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Poincare Conjecture Proof Completed 222

Posted by samzenpus
from the show-your-work dept.
Flamerule writes "A New York Times article has finally provided an update on the status of Grigori Perelman's 2003 rough proof of the Poincaré Conjecture. 3 years ago, Perelman published several papers online explaining his idea for proving the conjecture, but after giving lectures at MIT and several other schools (covered on Slashdot) he returned to Russia, where he's remained silent since. Now, mathematicians in the US and elsewhere have finally finished going over his work and have produced several papers, totaling 1000 pages, that give step-by-step, complete proofs of the conjecture. In addition to winning some or all of the $1,000,000 Millennium Prize, Perelman now seems to be the favorite to receive a Fields Medal at the International Mathematics Union meeting next week, but it's not clear that he'll even show up!"
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Poincare Conjecture Proof Completed

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  • Goddamn I love freaky misfit mathematical geniuses. They're even better than their nerdier cousins, the chess geniuses. The ones from Central/Eastern Europe and South Asia always seem to be the most fun.
  • by tonyr1988 (962108) on Wednesday August 16, 2006 @12:48AM (#15916787)
    Now, mathematicians in the US and elsewhere have finally finished going over his work and have produced several papers, totaling 1000 pages
    Someone's going to have to post a printer-friendly on that one.
    • Simple - just wave your hands and blather on for a page or so about how obvious the proof is... and in the footnotes reference the 1000 page version.

      Trust me, 99.9999% of the folks will never follow the link if your short blather is at all close to an accurite summary.

    • Re:Too Many Pages (Score:3, Interesting)

      by G3ckoG33k (647276)
      I will wait for the reader-friendly version. Reader's Digest, Simon Singh, Mario Livio where are you all?
  • by Vellmont (569020) on Wednesday August 16, 2006 @12:52AM (#15916799)
    What kind of strange rabbits have these topologists seen? The rabbits I've seen have a hole from end to end through them called the digestive tract.
  • by davidwr (791652) on Wednesday August 16, 2006 @12:53AM (#15916803) Homepage Journal
    $1,000,000, 1,000 pages, those numbers are apprpriately round for the occasion.
  • If I were known for proving Poincare Conjecture, I wouldn't give a damn to be known as a Fields medal winner. (They'll give it to him anyway, whether he's there personally to receive it.)
  • by Anonymous Coward
    The chinese press distorted the news:
    http://news.xinhuanet.com/english/2006-06/04/conte nt_4644754.htm [xinhuanet.com]
  • I remember that is was important to string theory, I just don't remember why. I did a search and found nothing. Can anyone elaborate?
    • Google is still in business: see here [umich.edu] for example.
    • by S3D (745318) on Wednesday August 16, 2006 @01:42AM (#15916971)
      Google your friend. ANAM (I'm not a matematician), but I'll try.
      According to string physicist Lubos Motl [blogspot.com] the proof indeed important to string theory. The proof based on the flow on the manifold (surface), analogous to heat dissipation - Ricci flow [wikipedia.org]. This flow deform metrics (distance between points of the surface). But this process also describe renormalization [wikipedia.org] of worldsheet - how the physics of the worldsheet [wikipedia.org] (surface which string drawing, moving in space and time) change with changing of the observation scale. That is how phisics of string change then the scale of calculation changed.
      • ... ANAM (I'm not a matematician) ...

        IANAA (I am not an acronymist)

      • by althai (992172) on Wednesday August 16, 2006 @03:39AM (#15917269)
        I'm not a geometer, but here is my understanding of the proof:

        The Ricci Flow [wikipedia.org] was defined by Richard Hamilton in 1981 as a step towards classifying topological compact 3-manifolds. Classifying 3-manifolds would certainly decide The Poincare Conjecture, as it states that all simply connected compact 3-manifolds are homeomorphic to the sphere. This is an important special case: most proofs of the classification of compact 2-manifolds start out by proving the an analogous statement for the 2-sphere. The Ricci Flow is a differential equation which defines how the shape of a manifold changes in time: given an arbitrary manifold M(0), you can apply the differential equation to it to get manifolds M(t) for (some) positive t, which gradually change shape. However, the Ricci Flow is not volume preserving, so you "renormalize" so that M(t) has constant volume.

