[Major updates to this post are now all complete. 2014.08.31 12:10 PM.]
Everybody knows about The Clay Maths Problem. There are claims, and then there are counter-claims. … First, there are some claims regarding The Clay Maths Problem. Then there are some claims going counter to them… And then, there also are claims about the claims about The Clay Maths Problem. Here is an example of a claim of the third kind.
If what Prof. Scott Aaronson often writes on his blog [^] about this issue is to be taken even semi-seriously, then he routinely receives something like [the particular estimates being mine] emails per week, all seeking his opinion about what he thinks of the arXiv article submissions per week claiming to have proved the P-vs-NP problem one way or the other, where is a very, very large number; who knows, it might even be approaching .
[Since this is a problem from theoretical computer science, for all estimates, the base has to be `2′; any thing else would be unacceptable. Aaronson is usually silent on precisely where the partition lies: whether the number of claims proving P = NP is statistically equal to those proving P != NP. However, he seems to hint that the two are equal. If so, then the CS-favorite number 2 would slip in once again, now as a divisor.]
The P-vs-NP problem is, thus, THE well-known Clay Maths Problem. Everyone knows about it.
Few people also know that there also is/was one more Clay Maths problem. … For example, they know that some decidedly crazy guy fooled them all—first, the mathematicians, and then, also himself. They—the mathematicians—accepted his solution, but he declined to accept the award, even the $1 million prize money that goes with it [^]. [Since the definition of a proper solution for the award is acceptance by mathematicians, it is easily conceivable that someone manages to fool them for two years and collects the prize.]
Relatively fewer people still know that as many as seven such million dollar problems were announced by the Clay Institute at the turn of the millennium.
As to the other problems, still fewer people ever bother to get past talking about the Yang-Mills problem. Even when it comes to this problem, as usual, they do not entertain any hope about seeing its resolution in the near future, where the “near” is left unquantified. But they all agree that it is a problem from mathematics—not physics.
What theoreticians agree on is always more interesting than what they disagree on. And, guess it was Ayn Rand who said it: also more dangerous.
Then, there are even fewer people who at all know anything about the Navier-Stokes problem—the mathematical version of it.
And, from my Web searches yesterday, there are very, very, very few people, at most a handful, who do something serious about it. This post is about them.
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The first gentleman who purportedly continues to remain concerned about this Navier-Stokes problem is: Prof. Charles Fefferman. He should have no choice in the matter, I suppose, because it was he who wrote the official problem statement for Mr. Clay in the first place. Prof. Fefferman is an American mathematician [^].
man person to stay worried about this problem for as long a period as one entire month, was one maths professor from Lehigh, one Ms. Penny Smith [^]. She—an American—soon later retracted her solution [^]. Judging by the dates of the v1 and v5 versions on abstract page of her arXiv paper, the retraction took about half a month. (The number two, again!) Most of the ‘net discussions regarding her solution seem to have since then undergone a process of typographical mistake-making [^], or of plain vaporization [^]. Peter Woit, an American professor of Mathematics Physics Mathematics at Columbia, often better known on the ‘net for his proclivity to arbitrarily delete others’ replies on his blog even after having first having allowed their publication there, however, tenaciously holds on to some of that discussion [^]. (Here, I read only the first and the last still-published comments.)
The third person to stay sufficiently bothered about this problem so as to go to the extent of writing a significant paper on it, seems to have been Prof. Mukhtarbay Otelbayev of Kazakhstan [^].
The first guy to venture discussing Otelbayev’s solution on the Mathematics StackExchange forum, chose to do so anonymously [^]. There must be something about the character of this problem that makes even people from Berlin, Germany, behave this way—writing anonymously—even on the Mathematics part of the StackExchange forum. Indeed, the question to strike this “Unknown” Berlin-based guy wasn’t the correctness or otherwise of Otelbayev’s solution; it was: Did Otelbayev solve the same problem as was posed by the Clay Institute? [^].
There were other follow-up discussions, but soon enough [within a month, of course] the author admitted that there was a mistake in his proof. The ‘net discussion on his proposal is still available [^]. [Professor Otelbayev is not an American.]
To my utter and great surprise, I also found (during an Internet search right this week) that in the meanwhile, there also has been none other than Terry Tao himself jumping into the
frey issue. (No one calls him Professor Terence Tao. That’s exactly like how very, very few people, if any one, anywhere, ever, calls Prof. Aaronson by his last name and/or profession.)
My surprise was not entirely baseless. Terry Tao is a Fields medallist [^]. It was plain inconceivable that someone who already is a Fields medallist, would directly take on a(ny) million [American-] dollar problem.
