Penny Smith, a mathematician at Lehigh University, has posted a paper on the arXiv that purports to solve one of the Clay Foundation Millenium problems, the one about the Navier-Stokes Equation. The paper is here, and Christina Sormani has set up a web-page giving some background and exposition of Smith’s work. I should emphasize that I know just about nothing about this kind of mathematics, but I’m reporting on this here for two reasons:

1. It looks plausible that this really is important.

2. Penny Smith tells me that she is a regular reader of this weblog.

**Update**: There’s an informative news article about this on the Nature web-site.

Michael,

If you want to sneer at Peter, he has given you plenty of opportunities. But you as everybody else here know that this is not the place. Can you please show some respect for the decencies?

Don’t worry, Penny, no reasonable being will think ill of you for making this mistake. On the contrary, they should be impressed at your ability to accept it and move on.

Good luck, Penny.

A whole lot of annoying aphorisms spring to mind, like the fact that a theoretician’s best instrument is his/her waste paper basket, and that nothing of value comes easily.

Don’t give up!

Someone expressed reservation earlier about the whole concept of a Millenium Prize, and I agree with these reservations. A lot of problems remain unsolved because they are – with existing techniques, anyway – insoluble. Encouraging people, using large cash prizes, to continue to bang their heads against brick walls when they might otherwise be going off in orthogonal, and perhaps more productive directions may not be such a good idea.

On a wholly other note, I wish Michael would spend a few days pondering Marty Tsysanner’s wise and well-meaning advice. I’m sort of amazed at the careful thought Marty put into writing that.

I agree with Baez (2 posts above): in my view scientists that admit their mistakes are more trustable than ones that don’t. Unfortunately withdrawing wrong papers is not the common-practice among us physicist.

Why is P. Smith publishing all of her results in JMAA.

I just noticed that P. Smith is on the editorial board of JMAA.

Does that help get a paper accepted without a thorough review?

I do have serious issues with this.

Maybe, they trusted me too much because of that.

Not my doing.

I have plenty of papers not published at JMAA

Penny

@Editor:

I don’t know any of the facts in this case. However, in general people are on the boards of journals in subject areas they understand well (even though they may not be detailled reviewers themselves any more). The areas they understand are likely to be the areas they are working in, so it’s quite likely that they’ll be submitting to those journals.

As to submitting to journals with less thorough reviews, I don’t know of anyone who believes the work they do isn’t actually correct, so thoroughness isn’t a reason for choosing where to submit. (There are more “debatable” reasons guiding submission choices, such as not submitting to journals that are so popular that they can afford to reject papers that are merely “good” and not “exceptional”. But that’s different from hoping incorrect work will be accepted.)

I think the saddest thing about this is as an instance of the “I was there” phenomenon: we want to have been there when something amazing happened, so we force the interpretation that something groundbreaking _has definitely_ happened rather than leaving it to experts to figure out “at a natural pace”. Given the history of things like Kempe’s flawed proof of the four colour theorem, I’m not surprised that it takes a while for erroneous steps to be spotted; however Kempe chains are (so I’m told) an important element used in the correct proof of the four colour theorem.

So lets let things “cook” for a few months and then make a considered judgement of things.

Dear John,

Well, first thing is to fix any wrong results, and get that published.

Also, I have other papers to write.

Thanks for the supportive comment.

All that really matters is getting the correct results in print, because that is SCIENCE.

It’s going to take more than a few months, because most math papers are reviewed VERY slowly.

I wish JMAA had been much slower.

(Once more, I apologize to others for my part in keeping this subthread alive. I expect this will be the last time.)

