put forward by the landscapeologists, all evidence is now that the set of ground states they identify is so large as to make predictions impossible. The argument against string theory is that there are two possibilities here: either the theory is too poorly understood to tell us what its ground states are, or it does tell us something, and there are too many ground states to make useful predictions. Either possibility leads to the same conclusion that this is a failed idea.

It is rarely acknowledged just how serious the problem of a lack of a definition of “string theory” really is. To get some idea of how bad this problem is, one can consult one of the main references in this paper, a survey of the Landscape and the Swampland, based on Vafa’s 2017 TASI lectures (this paper also discusses the idea that deSitter is not in the Swampland). Claims are often made that AdS/CFT resolves the problem of defining a non-perturbative string theory of quantum gravity, but in the paper one finds:

We can now ask the question if using this AdS/CFT correspondence gives a non-perturbative definition of string theory. The motivation for this is that we can give a non-perturbative definition of SYM theory, for example by lattice regularization, whereas the holographic quantum gravity dual theory in AdS has no complete definition. The fact that the CFT side, i.e. the non-perturbative definition of SYM, gives in principle, a non-perturbative definition of the AdS side, is of course true. But this may be not very useful for deeper questions of quantum gravity. In fact the regime that the gravity side is weakly coupled is big corresponds to when the SYM is strongly coupled. In fact ‘t Hooft was trying to use string theory as a solution to the gauge theory question at strong coupling and not the other way around!…

we find ourself back at the beginning: we want to know fundamentally, what is quantum gravity? It should describe the quantum fluctuations of the metric. From a brief analysis of the standard Einstein-Hilbert action, we see that fluctuations of the metric at the Planck scale should become very violent, leading to potential changes in the topology of the spacetime [103, 104]. This leads naturally to the idea that quantum gravity should be equivalent to summing over all spacetime topologies and geometries:
$$Z_{QG} \sim \sum_{\text{top. and geo.}} e^{-S}$$
In general we have no idea about what description will lead to the correct sum over geometries and topologies. We only do know that there should be some mechanism that washes out the Planck scale fluctuations to produce a smooth space at lower energies. It seems that this description must come from some new fundamental principle, rather than from some duality such as mirror symmetry or AdS/CFT. This lack of knowledge of describing the gravity side quantum mechanically is “the missing corner” in our understanding of string theory.

In both this survey and the new paper, the tactic of trying to remove the Landscape to restore the predictivity of string theory hits up against the obvious problem: you’re left with no theory at all (the equation above, with an undefined sum and an undefined action, is the essence of no theory).

Posted in Swampland | 21 Comments

Raoul Bott: Collected Papers, volume 5

I noticed today that the final volume of Raoul Bott’s collected papers, which Loring Tu has been working on editing for quite a while, has finally appeared. The Springer webpage for the book is here, and its content is available at institutions with a Springer subscription here here. Many of the articles within can be accessed by other means, for example Tu’s article about Bott’s life and work is available here (updated version here) and the collection of articles by others about him is here.

After Bott’s death in 2005 I wrote a blog post here, and won’t repeat what’s there. Much of the material in this last volume is related to mathematics that grew out of Witten’s work on Chern-Simons theory. Unlike earlier volumes, which are devoted mostly to Bott’s important research papers (almost all of which are well-worth reading!), here many of the articles are of an expository nature, written for various lectures that Bott gave. The volume also includes various incidental material, such as speeches written for various events. Reading these gives a good idea of the wonderful human being that he was, it’s great to have them finally collected and available.

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From Princeton to Prison

One of my graduate school classmates today sent around a link to a story about someone many of us remembered, Dragoljub Cetkovic. I somehow missed it last year when it appeared, it’s by Paul Halpern and entitled From Princeton to Prison: The ‘Boy Genius’ Who Was Recruited by John Wheeler and Sentenced by Trump’s Sister. As the story explains, Drago Cetkovic was a precocious young physicist, who John Wheeler brought to Princeton in 1974 (Barry Simon says it was actually Remo Ruffini, Wheeler’s postdoc, who had recruited him). Drago was studying quantum field theory, but wasn’t much interested in doing things like passing exams or working as a Research Assistant (the usual assignment for first-year grad students). This led to trouble and him getting kicked out of the graduate program.

By the time I arrived in Princeton, Drago was a mysterious fixture around the department, often to be found in the library. I recall a couple times asking him about what he was working on, receiving a dense and, to me, utterly incomprehensible explanation. There were rumors that he was somehow being supported on some grant or money from the IAS, but I have no idea how he made ends meet (Simon says he at times was sleeping on John Milnor’s floor). I left Princeton in 1984, and things evidently went downhill for him after that. As explained in the Halpern piece, as well as in this story from the time, in 1987 Drago for some reason left a cyanide-laced tea bag in a Princeton grocery store, and called in some sort of threat. He ended up being sentenced to five years in prison, I don’t know what happened to him after that.

