Master of chaos wins $3M math prize for ‘blowing up’ equations

Frank Merle is used to facing a messy world. He works on the mathematics of highly nonlinear systems, those that respond dramatically and unpredictably to even the smallest changes. It’s the same calculation that explains how, under the right conditions, the atmosphere above an arid plain can produce a raging tornado.

A linear equation is something like Yes = 2xwhich indicates that the value of Yes double every time you double the value of x. But most equations are much more sensitive to changes in their inputs. A highly nonlinear system is defined by equations that can go from zero to infinity almost out of nowhere. Determining whether a system of equations can exhibit this type of extreme behavior, called a “singularity” or “explosion,” is a difficult task for mathematicians.

Merle had enormous success mastering these explosions in the equations describing lasers, fluids and quantum mechanics. His trick is to embrace the non-linear. While most researchers before him treated these phenomena cautiously by making small modifications to a linear, well-behaved world, he focused them, directly studying their mathematical consequences. “I have a slightly different world view,” he says. “I see the world as a more catastrophic place to live.”

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By engaging in chaos, Merle discovers simplicity. Much of his work focuses on special structures, called “solitons,” that persist amid the chaos of nonlinear systems. Solitons are able to maintain their form and energy while moving through realms where the most complicated mathematics reign like a single rogue wave crossing an entire vast swirling ocean completely untouched. Merle believes that all nonlinear systems can be treated as a collection of solitons joined together – a chaos that hides simplicity.

Merle was today awarded this year’s Breakthrough Prize in Mathematics for his achievements. The prize comes with a $3 million reward. Scientific American spoke with Merle about how he managed to tame some of nature’s most tangled sets of equations.

[An edited transcript of the interview follows.]

What does this award mean to you?

It was a shock: it took me a while to recover. It’s a great honor. And it’s exciting, because when I discovered this new way of looking at these problems, most people weren’t convinced that I could produce anything interesting. Then one problem came up, and then another, so of course all this work now gets a lot of recognition.

What was your “new way of seeing problems” in nonlinear dynamics?

I was only focusing on the nonlinear structure. Most previous work has taken something we understand – linear things – and pushed it slightly towards the non-linear. But my starting point has never been linear structure; that was the non-linear thing.

And that led you to bring solitons to the forefront.

Yes, because solitons are a completely non-linear concept. A soliton is a special solution to nonlinear equations, such as fluid equations, that does not send energy to infinity: it keeps all of its energy contained and maintains the same shape.

When you look at physical quantities in nonlinear systems, they appear to oscillate and change chaotically. But if you look long enough, an emergent structure emerges that doesn’t depend so much on how things started. This emerging structure is the soliton. From a mathematical point of view, it is not initially clear why this will appear, and yet, somehow, it does.

Solitons seem much simpler than the crazy, chaotic behavior of nonlinear systems. Yet you think that the behavior of these systems comes down, one way or another, to solitons.

Yes, a family of interacting solitons. This is called the “soliton resolution conjecture”.

This has been a widespread belief since the 1970s, but at the time, people didn’t really see the nature of this phenomenon – why exactly it had to be true. And mathematically, there is no way to solve this problem except for a few specific types of nonlinear equations.

But the idea is pure beauty. You’re looking at a very complicated situation – your problem is chaotic, with infinite parameters – but in the end everything becomes simple, with a finite number of parameters that you can find and calculate.

The equation you discover at the end may be even simpler than you think. There is a simplicity that is very hidden, very difficult to see even through experience, but it appears. There’s a bit of magic in that.

You used solitons to study explosion, the phenomenon by which nonlinear equations break down and suddenly become infinite. Why is this important?

For different nonlinear equations, explosion can be good or bad: you either want an explosion or you don’t. But knowing how it works is important in any case. In the equation of focusing a laser, you want a blast because you want to focus your laser as much as possible.

And you have proven that laser equations can explode under certain conditions. Does this mean the laser actually becomes focused to infinity?

Not really. The math equation says it goes to infinity, but in reality it doesn’t. It will just become very focused and stay very focused for a long time.

But the equation is only an approximation. In fact, throughout physics, equations are always approximations. Different physics appear when the laser is very focused: sometimes known physics and sometimes completely unknown physics.

You also worked on explosion for fluid equations. How is this different?

In fluid equations, you want to avoid explosions because they are related to turbulence. But in real life, there is turbulence everywhere, so you have to at least understand it.

I worked on compressible fluids, governed by the Navier-Stokes equation. We already knew that a simplified version of the equation, without any friction, could produce singularities.

But the question was whether friction could at least slow down the formation of singularities or [even] stop it. Our result was to prove that it didn’t stop it, that friction didn’t stop the explosion.

Isn’t the Navier-Stokes explosion one of the Clay Mathematics Institute Millennium Prize Problems? Does that mean solving it is worth $1 million?

Clay’s problem is the same question for incompressible fluids. This was for compressible fluids: compressibility helps you in a certain sense. Clay’s problem therefore remains open.

You also worked on the nonlinear version of the Schrödinger equation governing quantum mechanics. What was the breakthrough there?

You have a linear part of the Schrödinger equation and a nonlinear part. Usually the linear term is the most important, but sometimes – the so-called “supercritical case” – the nonlinear term can have its own craziness.

Everyone – even me – thought for a long time that the solutions to the Schrödinger equation would never explode, because any singularity would disperse after a certain time. For a while we tried to prove it.

In mathematics, sometimes you almost prove one thing in several different ways, and each time there is a key point missing, something you can’t tame. Maybe you think it’s small.

But after a while, you get the feeling that maybe this is an indication that the opposite might be true. And this little piece turns out to be dramatic, the key element of what becomes your evidence for the contrary claim. This is what happened in this case. So the mathematical process itself is often non-linear too, at least for me.