        The Ricci Flow has the useful property that it tends to make manifolds smoother and smoother. For example, if you started out with a lumpy ball, you would eventually get a smooth ball. It was hoped that it could be proved that if the initial manifold was a compact simply connected 3-manifold, then as t increased, the manifold would tend towards a 3-sphere. Unfortunately, while locally solutions to differential equations always exist, they don't necessarily exist for all time, and for some starting manifolds, eventually you would get to a road-block: a t for which M(t) could not be defined. What Perlman (hopefully) showed was that all road-blocks were of certain types, and that a surgery could be formed that would modify the manifold but not it's topological nature, and then you could again apply the Ricci Flow, until the manifold became a sphere.

        Note that this method is useful beyond proving the Poincare Conjecture, as it (again, hopefully) describes all road blocks to extending the Ricci Flow, so that the same tools can be applied to any 3-manifold, and not just simply connected ones. In this manner, assuming Perlman made no mistakes (or that any mistakes can be corrected), it is possible to apply the same arguments to prove the Geometrization Conjecture of Thurston, which classifies 3-manifolds.
        • by polv0 (596583) on Wednesday August 16, 2006 @11:57AM (#15920089)
          I'm trying to glean what some of the practical implications could be of this discovery.

          It seems to me at this Ricci Flow differential equation could be quite useful practically. For example, in pattern recognition, if a computer could build a 3d model of an object using multiple vantage points, then simplify the object to one of the handfull of object types described by Perlman using the Ricci Flow, then this simple catagorization might help in the identification of complex objects (e.g. a donut really is a donut, even if it's been heavily frosted).

          Do you know if Perlman's technique for handling the singularities will help with the numerical implementation of this process? Or are these issues numerically simple to solve - but only challenging to solve in proof?
  • by blueZ3 (744446) on Wednesday August 16, 2006 @01:24AM (#15916913) Homepage
    The incredulity that this mathematician might have been more interested in the challenge of the work than fame and fortune in the Western world practically oozes from each sentence.

    I'm all for capitalism and the idea of "prizes" to encourage research, but have we really become so jaded that it's a complete shock when someone does something worthwhile merely for its own sake? Perhaps he's gone on to other challenges, or he's wrapped up in some research that has his complete attention. Heck, perhaps he just enjoys math for its own sake and doesn't want to deal with all the side-effects of notoriety.
    • Are you sure it's not just delight rather than incredulousness? Tone is rather hard to pick out with just text so you're assuming a lot in your conclusion...
    • by smallpaul (65919) <paul@@@prescod...net> on Wednesday August 16, 2006 @02:08AM (#15917042)

      I'm all for capitalism and the idea of "prizes" to encourage research, but have we really become so jaded that it's a complete shock when someone does something worthwhile merely for its own sake?

      It isn't a shock that he did it for its own sake at all. Look at the thousands of open source programmers. The shock is that he's been given a million dollars and seem uninterested. Linus Torvalds does Linux for its own sake but if someone gave him a million dollars, he'd take it. Even someone who is not materialistic might think: "hmmm. A million dollars might help many Russian orphans or deliver AIDS drugs to Africans or ..." It is strange for a single person to be neither greedy, nor ambitious nor altruistic ... merely obsessed.

      Yes, that's strange. It's rare and therefore strange.

    • by aiken_d (127097) <brooks@tangenBLUEtry.com minus berry> on Wednesday August 16, 2006 @02:09AM (#15917044) Homepage
      Oddly enough, people tend to form their expectations based on past experiences. Is it so unreasonable for the tone of the article to be incredulous when the situation is unprecedented?

      Where you see value judgments and a jaded reporter, I see a pretty reasonable surprise. I don't see anything in the article where the reporter suggests that Perelman "should" do anything other than what he is. Surprise, and remarking on an unusual behavior, is *not* approbation.

      -b
    • I'm all for capitalism and the idea of "prizes" to encourage research, but have we really become so jaded that it's a complete shock when someone does something worthwhile merely for its own sake?