Some of the stories about him refer to his supposedly threatening Barry Simon’s life (Simon was the faculty member in charge of graduate students who had to ask him to leave the program). I had heard this story, and one day when I was talking to Drago, asked him something like:

“Hey, I heard you threatened Barry Simon’s life, and that’s why he left Princeton to go to Caltech. Is that true?”

His answer:

“No, that is not true. And, if it were true, Caltech would not be far enough!”

I do wonder what happened to him, hope that he sooner or later ended up in a new and much better phase of life.

Update: My memory of those days is rather faulty, it quite likely was Nathan Myrhvold who asked him about Barry Simon. Nathan also recalls that he was working on spinor geometry. Maybe now I’d understand what he was trying to explain to me back then.

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Various News

  • Reader Chris W. pointed me to this story about what Cédric Villani, aka the Lady Gaga of French mathematics, has been up to. I see that the report of the “Mission Villani” is now available (in French or English) and it’s front page news at le Monde. There’s also an AI for Humanity website now up, and plans for all sorts of events tomorrow (video here) involving Villani and French president Macron.

    For insight into what this means, you’ll need an AI expert. I’m curious to hear if there’s anything really surprising in the report.

  • Neil Turok and collaborators have a new proposal for how to understand the Big Bang, with the headline version “The universe before the bang and the universe after the bang may be viewed as a universe/anti-universe pair, created from nothing.” There’s a short summary here, a longer paper with details here.

    The papers make various claims of predictions, I’m curious to hear from cosmologists what they think of these. Much of the papers does look like fairly straightforward QFT calculations, which I’ll try to look at more carefully when I find time.

  • The LHC is now in a machine checkout phase, ready for resurrection around Easter Sunday, with the start of beam commissioning for the 2018 run.

Update: A wealth of analyses of physics papers is available at this website and this preprint.

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What is Real?

There’s a new popular book out this week about the interpretation of quantum mechanics, Adam Becker’s What is Real?: The Unfinished Quest for the Meaning of Quantum Physics. Ever since my high school days, the topic of quantum mechanics and what it really means has been a source of deep fascination to me, and usually I’m a sucker for any book such as this one. It’s well-written and contains some stories I had never encountered before in the wealth of other things I’ve read over the years.

Unfortunately though, the author has decided to take a point of view on this topic that I think is quite problematic. To get an idea of the problem, here’s some of the promotional text for the book (yes, I know that this kind of text sometimes is exaggerated for effect):

A mishmash of solipsism and poor reasoning, [the] Copenhagen [interpretation] claims that questions about the fundamental nature of reality are meaningless. Albert Einstein and others were skeptical of Copenhagen when it was first developed. But buoyed by political expediency, personal attacks, and the research priorities of the military industrial complex, the Copenhagen interpretation has enjoyed undue acceptance for nearly a century.

The text then goes to describe Bohm, Everett and Bell as the “quantum rebels” trying to fight the good cause against Copenhagen.

Part of the problem with this good vs. evil story is that, as the book itself explains, it’s not at all clear what the “Copenhagen interpretation” actually is, other than a generic name for the point of view the generation of theorists such as Bohr, Heisenberg, Pauli, Wigner and von Neumann developed as they struggled to reconcile quantum and classical mechanics. They weren’t solipsists with poor reasoning skills, but trying to come to terms with the extremely non-trivial and difficult problem of how the classical physics formalism we use to describe observations emerges out of the more fundamental quantum mechanical formalism. They found a workable set of rules to describe what the theory implied for results of measurements (collapse of the state vBeyond Weirdector with probabilities given by the Born rule), and these rules are in every textbook. That there is a “measurement problem” is something that most everyone was aware of, with Schrodinger’s cat example making it clear. Typically, for the good reason that it’s complicated and they have other topics they need to cover, textbooks don’t go into this in any depth (other than often telling about the cat).

As usual these days, the alternative to Copenhagen being proposed is a simplistic version of Everett’s “Many Worlds”: the answer to the measurement problem is that the multiverse did it. The idea that one would also like the measurement apparatus to be described by quantum mechanics is taken to be a radical and daring insight. The Copenhagen papering over of the measurement problem by “collapse occurs, but we don’t know how” is replaced by “the wavefunction of the universe splits, but we don’t know how”. Becker pretty much ignores the problems with this “explanation”, other than mentioning that one needs to explain the resulting probability measure.