Michael,

You said that

What I despise is the way Peter makes no scientific contribution of his own while exploiting his populist criticism for attention and personal gain.Clearly you are putting yourself in the position of being a judge of Peter and his character. Who made you a judge of others? What makes you so certain of your infallible judgement of others and their motivations that you can make such pronouncements without any doubt? Clearly there are many others here who don’t share your conclusion about Peter’s motivations by the fact that they are participating in various discussions under his auspices. Even Penny, whose efforts you praised, is apparently a regular visitor, and also someone who felt that the quality of the audience here was sufficient to discuss her results. Do you think she and others would have done this if they thought Peter was doing all of this just for his own personal gain? Yet you seem to place yourself above others here in your belief that Peter is really just a fraud, and furthermore that your True Judgement empowers you to use disruptive tactics and otherwise reflect badly on string theorists in general. Your behavior may negatively affect a lot of people, but Peter is only one of them.

You said,

Nobody, of course.in response to my question about who you were trying to sway with your tactics. If this is true, then your entire participation here is completely pointless and a waste of your time. But I don’t believe that you don’t care about swaying anyone, because in your first response to me you said,

I just do not believe that he deserves the same kind of courtesy — next best thing is to try and expose him for what he really is.If you want to expose him, then you clearly want to expose him to others, to convice them that Peter is the fraud that you think he is. At least that’s the way I read what you wrote. So I think you still need to think about who you are trying to convince, and what is the best way to go about it. I have already explained why I think your past tactics are a completely counterproductive approach, so I guess I really don’t have anything else to say on the subject that is any more likely to reach you.

Dr Smith,

Please tell us what is error in your previous publication!!

>> Clearly you are putting yourself in the position of being a judge of Peter and his character. Who made you a judge of others?

I did. What do you want? You criticize me — what’s your entitlement?

>> What makes you so certain of your infallible judgement of others and their motivations that you can make such pronouncements without any doubt?

You are getting weirder by the minute. I’m not infallible. Just voicing an opinion and acting on my personal convictions.

>> your entire participation here is completely pointless and a waste of your time

No. It makes Peter’s unethical behavior less rewarding. I find it quite worthwhile…

>> But I don’t believe that you don’t care about swaying anyone, because […]

Swaying people’s opinions in a scientific matter is just completely different from exposing a crackpot. I find it hard to take you seriously if I have to explain this to you.

>> I have already explained why I think your past tactics are a completely counterproductive

Yes, in essence it’s because you believe in persuasion and democratic voting on scientific matters. And I don’t.

Michael,

OK, but why do you have to be a jerk?

Peter, this off-topic discussion is annoying. If you do not want to censor Michael, what about creating a separate posting where people can insult you without disturbing other readers?

a,

Sorry about this, presumably “Michael” will get tired of this soon, and in the meantime he’s an excellent example of the reaction of the string theory community to the appearance of my book. The unwillingness to deal with what I have to say there, coupled with personal viciousness and cowardice, are quite telling. It seems that “Michael” divides his time between UMass in Amherst and the HEP group at Brandeis. Someone familiar with one of these places might want to tell us who he is. I strongly suspect that if he had to sign his name to what he writes, he’d stop doing it.

I don’t understand what is the rational reason for putting other people down because they’re not making good progress in research or hit a snag when solving a very difficult millennium problem. Such actions do not enlarge a person, further their own career, or enlighten them in any shape or form. So why do it? To get a cheap kick out of insulting others that lasts maybe 5 seconds? And then you’d have to pile on more insults to get the same “high” the next time; this is addictive. Nip it in the bud, Michael.

Wow! Not hard to find every one of the seven deadly sins on this blog. Who says math is boring?

From my experience, making mistakes is almost inavoidable in research. Don’t give up, Penny!

Dear TruthSeeker,

why don’t you try reading this:

http://www.math.columbia.edu/~woit/wordpress/?p=470#comment-17132

You might just advance to OccasionalTruthFinder.

Michael,

You must be talking about the first sentence of my previous post. Well, the first part of it refers to people such as yourself, and the second part of it refers to people who posted rude things that subsequently prompted the Oct 8th comment above by Peter. (That rude person has apparently stopped posting here.) I hope that clears up your misunderstanding.

Penny, never ever quit. If you never quit, you’ll be astounded at what you can accomplish.