      Lots of people do things for their own sake (as long as they can pay their bills and get some food). But when someone got a prize of a million dollar as a bonus (for what you enjoyed doing anyway), can you really imagine someone turning this down? Well, Perelman hasn't done this (yet), but lots of people could im

    • by eddy (18759) on Wednesday August 16, 2006 @02:14AM (#15917061) Homepage Journal
      I think that never is this more amply examplified than when the people who manage 'rights holders' "explain" how, if it weren't for copyright, there would exist no art.
    • Maybe he... (Score:2, Funny)

      by rolandog (834340)
      just found a girlfriend? //I keed.
    • by Thisfox (994296) on Wednesday August 16, 2006 @02:23AM (#15917087)
      Sadly, yes, doing something for it's own sake rather than for monetary gain is frowned apon, and sometimes viewed with fear and confusion, not that I'm saying this review goes THAT far (if you don't believe me, try smiling at someone while in a subway one of these days: the person will generally check that you haven't got someone stealing their wallet while they are distracted. Or busk without a hat out: no one realises that an orchestral musician might just enjoy playing music in the sun in winter, and they search madly for a way to throw a coin into my closed music case). Perhaps he sees the money as a complication rather than a useful item: instead of assuming he could donate it, there would be all the trouble of getting the money into his country, bank balances, taxes, and more questions and papers to fill out to get it donated, and all the rest of it. All of which is time he could have been spending on solving another interesting question, or gathering mushrooms, or whatever. Coming into a fortune is not always fortunate.
      • OTOH, doing something openly for monetary gain is frowned on in academia. It just seems to me that everyone is behaving as their stereotyped role here.
      • Sadly, yes, doing something for it's own sake rather than for monetary gain is frowned apon

        That is not correct. Look at the hoopla around both Gates and Buffett giving way their money. Look at the adoration of Mother Teresa. Look at the army of fans for Linus Torvalds and Richard Stallman.

        and sometimes viewed with fear and confusion,

        Sure: anything out of the ordinary will engender fear and confusion. There is a difference between suspecting that someone MAY NOT BE altrustic and "frowning upon" the

    • >The incredulity that this mathematician might have been more interested in the challenge of the work than fame and fortune in the Western world practically oozes from each sentence.

      That and also while he did the hard work, that he didn't really contribute to the full proof, which is also weird.
    • BlueZ3: Heck, perhaps he just enjoys math for its own sake and doesn't want to deal with all the side-effects of notoriety.

      Quite.

      Perhaps he just hates parties. It's not like he'd be the first mathematician to do so. I and many other Slashdotters can sympathise with this, surely.
    • Meanwhile, what no one realizes is that Perelman is neither a recluse nor modest. Grigori Perelman is, in fact, a WoW addict.

      -Eric

    • the submitter seems to have misplaced the incredulity. the important thing is that other mathematicians are amazed that someone would throw around important parts of the proof, not wait for credit and leave it to others to write it up. then again, knowing perelman they are not incredulous.

      in mathematics, the trend has mostly been to keep the insights of a big result under wraps until the proof is written down properly and checked for bugs. that is the way to get yourself into the hall of fame [st-and.ac.uk]. it is almos
  • Quite an interesting character, this Perelman, and his proof could turn out to be a real landmark for mathematics.

    I liked this bit:

    Asked about Dr. Perelman's pleasures, Dr. Anderson said that he talked a lot about hiking in the woods near St. Petersburg looking for mushrooms.

    Whatever he's smoking, I want some!

    • Side note: the Millenium Prize is a cool million. Which is $24 million less than Adam Sandler makes per movie.

      Hurray for the free market! The true value for a personal accomplishment has once again been properly determined and awarded!

      • Sure he makes $25 million per movie now, but I'm sure he didn't make $1 million for his first movie.
      • You know...I think you're trying to be sarcastic, but you shouldn't because you're actually correct.

        Want to make a lot of money, do something the generates a lot of money. I can understand your point of view, but get real...
        • Want to make a lot of money, do something the generates a lot of money. I can understand your point of view, but get real...

          Innovation in math and science generates more money than any movie.
          Consider something obviously fundamental to the way we live, like calculus or Fourier transforms.

          It is very foolish to think that the direct and immediate monetary rewards a person receives are any real inidcation of the value their work provides to society.
          • Consider the impact that paid licensure of Fourier transforms would have on science and engineering. Or was the money supposed to come from the Magic Unicorn Cave?
            • So science and engineering don't make money, or they can't spare any for the poor mathematician that made it possible? Consider the impact of movie licensing on the poor theaters: do you know that the producers/distributers take 100% of the opening weekend box? We should just give it all to the theater ans let the people who produceed it go hang.
      • From your sarcasm it seems that you have no idea how free markets work... There is no such thing as innate value, the only value that something has is the demand for that thing.

