## Notes on Current Affairs

Blogging has been light recently, partly due to quite a bit of traveling. This included a brief trip the week before last to Los Angeles, where I met up with, among others, Sabine Hossenfelder. This past week I was in Washington DC for a few days, and gave a talk at the US Naval Observatory, one rather similar to my talk earlier this year in Rochester. In addition, I’ve been trying to spend my time on more fruitful activities, especially a long-standing unfinished project to make sense of the relationship between BRST and Dirac cohomology. Optimistically, I’ll have a finished (or at least finished enough to make public) version of something later in January, after taking a couple weeks at the beginning of the month for a vacation in France.

I’m completely in agreement with Sabine about the sad state of high energy particle theory, and glad to see that she has been forcefully trying to get people to acknowledge the problem. I don’t agree though with her “Lost in Math” characterization of the problem, and my talks in DC and Rochester tried to make the case that what is needed is more interaction with mathematics, not less.

Various things that might be of interest that have to do with the state of high energy physics theory are the following:

• A Y Combinator blog interview with Lenny Susskind. I think it’s fair to say that Susskind now admits that string theory, as currently understood, cannot explain the Standard Model, and that as a result he has given up on trying to make any progress on particle theory. He says:

My guess is, the theory of the real world may have things to do with string theory but it’s not string theory in it’s formal, rigorous, mathematical sense. We know that the formal, by formal I mean mathematically, rigorous structure that string theory became. It became a mathematical structure of great rigor and consistency that it, in itself, as it is, cannot describe the real world of particles. It has to be modified, it has to be generalized, it has to be put in a slightly bigger context. The exact thing, which I call string theory, which is this mathematical structure, is not going to be able to, by itself, describe particles…

We made great progress in understanding elementary particles for a long time, and it was always progressed, though, in hand-in-hand with experimental developments, big accelerators and so forth. We seem to have run out of new experimental data, even though there was a big experimental project, the LHC at CERN, whatever that is? A great big machine that produces particles and collides them. I don’t want to use the word disappointingly, well, I will anyway, disappointingly, it simply didn’t give any new information. Particle physics has run into, what I suspect is a temporary brick wall, it’s been, basically since the early 1980s, that it hasn’t changed. I don’t see at the present time, for me, much profit in pursuing it.

• Susskind instead spends his time on highly speculative ideas relating geometry and quantum theory, with the idea that while this has no connection to particle physics, it might somehow lead to progress in understanding quantum gravity. Natalie Wolchover at Quanta has a new story about Susskind’s latest speculations.
• This past week the Simons Foundation-funded “It from Qubit” collaboration has been having a two-part conference. This started at the IAS, talks available here, then moved on to the Simons Foundation headquarters in NYC (see here). Videos of the IAS part are available, for the NYC part, there’s Twitter. George Musser reports that Juan Maldacena has figured out how to construct (in principle) traversable wormholes, and that he’s arguing that “quantum computers are so powerful that they create spacetime”. For a tweet showing a summary of what has been achieved, see here.

Personally I’ve always been dubious that we’ll ever have a useful “quantum theory of gravity” unless we have some sort of unification with the standard model, which would provide a connection to things we can understand and measure. Lacking such a connection, another way to go would be to try and evaluate a “quantum theory of gravity” proposal based on its mathematical consistency, coherence and beauty. My problem with the “It from Qubit” program is that, ignoring the way it gives up on connecting to what we understand, I’ve never seen anything coming out of it that looks like an actual well-defined theory of quantum gravity that one could evaluate as a mathematical model consistent with quantum mechanics and what we know about 3+1d general relativity.

• For something about quantum computation and its relation to fundamental physics that I can understand, John Preskill has a wonderful article on Simulating quantum field theory with a quantum computer. Nothing there I can see about quantum gravity.
• Given that its founder Susskind and the other leading figures of the field at the IAS have pretty much given up on the project of relating string theory to particle physics, an interesting question is that of why the pathology of so many researchers still working on this failed project? The source of this pathology and the question of what can be done about it I think are at the center of Sabine Hossenfelder’s book and recent blogging. An important question that is getting raised here is that of the damage this situation is doing to the credibility of science. If you want to fight the good fight against those who, because it threatens their tribe, want to deny the facts of climate science, is it helpful if many of the best and brightest in science are denying facts that threaten their tribe? Scientific American has a story about Why Smart People Are Vulnerable to Putting Tribe Before Truth, but it doesn’t make clear the depth of the problem (i.e. that some of the smartest scientists around are doing this).
• For an example of the problem, see an interview with Gabriele Veneziano about the history and current state of string theory. He’s in denial of the obvious fact that string theory makes no predictions:

People say that string theory doesn’t make predictions, but that’s simply not true. It predicts the dimensionality of space, which is the only theory so far to do so, and it also predicts, at tree level (the lowest level of approximation for a quantum-relativistic theory), a whole lot of massless scalars that threaten the equivalence principle (the universality of free-fall), which is by now very well tested. If we could trust this tree-level prediction, string theory would be already falsified. But the same would be true of QCD, since at tree level it implies the existence of free quarks. In other words: the new string theory, just like the old one, can be falsified by large-distance experiments provided we can trust the level of approximation at which it is solved. On the other hand, in order to test string theory at short distance, the best way is through cosmology. Around (i.e. at, before, or soon after) the Big Bang, string theory may have left its imprint on the early universe and its subsequent expansion can bring those to macroscopic scales today.

This take on how you evaluate a theory by comparing it to experiment is not one that will give the average person much understanding of the scientific method or much confidence in scientists and their devotion to it.

Posted in Uncategorized | 12 Comments

## The End of (one type of) Physics, and the Rise of the Machines

Way back in 1996 science writer John Horgan published The End of Science, in which he made the argument that various fields of science were running up against obstacles to any further progress of the magnitude they had previously experienced. One can argue about other fields (please don’t do it here…), but for the field of theoretical high energy physics, Horgan had a good case then, one that has become stronger and stronger as time goes on.

A question that I always wondered about was that of what things would look like once the subject reached the endpoint where progress had stopped more or less completely. In the book, Horgan predicted:

A few diehards dedicated to truth rather than practicality will practice physics in a nonempirical, ironic mode, plumbing the magical realm of superstrings and other esoterica and fret­ting about the meaning of quantum mechanics. The conferences of these ironic physicists, whose disputes cannot be experimentally resolved, will become more and more like those of that bastion of literary criticism, the Modern Language Association.

This is now looking rather prescient. For some other very recent indications of what this endpoint looks like, there’s the following:

• In today’s New York Times, in celebration of forty years of the Science Times section, Dennis Overbye has a piece reporting that Physicists are no longer unified in the search for a unified theory. His main example is the recent Quanta article by the IAS director that got headlined There Are No Laws of Physics. There’s Only the Landscape. The latest from Dijkgraaf is that string theory is probably the answer, but we don’t know what string theory is:

Probably there is some fundamental principle, he said, perhaps whatever it is that lies behind string theory.
But nobody, not even the founders of string theory, can say what that might be.

• Overbye also quotes Sabine Hossenfelder, who is now taking on the thankless role of the field’s Jeremiah. Her latest blog posting, The present phase of stagnation in the foundations of physics is not normal, is a cry of all too justifiable frustration at the sad state of the subject and the refusal by many to acknowledge what has happened. Well worth paying attention to are comments from Peter Shor here and here.

Another frightening vision of the future of this field that has recently struck me as all too plausible has turned up appended to a piece entitled The Twilight of Science’s High Priests, by John Horgan at Scientific American. This is a modified version of a review of books by Hawking and Rees that Horgan wrote for the Wall Street Journal, and it attracted a response from Martin Rees, who has this to say about string theory:

On string theory, etc., I’ve been wondering about the possibility that an AI may actually be able to ‘learn’ a particular model and calculate its consequences even of this was too hard for any human mathematician. If it came up with numbers for the physical constants that agreed (or that disagreed) with the real world, would we then be happy to accept its verdict on the theory? I think the answer is probably ‘yes’ — but it’s not as clear-cut as in the case of (say) the 4-colour theorem — in that latter case the program used is transparent, whereas in the case of AI (even existing cases like Alpha Go Zero) tor programmer doesn’t understand what the computer does.

This is based on the misconception about string theory that the problem with it is that “the calculations are too hard”. The truth of the matter is that there is no actual theory, no known equations to solve, no real calculation to do. But, with the heavy blanket of hype surrounding machine learning these days, that doesn’t really matter, one can go ahead and set the machines to work. This is becoming an increasingly large industry, see for instance promotional pieces here and here, papers here, here, here and here, and another workshop coming up soon.