String theory, inflation and the cosmological multiverse are then brought in as supporting Many Worlds (e.g. that probability measure problem is just like the measure problem of multiverse cosmology). There’s the usual straw man argument that those unhappy with the multiverse explanation are just ignorant Popperazzi, unaware of the subtleties of the falsifiability criterion:

Ultimately, arguments against a multiverse purportedly based on falsifiability are really arguments based on ignorance and taste: some physicists are unaware of the history and philosophy of their own field and find multiverse theories unpalatable. But that does not mean that multiverse theories are unscientific.

For a much better version of the same story and much more serious popular treatment of the measurement problem, I recommend a relatively short book that is now over 20 years old, David Lindley’s Where does the Weirdness Go?. Lindley’s explanation of Copenhagen vs. Many Worlds is short and to the point:

The problem with Copenhagen is that it leaves measurement unexplained; how does a measurement select one outcome from many? Everett’s proposal keeps all outcomes alive, but this simply substitutes one problem for another: how does a measurement split apart parallel outcomes that were previously in intimate contact? In neither case is the physical mechanism of measurement accounted for; both employ sleight of hand at the crucial moment.

Lindley ends with a discussion of the importance of the notion of decoherence (pioneered by Dieter Zeh) for understanding how classical behavior emerges from quantum mechanics. For a more recent serious take on the issues involved, I’d recommend reading something by Wojciech Zurek, for instance this article, a version of which was published in Physics Today. Trying to figure out what “interpretation” Zurek subscribes to, I notice that he refers to an “existential interpretation” in some of his papers. I don’t really know what that means. Unlike most discussions of “interpretations”, Zurek seems to be getting at the real physical issues involved, so I think I’ll adopt his (whatever it means) as my chosen “interpretation”.

Update: For another take on much the same subject, out in the UK now is Philip Ball’s Beyond Weird. The US version will be out in the fall, and I think I’ll wait until then to take a look. In the meantime, Natalie Wolchover has a review at Nature.

Update: There’s a new review of What is Real? at Nature.

Update: Jim Holt points out that David Albert has a review of the Becker book in the latest New York Review of Books. I just read a print copy last night, presumably it should appear online soon here [Review now available here].

Update: Some comments from Adam Becker, the author of the book.

I won’t try to rebut everything Peter has said about my book—there are some things we simply disagree about—but I would like to clear up two statements he makes about the book that are possibly misleading:

Peter says that I claim the answer to the measurement problem is that “the multiverse did it.” But I don’t advocate for the many-worlds interpretation in my book. I merely lay it out as one of the reasonable available options for interpreting quantum mechanics (and I discuss some of its flaws as well). I do spend a fair bit of time talking about it, but that’s largely because my book takes a historical approach to the subject, and many-worlds has played a particularly important role in the history of quantum foundations. But there are other interpretations that have played similarly important roles, such as pilot-wave theory, and I spend a lot of time talking about those interpretations too. I am not a partisan of any particular interpretation of quantum mechanics.

Second, I don’t think it’s quite fair to say that I paint Bohr as a villain. I mention several times in my book that Bohr was rather unclear in his writing, and that sussing out his true views is dicey. But what matters more that Bohr’s actual views is what later generations of physicists generally took his views to be, and the way Bohr’s work was uncritically invoked as a response to reasonable questions about the foundations of quantum mechanics. It’s true that this subtlety is lost in the jacket flap copy, but that’s publishing for you.

Also, for what it’s worth, I do like talking about reality as it relates to quantum mechanics. But I suppose that’s hardly surprising, given that I just wrote a book on quantum foundations titled “What Is Real?”. I’d be happy to discuss all of this further over email if anyone is interested (though I’m pretty busy at the moment and it might take me some time to respond).

Posted in Book Reviews, Quantum Mechanics | 74 Comments

This Week’s Hype

Many thanks to Sabine Hossenfelder for her efforts to debunk the attempt to use Hawking’s death as a platform for multiverse hype. See her posting at Backreaction for a good explanation of what is going on here.

To summarize the problem, there are loads of news stories out there telling the public that Stephen Hawking’s ‘breathtaking’ final multiverse theory completed two weeks before he died, Stephen Hawking’s Final Paper Proposes Way to Detect the ‘Multiverse’, etc., etc. Cosmologist Carlos Frenk and theorist Thomas Hertog seem to be among those encouraging this nonsense.