        The demand for comedy is higher than the demand for mathematical proofs. The recompense for either has absolutely nothing to do with merit, even if you believe a mathematical proof has more innate merit than comedy. BTW, if you do believe that, please define for us exactly how a mathematical proof is better (has more value or merit)
        • On the contrary... (Score:5, Insightful)

          by moly (947040) on Wednesday August 16, 2006 @07:10AM (#15917807) Homepage

          A Scottish physicist two centuries ago sees a strange bump-like waveform in a canal. It persists for over three miles, moving at nearly constant speed along the canal trench. He writes a paper, calling it a soliton wave and two Dutch mathematicians find a nonlinear partial differential equation that describes its motion. The equation, the Korteweg-De Vries Equation, proves fiendishly hard to solve. Finally, the crew working on the hydrogen bomb, finish the job early, so Ulam decides to use ENIAC to help him solve the Korteweg-De Vries Equation. He attains the first analytic solutions, and the study of soliton waves begins in earnest.

          How does this earn a quid? Well, solitons model the way that blips of light move down a fiber-optic cable. The military decides that DARPA-net could run on fiber-optic cables, and uses them in building the early internet. Cellular telephone companies begin using fiber-optic cables to pack 100,000 phone conversations into a single pipe in such a way that they all get separated on the other end of the pipe-- one of the great engineering marvels of our time. We owe the modern internet, cell phones, anything that uses fiber-optics, to the solution of the Korteweg-De Vries equation. There was a similar burst of technology earlier in the last century when some closed-form solutions of the Schrödinger Equation were found.

          Truth is, when we solve a major math problem like the Poincaré conjecture, billions of dollars of revenue are generated by new technologies that spring into being because of the new scientific understanding that the solution affords us. A thousand Adam Sandlers will not generate the amount of capital that the solution of the Poincaré conjecture will generate, especially considering that Perelman has shown the world that the Millenium Prize Problems are actually solvable.

          • "Truth is, when we solve a major math problem like the Poincaré conjecture, billions of dollars of revenue are generated by new technologies"

            Ah. No... The money/capital isn't generated. It's simply moved from one place to another, from low performing areas to higher performing areas. Only the governments can print money. Are you trying to tell me that money invested in the telecoms industry inherently has more merit than money invested in the entertainment industry?

            What makes a mathematical proof inher
            • It's a philosophical question, is a society where everyone is connected instantly to every one else but constantly working "better" than a society where everyone is happy and relaxed?

              I think you hit the nail on the head with that. If only I had mod points.

              At some point in the unseen future that might change, the solution might on the other hand sit gathering dust on a shelf as a mathematical curiosity until the universe dies.

              Exactly. Merely because something is learned doesn't mean it's valuable. All

            • You don't understand the free market. In any free market transaction, value is created. Both parties walk away from a trade feeling that they have something of greater value. Paper money is not value. The government does not create stocks, and most value in the world is represented by stocks. Please get your facts straight.
          • Then perhaps the solution is to allow mathematical proofs to be patented so that the original discoverer can benefit from the resulting technological innovations. Ultimately, we're a resource based society, if someone wants to be compensated for the abilities they have to produce a resource with those abilities that can be bought and sold (or licensed, at least). If you give away your knowledge for free, you really can't complain when other people become multi-billionaires because of your initial hard work.
    • Here [maps.org] you go :)
    • by Bigos (857389) on Wednesday August 16, 2006 @02:44AM (#15917151)
      In Eastern Europe we don't pick up mushrooms to get narcotic high. It is merely a popular ingredient in our cuisine. The guy got his priorities right. No matter how rich and famous you are, in the West you cant get exactly the same ingredients for East European food. As mushrooms based meals are so delicious, I wouldn't be bothered to travel somewhere to get some stupid price when there is high season for mushrooms.
  • by BoRegardless (721219) on Wednesday August 16, 2006 @01:27AM (#15916925)
    "Perelman now seems to be the favorite to receive a Fields Medal at the International Mathematics Union meeting next week, but it's not clear that he'll even show up!"