That, indeed, turns out to have been the actual case. Terry Tao didn’t directly tackle the Clay Maths problem itself. See the Simons Foundation’s original coverage here [^], or the San Francisco-based Scientific American’s copy-paste job, here [^]. What Terry instead did is to pose a similar, and related, problem, and then solved it [^].
The “Unknown” Berlin-er mentioned above, was absolutely on the right track. It’s one thing to pose a problem. It’s another thing to pose it well.
That’s the light in which you might want to examine both the Clay Institute’s formulation of the problem and Tao’s recent efforts concerning it. Realize, Terry didn’t solve the original problem. He solved another, well-posed, problem. And, as to the well-posed-ness of the original Clay Maths problem, there has been another notable effort.
Challenging the well posed-ness of the Clay Maths NS problem seems to have been the track adopted by Prof. Claes Johnson [^] for quite some time by now—several years or so.
Johnson is not a pure mathematician, but a fluid dynamicist. In fact, he is a computational fluid dynamicist, who has actually worked on some practical fluid dynamical problems [^][^]. He seems to be an interesting fellow. Despite having an h-Index of 52 [^], he has written against the climate-warming alarmism [(.pptx) ^][^]. Also see his response after having been selected for the Prandtl medal [^].
There is no Simons Foundation coverage on Johnson’s work. Naturally, any coverage by the Scientific American is plain inconceivable—especially if Johnson is going to write about his positions in springs and summers.
Johnson raises the issue of whether the Clay Maths formulation is well-posed or not. In simple words, can it at all be solved (by any one, ever) or not—whether the problem is in principle amenable to a solution or not. In case you don’t know, the “well-posed-ness” is a technical concept from mathematics [^].
Yes, that way, the issue of whether a problem is well-posed or not, does mean something like: “do you yourself know what you are asking, or not,” but the sense in which Hadamard meant it was certainly a bit more refined.
The distinction of the well-posed vs. ill-posed applies specifically to the solution of differential equations, and it means something like the following:
If you are going to throw a ball so as to hit a distant target (i.e., technically, a two-point BV problem for the second-order differential equation i.e. Newton’s 2nd law), you have the following choice: within appropriate limits, you can select the initial angle for the parabolic trajectory of the ball, in which case you have no choice about its initial speed—the horizontal distance to cover, together with the initial angle, would fix the value of the initial speed with which the ball must be thrown if it is to hit the target. (It would fix how tall the parabola should be, given the initial slope and a fixed horizontal distance.) Alternatively, you can choose the initial speed for the ball, in which case you have no longer have a choice about the initial angle. If hitting the target is your objective, you cannot arbitrarily specify both the auxiliary conditions: the initial angle and the initial speed, at the initial point. The nature of the differential equation is such that specifying both the auxiliary conditions at the same time at the same point renders this differential equation problem ill-posed. That, probably, is the simplest conceivable example of what it means for a problem to be ill-posed or well-posed.
As Hadamard pointed out, a differential equation problem, to be well-posed, must fulfill three conditions: (a) a solution must exist, (b) the solution must be unique, (c) the solution must change continuously with data (i.e., auxiliary conditions, i.e., the boundary and initial conditions).
Sometimes, the solution exists, but is not unique. For example, the diffusion equation problem is well-posed in the forward time direction but not in the reverse, in general. The diffusion process tends to smoothen out any initial sharpness. For example, if you place a drop of ink in the shape of a square on a blotting paper, it soon spreads and becomes a big, thin blot, growing ever rounder and rounder in shape as time passes by. Therefore, the information about the initial shape of the blot gets smeared all over the domain in such a way that starting from this later, bigger and roundish shape, and then going back in time following the diffusion equation, you cannot uniquely recover the initial shape of the blot. Whether the initial ink blot is square or hexagon, they both become round during diffusion. The resulting round shape doesn’t hold enough clue as to the number and locations of the sharp corners in the initial shape. In other words, the information about the initial sharpness is immediately lost during the diffusion process. And so, you can’t uniquely say whether it was a square or a hexagon: both (and infinity of other shapes) are possibilities. [As an aside, I do have some objections to this logic of the diffusion equation, but more on it, in a separate blog post, some time later.] So, the forward diffusion problem is well-posed, but the reverse one is not—no unique solution exists for the latter.
There also are other considerations for well-posed and ill-posed problems, which are more complicated. They refer to the continuous dependence of the solution to the auxiliary data. The auxiliary data, for the time-marching problems like diffusion and fluid flow, crucially means: the initial data.