Geez, this thread was begun to discuss the NSEs and Penny’s heroic attempt at it, and then this boring and boorish “michael” intrudes once again with his own agenda. Sorry Peter, but I think that “michael” behaves like a pathological crank caller, and he will never grow tired of this.

Dear Penny,

Thank you for telling us the truth. I have a question

and a suggestion:

1. Where did you get wrong? How did you find it out?

2. You possibly need someone to help you to keep the mistakes (I mean the serious ones) from you.

Best Wishes

According to http://comet.lehman.cuny.edu/sormani/others/SmithNavierStokes.html and http://comet.lehman.cuny.edu/sormani/others/smith.html an anonymous mathematician found an error and reported it to her in one of her published papers which her preprints “Immortal Smooth Solution of the Three Space Dimensional Navier-Stokes System” and “Eternal Continuous Viscosity Solutions of the Einstein Cauchy Problem” relie on. Based on the following Smith said in this thread, “It’s going to take more than a few months, because most math papers are reviewed VERY slowly. I wish JMAA had been much slower.”, I believe that the paper with the flaw is one that was published in the Journal of Mathematical Analysis and Applications.

Your seggestion does not make sense to me, because I do not think that mathematician such as Smith would benfit from not having the details on any flaws in their papers.

Prof. Penny,

I salute you & your brave efforts. You will come back to fight another battle. I wish you well.

I am currently researching the N-S from a physics perspective & have observed a slow-fast phenomenon which allows a suitable decomposition of the system into esentially 3 equation groups (eg. Reynolds decomposition).

How would your proposed solution affect such decomposition concepts? This is one of the typical approaches used in turbulent model development – bulk + fluctuating + closure.

Thank you & best of success…

desA

“Heroic attempt.”

Isn’t this a bit of an overstatement?

I don’t blame Penny for trying unsuccessfully to obtain global

solutions to NSE and other nonlinear equations. I don’t even blame

her for deluding herself into thinking she had done it.

But the sycophancy of this thread is starting to grate. The reason anonymous experts said early on that the proof did not look

plausible without knowing the exact line in which the error was

is that the proof did not seem to be based on any propert of NSE or

overcoming any of the known difficulties.

Everyday, there are experts with a deep knowledge of NSE looking

for ways of making incremental progress against those difficulties.

The reason they aren’t making bold announcements isn’t that they

are further away from a solution than Penny Smith. It is that they

better understand the field. Let us not forget their brave attempts

while commending the attempt by Smith.

AE

“anonymous expert”, you are very cruel & unkind.

desA

No, I don’t think AE is “very cruel and unkind”. I think he’s honest and fair. After all, this is science.

Good … but think how much people would have hated you if you had been right.

I was wondering if Navier Stokes had any relationship to Quantum Gravity?

Testy:

Navier Stokes (NS) describes momentum conservation in the classical limit of a quantum field theory (the diffusion coefficient in NS is a remainder of this quantum heritage); and quantum field theory is likely to be a low energy limit of quantum gravity. voila!

…and because of this connection it wouldn’t suprise me if turbulence itself is a phenomenon grounded in the quantum realm.

To do good research, you must have the utmost respect for those who have also tried to atack the same questions. And you must also have the utmost respect for those who will study at your work.

While you’re doing your research, you don’t have to know what others have done or what ideas they have tried. Usually, it’s better to know, but sometimes it’s good not to.

But the respect for others is critical after you think you have done something. At that point, you have to ask yourself, “why have I succeeded when others have failed”, and it is absolutely critical to educate yourself and answer this question carefully and honestly before you announce anything to the world.

Testing your work against the simplest examples (as described so well by Feynman and suggested in the comments above) is the first thing every serious mathematician does, and it demolishes 99% of our efforts.

And if that test passes, it is only the beginning. There is an illusion that the responsibility for checking a proof lies in the referee of your paper. But that’s just not true; very seldom do referees have the time to check a proof in detail. The ultimate responsibility lies in the author of the paper. So before you submit a paper to a journal or even arxiv.org, you have to sweat blood to find every error in your paper and correct it.

And after you’ve done all you can, you have to enlist as much help as you can manage to do another round of checking, because it is very difficult to find all the errors by yourself. But the help has to come from experts in the field, so posting your work on arxiv or putting up a web site is *not* the right way to do this.

And you do this for papers that don’t come within light-years of proving a Millennium Problem.

Doing anything less than this wastes the extremely valuable time of those who will study your paper.

I consider both Penny and Christine friends, but I have to say that I think they both got a little too excited by the signficance of the problem and strayed from the fundamental rules of doing research just when they needed them the most. Normally, the consequences of this error are small, but, coming right after Perelman’s proof, everything got blown out of proportion. I wish Penny the best and hope that everyone involved will proceed with a little more caution and level-headedness in the future.

Prof. Smith — Just wanted to put in my hopes that this will get cleared up. I’ve never been very interested in fluid dynamics or differential equations, but the news about your paper was, and still is, very exciting for me. I’m sure something good will come of this whether or not Navier-Stokes is solved, but in the meantime, I’m hoping for the best. I know we can all think of at least one famous result that had to be retracted and repaired before it was accepted. 😉 Good luck!

Dear Michael,

A good way to determine if some one is speaking nonsense is to take n -> infinity. If, after we take an argument to the extreme, we find nonsense. We have to conclude that, in the absense of some critical point, the whole argument is nonsense.

***********

A pogo-stick theorist decides that the best way for a human to visit mars is to hold ones breath, get on a pogo-stick, and jump as high as one can.

The naysayer says, “that’s crap, human lung capacity is to small and it’s impossible to generate enough energy to bounce all the way to mars on a pogo-stick. Everybody look at me and give me attention. I want to be famous.The pogo-stick theorist replys, “screw you, you don’t have a better idea and you are clearly seeking fame. Therefore, you can’t criticize my idea“………I leave as an exercise for the reader to find the nonsense.

BTW, Peter’s opinion is clearly an opinion. I could be correct or incorrect, but it is not illogical.

one year ago Peter posted about a big theoretical claim in cosmology: No Cosmological Constant?. In a comment Sean told that the claim seemed suspicious. Indeed, it turned out to be wrong for that reason. The wrong paper still is on arXiv.

Deane: I see something to be blamed here, not in the paper that Penny has promptly widthdrawn.

At least one Fields medalist first announced a false solution to the conjecture he later became famous for solving, then a serious flaw was found, and a year or so later he solved the problem in spectacular and career-making fashion. The jury is still out on Prof. Smith’s work and failing to solve a big problem is no big deal.

See also Alain Connes remarks in one of his papers on Riemann Hypothesis. As a young student, he got advice that by publically attempting a famous problem one risks embarrassment. Connes then writes, that as he got older, he realized that the opposite approach, avoiding the embarrassment by not trying the hard problems, was equally self-defeating.

The Navier-Stokes equations won’t give up their secrets without a fight, but people putting a lot of hard work into honestly trying to solve these very diffciult problems deserve a round of applause even if the work is found to be flawed. However, important new ideas often emerge in an incomplete or flawed form and, like anything worthwhile with big payoffs, it takes a lot of effort and pain to get there. The millenium problems also attract a lot of attention and a lot of media attention though and this can be detrimental. But I hope Penny gets back to work on the problem with renewed determination.

My own encounter with Navier-Stokes was in the 90s when I began a Phd on “Turbulence in Blood”, which was to be a detailed piece of applied math/mathematical biology. Unlike the “simple” problem of turbulence in water, the density and viscosity are variable and determined by the fractional violume of red cells in suspension–which are essentially deformable bi-concave discoids– and by the blood proteins and clotting factors/thrombin and their chemistry, or what state they are in. Blood tends to be remarkably laminar until it encounters an inhomogeneity like a clot and so on. I gave up after about 3-4 weeks and was allowed to switch to something infinitely more tractable!! (Maybe it could be simulated on a supercomputer, or has been). My knowledge of hydrodynamics is mostly in statistical hydrodynamics so I have interpreted turbulence as a stochastic process or stochastic flow with dissipation. (Does anyone know how this or the well-known Kolmogorov 5/3 law fits into this millenium problem?). Anyway Navier-Stokes=damn hard. I would like to learn more though about what people like Penny are doing in this field. More knowledge of the N-S equations would be useful in applied science, engineering and biology.

Someone said: “does anyone know how Navier-Stokes are related to quantum gravity”. Both the Navier Stokes and Einstein equations could be interpreted as nonlinear, large-scale continuum approximations to deeper underlying miscrocopic descriptions. The N-S equations are continuum approximations describing the dynamics of a fluid which is really a statistical mechanical ensemble at shorter distances. The Knudsen number for example is K=L_{m}/L, where L_{m}=molecular mean free path and L=distance scale of interest. When L_{m} > L_{p}. This is very much like the Knudsen number of a fluid. The idea is that quantum gravity (maybe string theory) will take over before things get too “turbulent” as we approach the Planck scale or strong coupling regime. (I believe Briane Green had some computer graphics demonstrated something like this in his nova series). There is a lot of talk of “emergent spacetime” and “emergent gravity” bandied about but the comparison with the N-S equations could be useful here since they are clearly “emergent” too. In this sense I have never thought it made any sense to take the nonlinear Einstein equations as a starting point (as in LQG) and “quantize” them–the Einstein equations/GR should arise naturally in quantum gravity as a large-scale or continuum limit of the theory with no apriori assumptions of what the classical gravitational theory is like. At any rate, both the N-S and Einstein equations are among the most fascinating and difficult in mathematical physics, with the fun coming from the nonlinearity and the possibilities of blowups and singularities.

I am certainly not advising people not to try to attack the big unsolved problems. I think everyone should! It’s a lot more fun than nibbling around the edges of a subject. What I am advising is that before you announce a solution, make sure you know where the critical issues are and understand why your approach deals with them properly (what is it that you know that no one else did?).

Penny’s effort (a proof of the Navier-Stokes problem with a one month effort without ever having attacked the problem seriously before) cannot be seriously compared to Andrew Wiles’ (a seven year effort, which probably included at least 3 man-years of checking his own work and which followed many more years of work torwards his proof).

To believe so easily that the Navier-Stokes equation can be solved with such little effort is, frankly, a huge insult to many top mathematicians who have sweated blood to prove much less, including but not limited to Fields Medalists such as Jean Bourgain and Terence Tao.

Unfortunately, there is a prevalent misconception that doing mathematics involves thinking up lots of ideas until the right one pops into one’s head. The top mathematicians are certainly usually brilliant, but they also devote day and night to uncountably many failed calculations and proofs before they stumble onto anything worthwhile. I would even go as far to say that many if not all top mathematicians are at the top not because they are smarter than everyone else but they are fearless and relentless in their work. In other words, they succeed not because they know a good idea when they see it but because they have simply tried to work out the details of a hundred times more different calculations and proofs than anyone else.

Neither Penny nor Christine should have dismissed Penny’s proof out of hand; it was certainl possible that the right idea had somehow popped into Penny’s head. But the relative ease with which she obtained the “proof” should have been a clear signal for proceeding with extreme caution and skepticism. It should also have been a clear signal to everyone reading about their announcements for extreme caution and skepticism (it certainly was for me).

Caution and skepticism are not the same thing as dismissal. You can be extremely cautious and skeptical and yet hopeful at the same time. But those of us who have made serious attempts to attack difficult questions in mathematics know all too well how often our brilliant and ingenious solutions turn out to be mud. But sometimes it turns out to be gold, and the feeling you get when it does is like nothing else in the world.

I second what Deane has said. In fact a few days ago I showed Penny’s proof to our local NS-expert and within 5 minutes he found the same difficulty with the proof that was later pointed out in this blog by Euler. Here is the rule I follow: before making any significant work public, I show the stuff a trusted expert, and if things go wrong, it just passes as a silly question…

Part of my post (3rd paragraph) seems to have gotton mangled/lost when I pressed submit.