This is all based on this recent paper by Hawking and Hertog, which contains nothing like a way to “detect the ‘Multiverse'”. It’s a toy model of bubble universe formation, one the authors admit they can’t even solve:

However, the setup we have considered does not allow us to describe the transition from the quantum realm of eternal inflation to a universe in the semiclassical gravity domain. This is because our duals are defined in the UV and live at future infinity. It therefore remains an open question whether the conjectured smoothness of global constant density surfaces impacts the eternity of eternal inflation. To answer this will require a significant extension of holographic cosmology to more realistic cosmologies

Their calculations inspire them to state: “… we conjecture that eternal inflation produces universes that are relatively regular on the largest scales”, but this is just an extremely vague conjecture without much backing it. Using it to get press stories published claiming to have found a way to “detect the ‘Multiverse'” is just absurd, and it’s sad to see Hawking’s passing memorialized with a cloud of ridiculous hype.

Update: Another detailed explanation of what is going on here, from Ethan Siegel. His summary:

There are no observable consequences; there is nothing to measure; there is nothing to test. There’s no prediction about the end of the Universe, and there are no robust conclusions we can draw about its beginning. There are tremendous limitations to the implications of this work, and there are few compelling reasons to believe that their toy model has relevance for our physical Universe. It is a seed of an idea that itself is controversial, based off of an also-controversial foundation, and this is a very small step in its development. Furthermore, all of what they do is based on the Hartle-Hawking no-boundary conjecture, which is still not generally accepted as true. The authors go so far as to admit, in the discussion of this paper, that even within their toy model, they have not shown that there is a non-Multiverse-inducing exit to eternal inflation:

“It therefore remains an open question whether the conjectured smoothness of global constant density surfaces impacts the eternity of eternal inflation.”

Posted in This Week's Hype | 7 Comments

Abel Prize to Langlands

The 2018 Abel Prize has been awarded to Robert Langlands, an excellent choice. The so-called “Langlands program” has been a huge influence on modern mathematics, providing deep insight into the structure of number theory while linking together disparate fields of mathematics, as well as quantum field theories and physics.

The Abel Prize site provides a wealth of information about Langlands and his work. Davide Castelvecchi at Nature appropriately describes the Langlands program as a “grand unified theory of mathematics” (Edward Frenkel’s Love and Math popularized this description).

Many blog posts here have discussed the Langlands program and ideas that have developed out of it. For a good example of how wide the impact of these ideas has been, this week the Perimeter Institute will be hosting a conference discussing the latest work on the geometric version of the Langlands program, as well as connections to gauge theory and conformal field theory.

For the original work of Langlands himself, besides the material at the Abel site, the AMS Bulletin has recently published a long article by Julia Mueller. For the original sources and a wealth of other material written by Langlands himself, see the IAS site that collects his writings.

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Stephen Hawking 1942-2018

Front-page on every news source today is the sad report that Stephen Hawking died yesterday at the age of 76. For the best description of his scientific accomplishments, I recommend the obituary in the Guardian written by his sometime collaborator Roger Penrose.

I was going to write a little bit about one time I heard Hawking speak (or rather, his student interpret for us his speech), which was at the IAS back in the early 1980s. I just noticed though that evidently John Baez was at the same talk (he was an undergrad, I was a grad student), and describes it well here.

At the time I remember that many thought that quantum gravity would be understood within a few years, and that Hawking would not be able to live longer than another year or two, given the nature of the disease he was suffering from. It’s wonderful that the second of these turned out to be so wrong.

While Hawking was already a star in the physics community back then, his celebrity with the wider public came later. Of all the scientists who over the years have achieved some degree of celebrity, I can’t think of another one who so much both deserved and enjoyed the public attention.

Update: There are dozens of articles appearing discussing Hawking’s life and work. One you may not have seen which I enjoyed is from Nathan Myhrvold.

Update: Another piece by someone who worked with Hawking, Marika Taylor. It includes some discussion of his views on M-theory.

Update: Hawking has inspired some new theorizing from Niall Ferguson.

Posted in Obituaries | 13 Comments

Recent Developments in Constructive Field Theory

This week there’s a mini-workshop here at Columbia organized by the probabilists, on Recent Developments in Constructive Field Theory. I’ll be attending some of the talks, will write more here if I can come up with something constructive to say.

Update: One of the talks today was Sourav Chatterjee on Yang-Mills for Probabilists, explaining what Yang-Mills and lattice gauge theory are, along the lines of this preprint. The discussion brought back memories of my grad school days and early research, but I fear didn’t give much cause for optimism that there will be much progress anytime soon on a rigorous understanding of 4d Yang-Mills theory.

Update: Today Scott Sheffield gave a talk on Gauge theory and the three barriers. He started with the initial motivation of finding a continuum theory of random surfaces giving an alternative representation of gauge theory in 4d (this goes back to Polyakov and others around 1980). There’s not much progress on this, but the question has inspired a wealth of results involving in some sense random surfaces. For some details of all of this, one place to look is the website of Sheffield’s recent graduate course.

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