Thus, the additional relevant consideration concerning the NS system has to do with the smoothness or otherwise of the initial velocity field. Johnson and a colleague rightly point in a paper [(.PDF) ^], that:
“If a vanishingly small perturbation can have a major effect on a solution, then the solution (or problem) is illposed [in the Hadamard sense], and in this case the solution may not carry any meaningful information and thus may be meaningless from both mathematical and applications points of view.
In this perspective it is remarkable that the issue of wellposedness does not appear in the formulation of the Millenium Problem. The purpose of this note is to seek an explanation of this fact, which threatens to make the problem formulation itself illposed in the sense that a resolution is either trivial or impossible.”
I agree. Do see the paper in original.
While Johnson and colleague’s technical paper may be out of the reach of many people—and in any case, at many places in the later half, it certainly is beyond my reach—the first half of the paper as well as Johnson’s blog entries are simple and clear enough to be understood even by any graduate engineer. See the series of his blog posts on this topic, here [^]. For ease of reference, the Clay Maths official problem description is here [(.PDF) ^].
[In this update of my blog post, I have edited a lot around here—from the meaning of the well-posed-ness, to what constitutes Johnson’s position. Some of my writing in the very first version of this post simply was some draft in (very rapid) progress, and so didn’t summarize Johnson’s position well. But then, I had noted that I was going to come back and edit this whole post. In particular, in the following couple of updates, I have deleted the line to the effect that Johnson meant that a blow-up won’t occur in the NS, as if this were to be his final, unqualified position, which, of course, it is not. ]
So, in summary, Johnson has been repeatedly pointing out some important considerations regarding this problem, for a long time. A summary post could be this one, his latest [^].
And then, tarries along Terry Tao, correctly poses and seemingly correctly solves a problem that is teasingly near the original Clay Maths problem. He shows that a blow-up does occur—but only in his system, not necessarily in the NS system.
Terry, obviously, is interested in only teasing his reader, but not yet quite willing to jump into the… [Ahem.] … You see, otherwise, he could have easily dressed up the same result in different terms [say, in fuller clothes] without making any reference to the Clay Institute or its problem. But he does.
So, that’s what Terry Tao does. He wants to be seen as both addressing and not addressing the Clay Maths Problem. [He is an American.]
And, of course, even though Terry Tao responds at his blog to many, many people, he curiously doesn’t at all respond to a well-established European professor with definitely impressive credentials, like Johnson [^]. [Interestingly, though, Tao also does not delete Johnson’s replies once he publishes them. [Tao also is an Australian—he is a dual citizen.]] But the lack of response in such a context takes the matter closer to “tantalizing” than plain “teasing.” At least to someone like me, it does.
The issue is not whether the particular arguments that Prof. Johnson forwards are in themselves general or powerful enough to settle by themselves the Clay Maths Problem, or not.
The real issue is: the broader, valid and extremely relevant point regarding well-posed-ness which he repeatedly raises. Professor Tao should have responded to that. … You simply can’t go on beating [dancing] around the bush [pole], you know! [Professor Tao works in California, USA.]
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Ok, enough of “cross-referencing.”
But if you ask me what my take on this whole issue is, let me stick my neck out and say (at least) this much:
I think that:
(i) Prof. Johnson’s broader point, regarding the well-posed-ness is absolutely relevant, and unless it is addressed in a forthright manner by the Clay Institute, this Millenium problem essentially remains muddy, and
(ii) Prof. Tao’s line of thought—the idea of fluid logic gates and machines to be built out of the ideal fluid, and the fluid computer, etc., etc., etc.—would eventually be regarded as not at all relevant to the very core of resolving the NS issue. Tao at best clarifies some abstract part of the nature of the problem, and this part, IMO, isn’t going to be relevant—not to the NS problem itself. His best contribution concerns the energy cascade across the scales, but he offers absolutely no insight as to how that cascade might respond in the actual NS problem settings.
Let me put the second point in a round-about and enormously hand-waving kind of way: If Tao’s current work is at all relevant to the NS problem, then it means that the mathematical community’s standards are sufficiently lax that a solution to the P-vs-NP problem could also be had without the mathematical community first coming to agree with some new and explicit clarity about some of the particular details about how the continuum hypothesis is to be interpreted in that context. But the second is not going to happen, simply because it is THE Clay Maths Problem. Ergo, Tao’s current work… QED.
On the other hand, to repeat, the points that Johnson raises are relevant, and they will stay relevant.
And, of course, all that is quite apart from another, “related” issue: even if there is or isn’t a blow-up in the NS system (and even if the Clay prize gets awarded for “proving” it either way (i.e. for someone getting the mathematicians to agree with him for two consecutive years)), the real issue would still remain: whether the Navier-Stokes system happens to be a good model for real fluids or not.
