Millennium Problem number four is my favourite, hands down. I’m probably not supposed to be biased but when you have an equation tattooed on your body the rules change. The Navier-Stokes equations describe the flow of every fluid you can possibly think of: rivers, water from a tap, waves, wind, air flow around an aeroplane, ice in glaciers, ketchup, honey dripping off a spoon, blood in your body… I could go on forever.
The fact that these equations can do all of this is great – it shows that things in nature behave similarly, and we may actually understand some of it. But, there is a downside. To be able to describe such a variety of different fluids all at once, these equations are super-complex. I’m talking the plot of Inception complex. And just like no-one really understands Inception (no matter what they might tell you), mathematicians don’t understand all the little intricacies of the Navier-Stokes equations.
The easiest way to think about it is using what we call a singularity. It might sound complicated but I’m going to explain it step-by-step so stick with me. Start with a number, let’s say 2. Divide 1 by 2 and you get 1/2. Now take a smaller number, say 1/4 and divide 1 by it – ‘dividing by a fraction is the same as multiplying by it upside down’ (sorry, I’m just hearing the voice of my primary school teacher). The answer is 4 though. Now take a smaller number, say 0.1 and divide 1 by it. You get 10. Take a smaller number 0.01 and divide 1 by it, you get 100. Continue this: divide 1 by smaller and smaller and smaller numbers and you will get a bigger and bigger and bigger answer. So what happens when you divide 1 by 0? Maths breaks is the answer, but we can think of it as infinity or in our case a singularity.
Singularities occur in nature too with perhaps the most famous example being a black hole. These guys are so complicated that even Stephen Hawking struggles to understand what’s going on, which gives you an idea of why singularities are such a nuisance. Going back to the Navier-Stokes equations and the motion of fluids, my favourite example involves bubbles. Let’s do a little experiment. Take two circular pieces of wire and holding them close together dip them in soapy water. Imagine those little bottles of bubbles you used to get as a kid, and the plunger thing with the circle bit on the end… that. Well, two of them close together. The idea is that when you hold them close together, dip them in soap and then take them out a bubble will form between the two. It should form a cylinder shape – like the centre of a toilet roll. As you move the two circular wires apart the bubble will stretch and grow taller (think toilet roll to kitchen roll). You can keep moving the wires further and further apart and the bubble gets longer and longer and then POP! You’ve moved them too far apart and the bubble breaks.
Thinking about this mathematically is a nightmare. It makes sense at first, the wires move further apart and the size of the bubble grows. Increase the distance between the wires and the bubble size increases – a nice simple mathematical relationship. Until you reach the point where the bubble pops. At this instant the increase in the distance between the wires causes a sudden and incredibly fast decrease in the bubble size to zero. It’s so fast you can call it infinite. This is your singularity. The video below shows a great example of the experiment I’ve just described and shows the moment where the bubble size suddenly goes to zero.
As I said above the Navier-Stokes equations model the flow of any and every fluid – this means they describe the bubble popping madness we’ve just looked at and most importantly the singularity. We don’t know how or why or what is going on with these guys – again, think of black holes – and that is the Millennium Problem. Can we improve our understanding of these equations? In Lord of the Rings it was one ring to rule them all, in the maths of fluids the Navier-Stokes equations are your ruler… now bow down and make some bubbles.
You can listen to me interviewing Professor Keith Moffat about the problem here.
I’ve written a series of articles on each of the 7 Millennium Problems which can be found here.
By Jacob Aron and Katia Moskvitch
Mathematics is a universal language. Even so, a Kazakh mathematician’s claim to have solved a problem worth a million dollars is proving hard to evaluate – in part because it is not written in English.
Mukhtarbay Otelbayev of the Eurasian National University in Astana, Kazakhstan, says he has proved the Navier-Stokes existence and smoothness problem, which concerns equations that are used to model fluids – from airflow over a plane’s wing to the crashing of a tsunami. The equations work, but there is no proof that solutions exist for all possible situations, and won’t sometimes “blow up”, producing unrealistic answers.
In 2000, the Clay Mathematics Institute, now in Providence, Rhode Island, named this one of seven Millennium Prize problems offering $1 million to anyone who could devise a proof.
Otelbayev claims to have done just that in a paper published in the Mathematical Journal, also based in Kazakhstan. “I worked on the problem on and off, for 30 years,” he told New Scientist, in Russian – he does not speak English.
Mathematical Babel fish
However, the combination of the Russian text and the specialist knowledge needed to understand the Navier-Stokes equations means the international mathematical community, which usually communicates in English, is having difficulty evaluating it. Although mathematics is expressed through universal symbols, mathematics papers also contain large amounts of explanatory text.
“Over the years there have been several alleged solutions to the Navier-Stokes problem that turned out to be wrong,” says Charles Fefferman of Princeton University, who wrote the official formulation of the problem for Clay. “Since I don’t speak Russian and the paper is not yet translated, I’m afraid I can’t say more right now.”
Otelbayev is a professional, so mathematicians are paying more attention to his proof than is typical for amateur efforts to solve Millennium Prize problems, which are regularly posted online.
The Russian-speaking Misha Wolfson, a computer scientist and chemist at the Massachusetts Institute of Technology is attempting to spark an online, group effort to translate the paper. “While my grasp on the math is good enough to enable translation up to this point, I am not qualified to say anything about whether or not the solution is any good,” he says.
Stephen Montgomery-Smith of the University of Missouri in Columbia, who is working with Russian colleagues to study the paper, is hopeful.”What I have read so far does seem valid,” he says “but I don’t feel that I have yet got to the heart of the proof.”
Otelbayev says that three colleagues in Kazakhstan and another in Russia agree that the proof is correct.
Burden of proof
Understandably, a high burden of proof is required to claim the $1 million prize. Clay’s rules say the solution must be published in a journal of “worldwide repute” and remain unchallenged for two years before it can even be considered. Nick Woodhouse, president of the Clay Mathematics Institute, declined to comment on Otelbayev’s proof.
“It is currently being translated by my students, and will be available soon,” says Otelbayev. He says that he will publish it again once it is translated into English – initially in a second Kazakh journal, and then perhaps abroad.
To date, only one Millennium Prize problem has been officially solved. In 2002, Grigori Perelman proved the Poincaré conjecture, but later withdrew from the mathematical community and refused the $1 million prize.
A possible solution for another problem, known as P vs NP, caught mathematicians’ attentions in 2010, but later proved to be flawed. Whether Otelbayev’s proof will share the same fate remains to be seen.