Here it is again in case anyone is remotely interested:)

Someone said: “does anyone know how Navier-Stokes are related to quantum gravity”. Both the Navier Stokes and Einstein equations could be interpreted as nonlinear, large-scale continuum approximations to deeper underlying microscopic descriptions. The N-S equations are continuum approximations describing the dynamics of a fluid, which is really a statistical mechanical ensemble at shorter distances. The Knudsen number for example is K=Lm/L, where Lm=molecular mean free path and L=distance scale of interest. When Lm is much less than L then the continuum approximation provided by the N-S equations holds, and so these equations (and similarly the equations of continuum mechanics) can be studied without worrying about the underlying molecular physics. When L is of the order of Lm, or L is less than Lm then you are in the realm of statistical mechanics.

Similary, the nonlinear hyperbolic Einstein vacuum equations ought to be a continuum or infrarad limit or consequence of a deeper microscopic description of spacetime structure. Of course, the Einstein-Hilbert action, coupled to additional fields, emerges as the long wavelength or low-energy effective field theory limit of string theory. (Regardless of what people here might think of string theory this remains a very interesting result). Also, the old idea of spacetime foam, which is a kind of “turbulence” occurs when Lp/ L is of the order of unity, where Lp is the Planck scale, and classical general relativity is recovered when L is much greater Lp. This is somewhat like the Knudsen number of a fluid. The idea is that a quantum gravity (maybe string theory) should take over before things get too “turbulent” as we approach the Planck scale or strong coupling regime. (I believe Briane Green had some computer graphics demonstrated something like this in his nova series).

There is a lot of talk of “emergent spacetime” and “emergent gravity” bandied about but the comparison with the N-S equations could be useful here since they are clearly “emergent” too. In this sense I have never thought it made any sense to take the nonlinear Einstein equations as a starting point (as in LQG) and “quantize” them–the Einstein equations/GR should arise naturally in quantum gravity as a large-scale or continuum limit of the theory with no apriori assumptions of what the classical gravitational theory is like. At any rate, both the N-S and Einstein equations are among the most fascinating and difficult in mathematical physics, with the fun coming from the nonlinearity and the possibilities of blowups and singularities.

If I understand this correctly, it seems that my name is being cited here because of my words of support to Penny Smith. I would like to mention that these words of support do not mean at all that I agree with her submition procedure to the arXiv. In fact, one can clearly notice from the records that I only post to the arXiv after my paper is accepted by a refereed journal. And it is clear that the responsability of the results published is of the authors. That is my procedure, and one can also verify that I am the first (or only) author in all my papers so far, so I am quite aware of the responsability envolved.

I do not wish to judge Penny Smith or anyone else’s scientific posture here. But as a scientist colleague, I understand her genuine efforts, and having recognised a mistake in a moment when all attention was over her was a very difficult situation. That is why I have given her my words of support and they stand.

Best wishes,

Christine

Deane Says:

I consider both Penny and Christine friends, but I have to say that I think they both got a little too excited by the signficance of the problem and strayed from the fundamental rules of doing research just when they needed them the most.As I read this again, it continues to be very strange. I have nothing to do with the research by Penny Smith. How can someone judge that I have “strayed from the fundamental rules of doing research”? Based on my words of support to Penny Smith?? What does it mean to consider “friends”??

Christine

Christine,

I don’t claim to have judged the situation correctly, but it does appear to me that you went to considerable effort to promote Penny’s work without the caution and skepticism that I think should have been warranted under the circumstances. I think your efforts to help Penny were well-intentioned but misguided.

Regards,

Deane

OK, after reading again Woit’s initial post, I can only conclude that Deane confused Christina Sormani with me, Christine Dantas. Well, at least I hope so.

Christine

Christine ,

Oops! I do mean Christina Sormani and not you! I should check names as carefully as I check my proofs! I am very sorry!

Deane

Dear Deane,

That is all right.

Christine