    The curse of the gifted is that niggling worry in the back of the mind that if one accepts praise, one may lose his focus, drive or muse, if you will.
  • by ucaledek (887701)
    I think the greatness of the prize isn't the mercenary value people seem to think it holds. The money just shows importance. The prize's value comes from the dialogue and new paths of discovery that are opened up. Remember that in the end Fermat's last theorem (proof of which is what prompted this, at least in part) wasn't important in its result. It was important because the search for a proof resulted in huge new areas of research that are much more fruitful both in the purely abstract mathematical se
  • name change? (Score:5, Insightful)

    by bark (582535) on Wednesday August 16, 2006 @02:00AM (#15917021)
    Now that the conjecture is proved, do they change the name to "theory"? Or does the name stay put because that's what everyone knows and refers to it as?
    • In all probability, once it has been vetted and accepted, it will be called a theorEM. Theories are for the inductive sciences. Little nitpick, sorry.
    • Re:name change? (Score:5, Informative)

      by Kjella (173770) on Wednesday August 16, 2006 @03:57AM (#15917320) Homepage
      Now that the conjecture is proved, do they change the name to "theory"? Or does the name stay put because that's what everyone knows and refers to it as?

      Things that are proven, are called theorems. They do depend on axioms, but those are defined as true. Sciences about the real world that can't put up axioms (because that'd require ex facto knowledge about the real world), so they can never be conclusively "proven". Hence well call them theories, like theory of gravity, theory of evolution. A few we've called "laws" as well because they have been so extensively tested, but it is not proven in a strict formal sense.
      • You are wrong. This is true for other fields, but in mathematics, theories are more a like set of definitions, propositions and theorems used in a particular field. Remember, in mathematics everything is proved except conjectures (which are basically theorems you don't have proof for, but you can't find a proof of the contrary either). Mathematics are a purely virtual world governed by logic rules. There is no place for observation or rough suppositions like in physics or biology. For example, the category
        • There is no place for observation or rough suppositions like in physics or biology.

          This is mildly incorrect. Theories also consist of a body of associated hard to solve problems. These problems often turn out to be observations and guesses or as you put it, "rough suppositions". For example, "Fermat's Last Theorem" has for centuries been a part of number theory even though it was proven only ten years ago.

          • What I meant is that in physics or biology, there are competiting theories on a given subject. If you read /., you know that a "no Black Hole" theory pops up regularly and some physicists propose their own theory to explain an *observation* of a natural phenomenon. In mathematics, you can prove things in different ways, but in the end a valid proof is a valid proof. Sometimes, it may be proven using intuitive logic so someone will want to re-do it using constructive logic, etc. There are different schools.
      • Actually, the terms "theory" and "law" are used in mathematical logic as well. In a given logic language, if you have a set of logic formulae called axioms, a theory is all that can be derived from these axioms by applying modus ponens. If the axioms eventually derive contradiction, then the theory is said to be the trivial theory, that is the theory that consists of all possible statements of the language. The smalles theory of the language is the one that contains only all taughtologies of the language. T
  • by Anonymous Coward on Wednesday August 16, 2006 @04:23AM (#15917390)
    If any of you had read the article you would have noticed that the 1000 pages is actually a very rough figure for the sum page count of all 3 articles by various people each of which explains Perelmans result in context, thus duplicating the other 2. So in fact the full articles are about 315-470 pages each. Also what Perelman infact did was show that using the Ricci Flow technique on the 3D shapes to solve the Poincare conjecture, an idea of Hamilton's from the 80's, can work. Up till now it was thought that certain structures might degenerate to singularities and fail, but Perelman showed that those singularities would in fact all turn out ok. Poincare's conjecture is for 3D shapes, and higher dimensional generalisations have previously been solved (5+ dim by Smale in 60's, 4 dim by Freedman in 80's, both got Field's medals).
  • It is said that the Poincare Conjecture proof is one of the most important proofs in Mathematics. But I never managed to understand why. What are the practical consequences of this proof? does it have any real-world applications?
    • The Poincare conjecture matters to basically any area of science where topology is important. i.e relativity and quantum mechanics. Also to the new(er) directions in physics like string theory, et. al.

