Good News Everyone! I learned just now from Good Math, Bad Math there may well be a successful proof of yet another of the great Millennium problems. This one pertains to solutions of the Navier-Stokes equations which are of central importance in fluid dynamics, for example. (The figure to the left -click for larger- is from a computer simulation, by Greg Ashford, of the airflow around an aircraft wing (in cross section) and the foil that it uses to change the overall shape of the wing for maneouvering. I got it from a site linked here.)
Penelope Smith, at Lehigh University has presented a preprint with her proof, which uses a set of techniques that she has been developing for a number of years. Christina Sormani has a website which describes aspects of the proof, and Smith’s website is here. Read also the commentary at Good Math, Bad Math
As usual, there will be a long process of independent checking and re-checking her results before the mathematics community declares this problem solved (should her solution pass those tets). So there is a bit of a wait to follow. Don’t be under any illusions about the importance of this problem. It is an incredibly difficult system of equations to solve, while at the same time they arise as governing the physics of so many situations (pretty much anything inolving the flow of fluids: water, air, etc), so they show up in fields as diverse as the study of the weather, the structure of the interior of a star, the design of more aerodynamic (and hence more fuel-efficient) cars, etc.
I can’t help but wonder if the Clay Institute, which will have do dole out another million dollars in such a short time, had budgeted for such a rapid rate of mathematical progress…
-cvj
Red,
My reason for the comment was twofold 1) I have worked on this problem on and off for nearly 35 years and have a pretty good insight into why the pattern “appears” random which directly relates to differences between addition and multiplication. and 2) this comment was made to me directly by a stunningly brilliant but no longer practicing mathematican who must remain anonymous at this time. He came up with this analysis after looking at the problem for about 10 minutes.
If you have something related to this I would encourage you to write it up. My sense is that there is a deep connection between the non-commutative aspects of mixing these operations and the non-commutative aspects of “mixing” space and time. But that is at this point just a wild speculation on my part.
Regards,
Elliot
Elliot stated: “The Collatz conjecture is connected to the deep differences between addition and multiplication.”
I find this to be a remarkable statement in itself because:
(i) it shows insight into the actual 3x+1 problem
(ii) I’ve not seen it expressed that way before
Is this actually well known or just a guess? The reason I ask is that I have an explicit analysis of this aspect but I’ve never bothered to write it up more formally. Should I? I’m not claiming this constitutes a proof at all.
My original motivation was a kind of “No Go” theorem for Wolfram’s claim that “complex” patterns (and by extension, physical phenomena) arise from “simple” rules. He uses 3x+1 as an exampe in his NKS book. However, when examined more closely one sees that addition is responsible for an amazing kind of ‘mix master’ effect. Thus, the 3x+1 rule can hardly be called “simple”, even though it looks simple.
Red
Since Erdos said of the problem “mathematics is not ready for such problems”, perhaps the person who solve it can claim an honorary low Erdos number.
Aaron,
The Collatz conjecture is connected to the deep differences between addition and multiplication.
While conjectures in integer arithmetic may be plentiful. The devilishly simple nature of this problem without current resolution makes it attractive in its own way.
I agree that it probably won’t make it on to the Clay prize list however. But whoever cracks it will at a minimum become a folk hero.
Elliot
Pingback: Giant Still Standing - Asymptotia
Wow!
-cvj
Actually I meant peek.
I just checked the website of the preprint, thinking that I might want to take a peep as I’m in kind of fluid (water) mechanics business and this is what I saw:
This paper is being withdrawn by the author due a serious flaw.
So what’s the deal?
It is rather fascinating though…. Thanks Chris W., and Aaron.
-cvj
Unless to Collatz conjecture is shown to be related to something deep, I can’t imagine it ever becoming a Clay problem. Conjectures in integer arithmetic are a dime a dozen, really.
Wikipedia has covered it in detail.
Any info on it? I do not know this…. Might have a go at it on the bus…. 😉
-cvj
I’m hoping that if a couple of the Clay problems are solved the “Collatz conjecture” will be added to the list. It is devastatingly simple to state but very difficult to prove. A number theory problem that will make you tear your hair out.
Cheers
Elliot
Oh… he did? Hey, someone tell them I can use the money…they should not keep it since it will only confuse the budget quite a bit. I’ll set up a little institute here. I’ll call it whatever they want me to call it. 😉
-cvj
Note that the Clay institute actually did not have to dole out the previous million dollar as Perelman refused it. So I would rather say that their budget is quite positive these days 😉
Christina Sormani’s summary discusses Penny Smith’s work on the Einstein field equation about as much as Navier-Stokes, since they are so interrelated. Note this: