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Business News/ Opinion / Columns/  Stokes takes Iyer to the fray
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Stokes takes Iyer to the fray

The Navier-Stokes equations - partial differential equations all - describe how a fluid moves through space, though our understanding here has limits

The motion of fluids - water, or a gas - through the air is very far from intuitive. (Photo: courtesy Ranjani Shettar and Talwar Gallery, New York I New Delhi)Premium
The motion of fluids - water, or a gas - through the air is very far from intuitive. (Photo: courtesy Ranjani Shettar and Talwar Gallery, New York I New Delhi)

Spend a few moments thinking about the way things move through the air. Because ... well, why not? Think of a plane, a cricket ball, a burst of water from a hose, a feather, whatever. Each one moves quite differently, you'll agree. The feather is more susceptible to air currents than the cricket ball. The plane relies on certain laws of physics to rise off the ground and stay in the air; other laws of physics dictate how the water disperses into drops. Of course laws of physics also explain the behaviour of the cricket ball and the feather.

What's the point here? Well, just that there are physical laws that govern all these motions and we can learn about them, even if they don't all seem straightforward or intuitive. For example, a cricket ball can drift or swing in the air, depending on how the bowler delivers it. What explains that motion? What gives a plane that lift into the air? How does the water break up into drops and how does each drop move?

In fact, the motion of fluids - water, or a gas - through the air is very far from intuitive. Bowl a cricket ball and you can be pretty sure where it will land - that's the foundation on which umpire referral techniques like Hawkeye work. But throw some water into the air, or stand atop a waterfall, and it's near impossible to predict with similar accuracy where the water will land - or really, where all the drops will land.

But there is a law that governs this motion of water. Actually, it is a system of equations and it's about much more than water. The Navier-Stokes equations - partial differential equations all - describe how a fluid moves through space, though our understanding here has limits. Solving the Navier-Stokes equations for specific cases of fluid flow often involves turbulence, which is chaotic and unpredictable. This makes it hard to design equipment for fluid flow. In a given situation, it is impossible to prove that fluid will flow without turbulence, and just as impossible to prove the opposite, that it will always be turbulent.

Which is why solving the Navier-Stokes equations is one of the great unsolved problems in mathematics. So much so that in 2000, the Clay Mathematics Institute listed it as one of its seven "Millenium Prize" problems, offering a million dollars each for solutions.

The Clay Institute listing is this sentence of mathematics-speak:

"In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations."

Never mind what that means. The task for mathematicians seeking to win that million dollars is to either prove or disprove this statement. Many have worked on Navier-Stokes, but without luck.

Until now.

A mathematician called Benjamin Stokes grew up thinking idly that with his name, he should be the one to solve the Navier-Stokes problem and win the million dollars. For years, this was just a nice dream. But as a graduate student and occasional cricketer at Durham University in England, he began working on Navier-Stokes more seriously, and eventually made progress towards a solution.

The exact mathematics involved is way over my head, but let me try to offer a flavour of what Stokes went through.

His first insight was that there must be a deep connection between Navier-Stokes and certain "elliptic" curves. The catalyst for this was the kinds of trajectories cricket balls trace as they fly through the air, moving from a height to the ground and swinging from side to side. Seen from any angle, the ball follows not a straight line, but a curve.

Stokes worked for some years on curves like these, using their subtleties in different ways. Intuitively, he felt this was the path to the Navier-Stokes holy grail, but he kept running into walls. (Mathematically speaking.) His colleagues hinted that he should give up his quest. In an interview, Stokes responded thus: "I've had so many people say stuff to me. I meet them, have a chat for five minutes and they think they can say what they like. I used to laugh it off, but now I think 'Why do you think you can say that to me? You don't know me.'"

In other words, the hints only got Stokes more determined to find his solution.

He had a breakthrough in 2018, while he was actually struggling with an obscure mathematical artifact known as the Hale-Ali Conjecture. It addresses so-called "happy" numbers, which are easily found: Take any number, square each of its digits and add the squares. Do the same with the result. Repeat. If you arrive at 1, you started with a happy number. (See my column)

Stripped to its essence, the Hale-Ali Conjecture is that happy numbers are not "natural". Now this is a long-running conundrum, but only in certain circles. In other circles, the Conjecture has long been considered false. Whichever it is, Stokes was working with certain happy numbers in 2018 and almost inevitably, found that he had to deal with Hale-Ali.

This was a significant obstacle to battle and overcome. And when Stokes thought he was done with Hale-Ali, he came up against a number of researchers who would not accept his efforts. It was not necessarily that they believed the Hale-Ali Conjecture, but that according to them, Stokes' methods were wrong. They amounted to "brute force", some remarked, and there is no place in mathematics for brute force. Some drew parallels to the solution to the Four-Colour Map Theorem that was found several years ago: in the end, the proof rested on several hundred individual cases that had to be checked by a computer, not by pure human logic. It may have solved an apparently intractable problem, but many mathematicians thought that resort to a computer amounted to inelegant brute force.

Still, Stokes eventually was able to show that his methods were not only justified, but the only ones that worked. And overcoming the Hale-Ali obstacle in the way he did was a good thing. For it gave him the tools he needed to make substantial progress towards the larger goal of Navier-Stokes.

He took the final steps over the last couple of years in collaboration with another PhD student at Durham, Narayanan Venkataratnam Iyer. More recently, Stokes moved from Durham to the West Indies, so the collaboration with Iyer thus had to continue long-distance.

Still, that didn't hinder them from getting, finally, to the broad root of the problem. Stokes worked on his beloved elliptic curves, while Iyer applied his results to particular kinds of happy numbers. Working together thus, they were able to find profound links between the early insights into elliptic curves, the Hale-Ali Conjecture and the phenomenon of happy numbers. And once those connections were spelled out, the deep connection Stokes had long suspected - between elliptic curves and the Navier-Stokes equations - became obvious. At least, mathematically obvious.

Thus it is that today, we have the NV Iyer-Stokes solution to the Navier-Stokes problem. Just remember, today is April 1.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun

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Published: 01 Apr 2022, 02:05 PM IST
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