For an idea of where this may be going, see Towards an AI Physicist for Unsupervised Learning, by Wu and Tegmark, together with articles about this here and here.

Taking all these developments together, it starts to become clear what the future of this field may look like, and it’s something even Horgan couldn’t have imagined. As the machines supersede human’s ability to do the kind of thing theorists have been doing for the last twenty years, they will take over this activity, which they can do much better and faster. Biological theorists will be put out to pasture, with the machines taking over, performing ever more complex, elaborate and meaningless calculations, for ever and ever.

Update: John Horgan points out to me that he had thought of this, with a chapter at the end of his book, “Scientific Theology, or the End of Machine Science” which discusses the possibility of machines taking over science.

Posted in Multiverse Mania | 43 Comments

## Langlands/Frenkel and Some Other Things

The Canadian publication The Walrus today has a wonderful article about Robert Langlands, focusing on his attitude towards the geometric Langlands program and its talented proponent Edward Frenkel. I watched Frenkel’s talk at the ongoing Minnesota conference via streaming video (hopefully the video will be posted soon), and it was an amazing performance on multiple levels. A large part of it was a beautiful explanation of the history and basic conception of what has come to be known as geometric Langlands. He then went on to explain carefully some of the ideas in the recent Russian paper by Langlands, basically saying that they worked in the Abelian case, but could not work in the non-Abelian case. He ended by describing some alternate ideas that he is working on with David Kazhdan. Langlands was in the audience and at the end of the talk rose to comment extensively, but I couldn’t hear his side of this since he had no microphone (that Frenkel was sticking to his guns though was clear).

Besides giving the talk, Frenkel has made available a manuscript which gives a much more detailed version of the talk. See section 3.5 for an explanation of what he sees as the fundamental problem with what Langlands is trying to do: even in the simpler case of G/B over the complex field, you can’t successfully define a Hecke algebra in the way that Langlands wants.

The conference is finishing up right now, with final remarks by Langlands coming up later this afternoon.

A few more items, mostly involving my Columbia math department colleagues:

• If you connect quickly to the streaming video from Minnesota, you may be able to catch Michael Harris’s talk on local Langlands.
• Quanta magazine has an article about a recent proof of an old conjecture by Dorian Goldfeld about ranks of elliptic curves. This is due to Alexander Smith, now a third fourth year graduate student at Harvard (he started working on this while an undergrad at Princeton, with Shouwu Zhang). His twin brother Geoffrey is also a math grad student at Harvard.
• Andrei Okounkov has been giving some talks recently at various places about developments in geometric representation theory with some connection to physics, under the title New worlds for Lie Theory. The slides from the ICM version of the talk are here.
• For those more interested in physics than mathematics the new issue of Inference has some articles you might enjoy. In particular, Sheldon Glashow is no fan (neither is Chris Fuchs) of the book I reviewed here

Update: Michael Harris is appearing via Skype from his home near here, since transportation out of NYC yesterday was mostly shut down (very early season unprecedented snowstorm, during rush hour…).

Update: I’m listening to the closing talk by Langlands. He is explaining his version of geometric Langlands, responds to criticism from Frenkel with “As far as I know there are no errors in the paper, no matter what you may see elsewhere”. He ends his talk with something like “At the last page I threw down my pen… It works and it works by a miracle. Don’t doubt it, it does work!”

Update: Another livestream, starting in moments: Alice and Bob Meet the Wall of Fire, a panel discussion with Quanta writers at the Simons Foundation.

Update: Videos from the Langlands Abel conference are now available, in particular Frenkel here and Langlands here.

Update: For another expository piece about the Langlands program, one that I somehow missed when it came out recently, see Sol Friedberg’s What is the Langlands Program? in the AMS Notices.

Posted in Langlands, Uncategorized | 14 Comments

Based on this preprint from Banks and Fischler, I added an update to the FAQ entry about why the ever-popular “string theory makes predictions, but only at high energies where they can’t be tested” argument is not true.

This preprint also updates the acknowledgments story discussed here, with the current version:

The work of T.Banks is NOT supported by the Department of Energy, the National Science Foundation, the Simons or Templeton Foundations or FQXi. The work of W.Fischler is supported by the National Science Foundation under Grant Number PHY-1620610.

## Langlands News

Various Langlands program related news, starting with the man himself:

• For the latest from Langlands about the geometric theory, best if you read both Russian and Turkish. In that case you can read this and this. For the rest of us, all we get are this commentary on the Russian and Turkish documents and these last or very well last thoughts on them.
• In a couple of weeks there will be a conference celebrating the work of Langlands, organized in conjunction with his Abel Prize. Perhaps there will be live stream here.
• I hear that at his lecture at the CMI at 20 conference Scholze made some new conjectures about possible ways of getting the Langlands correspondence in certain cases of the number field case. I haven’t however seen anywhere that one can read or hear more about these. It would be great if the Clay Mathematics Institute could make available videos of the talks at that conference.
• Scholze will be giving the Chow lectures in Leipzig next week. The program there includes some preparatory talks by others, including my ex-Columbia colleague Daniel Litt (now at the IAS). I see that Daniel has at least posted a problem set you can get started on.
• Also coming up next week is the Breakthrough Prize Symposium at Berkeley, where Vincent Lafforgue will talk about his (valued at $3 million) work on the Langlands program Monday morning (live stream here). On the physics side, in the evening a group of prize-winning theorists will talk about “Is Time Travel Possible”, live stream here. • A central idea conjecturally relating the geometric version of local Langlands to the number field version is the Fargues-Fontaine curve, which Jacob Lurie has been giving a course about at Harvard UCSD this fall. This fall in Bangalore there will be a meeting devoted to the Fargues-Fontaine curve, about which the organizers tell us: “This field will unravel in the coming years…” • On the local geometric Langlands front, there’s something new from Dennis Gaitsgory. I’ve always been fascinated by the way BRST appears in this story. • I’m told by experts that one of the best recent results in the Langlands program is this work, which doesn’t seem to have yet made it to the arXiv, but was explained in some detail in a blog post last year by Frank Calegari. Update: Slides from the Chow Lectures are becoming available, see here. Remarkable in particular is Peter Scholze’s wonderful introductory lecture on Numbers and geometry, which includes something one sees all too rarely, a set of drawings showing the sort of pictures arithmetic geometers have in their minds for how to think about number theory geometrically. Update: I just watched Vincent Lafforgue’s talk at the Breakthrough symposium. It included basically thanks to the CNRS for providing him a permanent position with freedom, a survey of Langlands, mainly talking about the topology of algebraic varieties, and comments on the ecological crisis. He says he’ll put up the slides on his website http://vlafforg.perso.math.cnrs.fr/ He made one (to me) very striking claim, that the functoriality conjecture could be thought of as a quantization problem, how to pass from a classical system to a quantum system. Can an expert enlighten me on what exactly he was referring to here? Update: Lafforgue’s slides are here. James Milne has provided a Google-translated version of the Langlands Russian article here, with the comment: This may help readers gain some idea of what the manuscript is about until there is an official translation. Given that even native Russian speakers (not just google) have trouble understanding Langlands’s Russian, this would best be done by the author. Update: Edward Frenkel gave a talk at the Langlands Abel Prize conference discussing the geometric theory and a bit about recent ideas of Langlands on this topic. He has written up some detailed notes on his take on this, available here. Update: Videos of the CMI-20 talks are available, with Scholze’s here. Posted in Langlands | 11 Comments ## The End of LHC Run 2 and the Road Ahead Some experimental HEP news items: • Since 2015 the LHC experiments have been taking data from proton-proton collisions at 13 TeV. This is “Run 2” of the LHC, “Run 1” was at the lower energy of 8 TeV. The proton-proton Run 2 ended this morning, with the LHC shifting to other tasks, first machine development, later heavy ions. It will shut down completely in December for the start of “Long Shutdown 2 (LS2)”, which will last for over two years, into early 2021. During LS2 there will be maintenance performed and improvements made, including bringing the collision energy of the machine up to the design energy of 14 TeV. ATLAS is reporting 158 inverse fb of collisions delivered by the machine during Run 2, of which 149 inverse fb were recorded, the CMS numbers should be similar. Most data analysis reported to date by ATLAS and CMS has only used the 2015 and 2016 data (about 36 inverse fb) although a few results have included data through 2017 (about 80 inverse fb). My impression is that for many searches they have been waiting for the full run 2 dataset to be available. Perhaps results of searches with the full dataset might start becoming available by the time of summer 2019 conferences. The LHC run 3 is planned for 2021-2023, producing perhaps 300 inverse fb of data, results perhaps available in 2024. It will thus be quite a long time after run 2 results start appearing before better ones due simply to more data become available. • The Europeans are now starting a process that will lead to an update of the European Strategy for Particle Physics. Tommaso Dorigo has a blog post here, and there’s a website here. A first stage of this process will ask for community input, with deadline December 18, via a portal that will open November 1. The next stage will be an Open Symposium to be held May 13-16 in Granada. • This week there’s a Workshop on Future Linear Colliders being held in Austin Texas. The big question being discussed there is whether the Japanese will decide to go ahead with a plan to build the ILC, a 250 GeV linear electron-positron machine. The current situation is described in detail here, with the crucial next step a decision from the Science Council of Japan expected by the end of November. If the ILC project does go forward, a tentative schedule has construction beginning in 2026 and commissioning in 2034. • For a theorist’s recent take on future colliders, see this from LianTao Wang. One thing Wang reports is an “excuse to have fun” (since it’s based on an unrealistic assumption), a community study in particle theory being organized by Michael Peskin, which would address the question “What would we learn from an electron accelerator of energy 10-50 TeV?” Posted in Experimental HEP News | 34 Comments ## Last Night’s Hype If you’re a Friend of the IAS ($1750/year and up), you were invited to a talk last night, at which IAS member Thomas Rudelius promised to explain to you How to Test String Theory. The video of the talk is now available here.

After a long introduction involving large amounts of misleading hype, Rudelius in the last couple minutes finally gets to the promised explanation of “How to Test String Theory”. What is it? It’s his discovery that some versions of axion cosmology are incompatible with the Weak Gravity Conjecture, and thus conjecturally incompatible with string theory.

I assume that the IAS Friends in attendance, besides being financially well off, are also not so dim-witted that they wouldn’t notice that they’d been had (there’s no evidence for axion cosmology, so conjectures about whether or not various axion cosmology models are consistent or not with string theory are completely irrelevant to “testing string theory”). Any questions asked after the talk didn’t make it to the video, so it’s unclear if anyone bothered to complain about what had just been done to them.

Update: For a sensible, informative video about string theory (as opposed to the IAS one), see this from Sabine Hossenfelder.

Posted in Swampland, This Week's Hype | 3 Comments

## Breaking News

Two midday breaking news items:

• The ACME II experiment is reporting today a new, nearly order of magnitude better, limit on the electric dipole moment of the electron:
$$|d_e|\leq 1.1 \times 10^{-29} e\ cm$$
The previous best bound was from ACME I in 2014:
$$|d_e|\leq 9.4 \times 10^{-29} e\ cm$$

One significance of this is that while the SM prediction for the electron EDM is unobservably small, generically extensions of the SM predict much larger values. Already the 2014 bound was in conflict with typical SUSY models with LHC-scale supersymmetry, and was starting to rule out parts of the ranges expected for split-SUSY models (Arkani-Hamed’s current “best bet”) as well as the expected range for SO(10) GUTs (see for instance slide 25 here).

Today’s result pretty much completely rules out generic versions for both the most popular SUSY models still standing (Split SUSY), as well as the most popular class of GUTs. This provides another nail in the coffin of the SUSY-GUT paradigm which has dominated expectations for physics beyond the SM over the past forty years.

• The Breakthrough Prize people are having their usual sort of ceremony for the 2019 prizes on November 4, with an Oscars-like production, this year hosted by Pierce Brosnan. In a break with the past, this year they’re announcing the winners in advance, see here. The $3 million physics prize goes to Kane and Mele for their work on topological insulators. The$3 million mathematics prize goes to Vincent Lafforgue, for his work on the Langlands correspondence. The prize description has some information about him I was unaware of:

Deeply concerned about the ecological crisis, Lafforgue is now focused on operator algebras in quantum mechanics and devising new materials for clean energy technologies.

Update: The promotional videos for the Breakthrough Prize winners that will be shown at the November ceremony are already available on Youtube.

Update: Those phenomenologists work fast! A detailed study of the implications of the ACME result for SUSY models is on the arXiv tonight. For a precise version of the crude claim that “generic split SUSY is now ruled out”, look at the top two plots in figure 4.

Posted in Uncategorized | 9 Comments

## This Week’s Hype

The story of string theory as a theory of everything has settled into a rather bizarre steady-state, with these three recent links providing a look at where we are now:

• At his podcast site, Sean Carroll has an interview with string theorist Clifford Johnson. It’s accurately entitled What’s So Great About Superstring Theory, since it’s an hour of unrelenting propaganda about the glories of string theory, save for a short mention that there had been some criticism from (unnamed) sources a decade or so ago.

The truly odd thing about the discussion though was the way it seemed frozen in time back in 1998 just after the advent of AdS/CFT duality, with almost no discussion of developments of the last twenty years. Nothing about the string theory landscape and the controversy over it, nothing about the negative SUSY results from the LHC. The attitude of Carroll and Johnson towards the failure of string theory unification seems to be to simply refuse to talk about it, and try to keep alive the glory days just after the publication of The Elegant Universe. They’ve taken to heart the post-fact environment we now live in, one where if you keep insisting something is true (string theory unification is a great idea) despite all evidence, then for all practical purposes it is true. Johnson has famously admitted that he refuses to read my book or Lee Smolin’s. As far as he’s concerned our arguments do not exist, and Carroll goes along with this by not even mentioning them.

• For the latest on the Swampland (for background, see here), there’s String Theorists’ Heads Bobble Over Potential Dark Energy Wobble, where we’re told that string theorists are claiming “huge excitement” over the possibility that string theory might make a “prediction” about dark energy. Over the years there have been endless claims about “predictions” of string theory, none of which have ever turned out to actually exist, and this is just one more in that long line. The rather odd aspect of this latest prediction is indicated by how it is described in the last paragraph of the article:

The real excitement comes from how soon we might know whether Vafa’s work has produced a testable prediction of string theory—which would be a first. Experiments like the Dark Energy Survey or the upcoming WFIRST telescope could possibly detect whether dark energy is constant or changing over time, and could perhaps do so within the next few years.

Reading this, one gets the impression that we’ll know what string theory “predicts” about dark energy just when there’s a measurement. This actually does describe what’s going on here: for some, string theory is a theory of everything as a matter of faith, so to them any new measurement tells us more about string theory, in particular that string theory “predicts” that measurement.

• Finally, there’s an article out by Thomas Hertog, which contains more about his work with Hawking that was widely advertised after Hawking’s death (see here). Hertog claims another sort of “prediction” of string theory:

String theory predicts that our universe is fundamentally a hologram that reveals itself only in the most extreme conditions, such as those at the Big Bang.

For the implications of this prediction, see String Theory Summarized.

Posted in Swampland, This Week's Hype | 14 Comments

## High Life

I spent yesterday night at the New York Film Festival, watching Claire Denis’s new film High Life. For a detailed and accurate review of the film, see the one at Variety.

This film is about a voyage to a black hole, in some sense an anti-Interstellar. Where the scientific plot of Interstellar was inspirational and made no sense at all, in High Life you get a plot that is all too plausible, and completely depressing. There’s a spaceship headed on a mission to a black hole, but this one doesn’t have brilliant scientists, traveling in a clean and shiny environment, and out to save the world. Instead, the crew is a bunch of ex-Death Row inmates, stuck on a dead-end trip in a filthy spacecraft swarming with recycled excrement, being subjected to grotesque sexual experiments, with periodic violent assaults, murders, and screaming babies to liven things up.

The supposed mission of the spacecraft is to travel to a nearby black hole and test whether energy can be extracted by the Penrose process. Because of all the murdering and such, that doesn’t work out too well. The ending involves another trip into a black hole, with discussion of whether they’re going to hit a “firewall”. One character thinks not, but that sure looks like one to me at the end. Theorist Aurélien Barrau is listed as “Cosmic Companion” or some such, and must have been responsible for providing the higher level of scientific verisimilitude than that of Interstellar (one of the images of a black hole does look like the famous one Kip Thorne provided for the earlier film).

I can’t really recommend this film to the average viewer seeking enlightenment or entertainment. On the other hand, if you’re looking for something unrelievedly grim, grotesque and disturbing, and really like black holes, maybe you should check it out.

Posted in Uncategorized | 15 Comments