Just for the starters, as every one knows, at appropriate Knudsen numbers, the no-slip condition is no good. If you are going to complicate the auxiliary conditions to such an extent that the complexity of their effects exceeds that of the basic governing differential equation, then, you sure gotta ask of what good use your basic differential equation is, in the first place. In physics and engineering, we adopt the differential equation paradigm only because it has an epistemological value: it helps reduce cognitive load. If the bells and whistles are going to weigh a ton, how do you expect a cart powered by a toy-spring (or a 12 V 0.4 A Watt stepper motor) to get off to any start?
Or, for another matter, if what happens at the small, local, scale is extraordinarily different from what happens at the large, global, scale, or if pathological connections exists between the two scales, then, you have gotta ask some time: Why elevate some scale-based parameters that only give kicks to the mathematicians but does not make life any simpler to anyone else? Why stick to this continuum-based description for all the scales, in the first place? After all, even simplest phenomena like droplet formation or coalescence are in any case beyond the reach of this “basic,” “fundamental,” etc. etc. Navier-Stokes formalism anyway. Why elevate a theory of such obvious flaws to such a high pedestal? Why keep such a narrow mind that it can’t even deal with some of the simplest phenomena of nature? And, if you must do that, then why not take just one more step and declare another million dollar prize for some latest parlour game? What ultimately does distinguish mathematics as we know it, from the parlour games?
But, of course, we need not go all that far, really speaking. In many ways, the NS system is of enormous practical importance. All that we need to do is to bring some astute observations regarding differential equations, into the problem formulation.
Absent that one, no one has proved either a blow-up, or its absence, in the NS system, thus far. Not even Terry Tao. And, he must know that that’s because of the ill-posed-ness of the problem formulation itself. That’s why he must be choosing to remain silent to Johnson’s query, here [^]. Remember, he both works and does not work on the Clay Maths problem?
The current situation, in many, many ways, is something like this:
Suppose that mathematicians are busy building castles out of thin air (say, debating endlessly about whether the d’Almbert paradox is mathematically consistent or not). Suppose that the working epistemology of the culture has reached such a low level that no one can tell if the theories of thermodynamics and EM are consistent within themselves, let alone with each other. So, a rich guy steps in and declares a big mathematical prize for someone proving whether the Rayleigh-Jean blow-up really follows from the Maxwell system or not. Predictably, there is a flurry of activity, and thus there follow a few mathematicians who can’t get it right. And then, there steps in a young, brilliant mathematician, and declares that he can prove that a blow-up cannot occur but that his proof is limited only to a subcategory of the classical EM fields: the ones that are averaged in some sense. And then comes along some industrial physicist (i.e. actually a theoretical engineer, who has got a job declaring himself an applied mathematician). The industrial physicist points out that the mathematician’s argument can also be taken to imply the exactly opposite conclusion: namely, that these average fields must necessarily lead to the ultraviolet catastrophe. The mathematician chooses not to respond. The audience claps for the mathematician, and falls dead silent for the industrial physicist.
And, no one thinks of instituting a new prize that would reward such efforts as of building a new theory of physics which shows how to prevent the blow-up, even if doing so would involve breaking away from the clutches of some deeply held assumptions about the physical nature of reality, even if such a break-away is only a desperate last measure.
Sure, the situation is not exactly analogous. But if you go through the history of QM, and see how no physicist (or mathematician) ever left others’ valid queries unanswered—if you see how, on the contrary, they rapidly and openly communicated if not collaborated with each other—and then, if you see the kind of hype and blogging practices currently going on in our times, you will begin to see some pattern, and if not that, then at least some semblance, somewhere.
And, you will dearly feel something like—what? wistfulness?—about the good old times now so distant from ours: the times when the rational culture of science had already taken strong roots and it still was mostly a free, application-driven enterprise (the studies of cavity radiation, leading to the first physics Nobel, were sponsored precisely so as to help produce brighter bulbs more cheaply, for better business profits); the times when the state control of science was barely in its nascent stages or altogether absent. (Check out the history of Income Tax on Google, for instance. And, remember, that one—the Income Tax—is only for the starters, as far as the means of the state control of science goes.)
Guess I have made the most important two points which I had.
[Guess my major editing is over, except perhaps for a typo here and there. This long week-end for the Ganesh Chaturthee ends today, and from tomorrow, I will be back to my heavy class-room teaching duties—i.e. away from blogging. (In fact, factoring in the preparations for lectures, I would be into my heavy teaching duty starting right this afternoon.) So, bye for now.]
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A Song I Like:
(Hindi) “purvaiyya leke chali…”
Music: Ravindra Jain
Singers: Lata Mangeshkar and Shailendra Singh
Lyrics: ? (Ravindra Jain?)