      Even if the PC has no direct bearing on some of these fields, the techniques used in the proof will probably end up deeply influencing their research methods.

  • by ed_g2s (598342) on Wednesday August 16, 2006 @04:49AM (#15917447)
    According to The Guardian [guardian.co.uk]
    • What about charity? I mean, if he doesn't want the money for himself, he should find a better use for them.
      • That's stupid. He might turn money down. Now it's announced that proof is correct and that makes him candidate for that prize.

        Even Russian newspapers do not have any official reaction of Perelman himself yet.

        His (western) colleagues speculate that he might turn the award down. He is too far from normal life and money would distract him - so his friends say. That's speculation.

    • Would you blame him? He obviously poured a lot of time and energy into this. I'm sure there was no shortage of nose-thumbing, pride, and jealousy if my experience of SOME people has proven.

      Is it thanks to receive some money and a medal after your peers roasted you for a couple of years?
      • Ten or twenty years' worth of academic wages ain't something to sneeze at, nor is the single most prestigious award in mathematics. I could see turning the Fields down just to make a point, but the million dollars can free you from financial obligation so you can distance yourself from your peers however long you want while doing what you love doing. Otherwise the money just goes back into feeding the system that you apparently hate.

    • TFA mentions he has distanced himself from others in the Math community because he has become disillusioned. I read into that my own experience, which involved professors trying to hit on me, others trying to get me to write/edit their papers and then taking the credit, others who weave tall tales with just enough truth to fool grant money providers.

      One of my colleagues now believes that Science is actually performing a random walk on the landscape of Truth. Occasionally, the walk stumbles over something
  • I understand
    [/lie]
  • Something that's intrigued me since I noticed it, relating to hypersphere volumes. In 1D it's 2r, in 2D it's pi.r^2. 3D is 4/3 pi.r^3. The sequence continues: const.pi^2.r^4, const.pi^2.r^5, const.pi^3.r^6, const.pi^3.r^7 (can't remember offhand what the consts are but they can easily be found).

    Obviously you're going to get an extra r with each dimension, buy why do you only get another pi every other dimension?

    While I'm at it, on a related subject it seems to me there are two possible ways of constructi
    • Obviously you're going to get an extra r with each dimension, buy why do you only get another pi every other dimension?


      The Jacobian, or unit volume if you will, of a hypersphere [wikipedia.org] has a a highest term of sine, or cosine, which grows as you increase dimension. Specifically, for an n dimensional sphere, the highest power of sine or cosine will be sin^(n-2).

      Anyway, to answer your question, integrals of sine or cosine to odd powers produce only functions of other sines and cosines. However, integrals of sine or cosine to even powers produce functions of sin(x), cos(x) and x. The x part gives you your pi, but only does so every second dimension, when the highest power is even.

      Here's the integrals of (sin(x))^n, for various n

      n=0: x
      n=1: - cos(x)
      n=2: x/2 - sin(2x)/4
      n=3: 1/3 * (cos(x))^3 - cos(x)
      n=4: (sin(4 x) - 8 sin(2 x) + 12 x)/32
  • by purplelocust (944662) on Wednesday August 16, 2006 @09:25AM (#15918498)
    I don't work in three-manifolds but my research has some connections with it so from time to time I'm at a conference or two in the area. Grisha Perelman is an interesting guy, even amoung the very driven math folks who tend to be an interesting lot, and his disinterest in the political/social aspects of his work is I believe genuine.

    1) I met him at the Mathematical Sciences Research Institute in Berkeley at a workshop sometime around 1994 and he at that point had ridiculously long fingernails and was quite unkempt, even by the quite weak standards applied to research mathematicians. That was a while ago, of course and that was probably one of his first visits to the US. He gave an incomprehensible energetic talk so what most people commented on was his nails.

    2) In 2003 or so, during a limited lecture tour about his proof of the Poincare Conjecture, he responded deftly and hilariously to a comment of Misha Gromov in the audience. Gromov is one of the most difficult people to have in a talk- he is a great mathematician with not much patience and has derailed or rerouted talks by many great researchers, who sometimes get quite flustered. I can't remember the exact wording of the exchange, which is too bad since it was precious, but Gromov asked something like "I don't see how that goes, I'd like to see some more details" and Grisha responded with something like "well, yes, you would" and carried on as he had intended.

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