Finding Beauty at the Piano and Elegance in Mathematics

Sylvain Carpentier, who is completing his tenure as a Simons Junior Fellow, has discovered new ways to promote the beauty of mathematics — both in its own right and as a practical tool for understanding the physical world.

Sylvain Carpentier in front of a keyboard — SSOF Junior Fellow Sylvain Carpentier. Image courtesy of Sylvain Carpentier.

When he is not studying complex equations, you’re likely to find Simons Society of Fellows Junior Fellow Sylvain Carpentier at the piano. The accomplished pianist was born into a family of musicians and has performed, most notably, with the Boston Pops.

Carpentier is an accomplished mathematician as well, and he speaks eloquently about the connections between math and music. In his work, he focuses on the algebraic roots of partial differential equations (PDEs), equations that represent the relationship between at least two variables. PDEs are usually very difficult, if not impossible, to solve, but Carpentier argues that efforts to do so can help us better understand the real-world dynamics of our universe — from unusual wave activity in the ocean to how light beams diffract at very high intensities.

Carpentier is now wrapping up his final year as a Simons Junior Fellow as a postdoctoral researcher at Columbia University with Andrei Okounkov, himself a Simons Senior Fellow. Carpentier earned his master’s degree in mathematics from the Université Paris-Sud in France and a doctorate at the Massachusetts Institute of Technology.

Carpentier and I recently spoke about math, music, what he’s gained from his fellowship and what comes next. Our conversation has been edited for clarity.

You study partial differential equations. What are they and why are they important?

A partial differential equation, or PDE, finds a relationship between the rates of change of a physical quantity with respect to several variables. That quantity must depend on two or more variables — say, space and time. By contrast, any differential equation with just one variable is called an ordinary differential equation.

PDEs are important for representing natural phenomena, such as how ocean waves rise and fall or how beams of light travel through space. These types of phenomena involve at least two variables. Ordinary differential equations, by comparison, don’t really help us understand the physical world, because it’s rare to find physical phenomena that only involve one variable.

Almost all PDEs are hard to solve, because the interconnecting relationships between variables that result in natural phenomena are complex and hard to tease out. Despite these difficulties, some PDEs do help us better understand the natural world. For example, a class of PDEs called Navier-Stokes equations describe the movements of viscous fluids. And the heat equations, as you’d expect, describe how heat moves through a given space.

This doesn’t mean that we’ve figured out absolutely everything about how viscous fluids or heat operate. A PDE solution might work in some contexts and not others; this is why PDEs are generally so hard to solve. But there’s a lot of interest in solving and understanding as many PDEs as we can. For example, anyone who can solve some variants of the Navier-Stokes equations will receive 1 million dollars!

How does your own work involve the study of PDEs?

One area of mathematics I’m interested in is called nonlinear optics. With light beams of regular intensity — like those coming from the lightbulbs you find in your home — we know a significant amount about the dynamics of how the bulb’s beam will naturally spread. This is due to our broad understanding of principles like diffraction.

But with more intense light, like those that can illuminate a football field even from 100 miles away, typical diffraction patterns can be altered by nonlinear effects that can’t be predicted in advance. That’s the broad problem, and I’m especially interested in understanding light that maintains its form over long distances, even at high intensity. One PDE that’s helpful for understanding this dynamic is called the nonlinear Schrödinger equation, but — given that it’s a PDE — it has not been entirely solved for every situation. I’m working on that, along with colleagues around the world.

Two equations. The first one is (∂_t)u = (∂_t^3)u + 6u(∂_x)u (Korteweg de Vries). The second equation is i(∂_t)Φ = -1/2(∂_x^2)Φ + κ|Φ|^2Φ (nonlinear Schrödinger) — Two key examples of integrable systems: the Korteweg de Vries equation and the nonlinear Schrödinger equation. Credit: Sylvain Carpentier

Are there other applications of this particular PDE?

Yes! It takes a bit of backstory to explain.

For centuries sailors have mentioned that some waves just spring up out of nowhere in the ocean and are much higher and more intense than the waves around them. This didn’t fit any known models of how waves should rise and fall. So, just like we understand the basics of how light travels, except for when that light reaches high intensity, we understand a lot about waves — except for these mysterious, suddenly forming, intense waves.

For a long time, scientists dismissed these waves because we didn’t have a theory that explained them and nobody besides sailors had ever seen them. We now know that these so-called rogue waves really do exist — thanks to direct observation in 1995 off the coast of Norway. Since then, mathematicians have worked to develop PDEs that explain the dynamics of rogue waves. It turns out that the same nonlinear Schrödinger equation that helps reduce the mysteries of nonlinear optics also helps us better understand the workings of rogue waves. Note that I said, “better understand,” not “fully understand”! PDEs are hard to solve entirely.

Are there any PDEs that can be solved entirely?

There’s a subclass of PDEs that indeed can be solved, called integrable PDEs, though technically these solvable equations are part of a larger, virtually unsolvable PDE. However, it is nice to know that at least the integrable components have an exact solution!

One example of an integrable PDE that also relates to wave dynamics is known as the Korteweg-de Vries equation. This equation can explain exactly why shallow waves called solitons can maintain their shape over long distances. The soliton equation is one part of the Navier-Stokes equation that we mentioned earlier, which in turn is part of a much larger set of PDEs.

Though integrable PDEs are rare, they actually are quite important as a first approximation for how to solve more complex problems. They can help to develop a test that distinguishes between what is and is not an integrable PDE.

Of course, it’s good to try to solve equations, and that’s part of what I do. But I’m actually more interested in the algebraic roots, the skeleton that underlies the solution to any equation that we’re working to solve. Just as an integrable PDE can be a guide toward solving harder problems, an understanding of an equation’s roots can help solve deeper challenges as well.

You are about to wrap up your Junior Fellowship with the Simons Society of Fellows. What do you think about the experience, and where might you go next?

The best thing about the fellowship is how multidisciplinary it is. I’ve heard fascinating talks from people in fields very different from mine and have been inspired by the brilliance of the other junior fellows.

I’m also deeply grateful to the Simons Foundation for its generous support during the pandemic. Because my fellowship was set to end soon after the pandemic began, the foundation provided for another year of support. That additional support was priceless.

As to what happens next: All I can say right now is that I’m exploring offers from various institutions, both in the U.S. and abroad. My passion for research is as strong as ever, and I’m also looking forward to supervising student work. Even though I can’t say yet where I’ll be come September, I’m looking forward to whatever opportunities come my way.

And of course, every mathematician dreams of finding an equation that nobody else has ever seen before, and then that equation is named after them!

Excerpt from Frédéric Chopin’s Andante Spianato et Grande Polonaise Brillante for piano and orchestra, performed by Sylvain Carpentier.

Finally, we must discuss your accomplished piano career. Many of your concerts are on YouTube, and you’ve performed in very prestigious concert halls. How does your music relate to your math, or is it another dimension of your personality entirely?

There are definitely connections between math and music. Musical harmony, for example, derives from precise ratios of sound frequencies.

With music, my aim is to find beauty in a way that transcends the literal notes that are written on sheet music. Whenever I perform, I hope that every member of the audience understands what I’m feeling as I play the notes. And similarly, there’s a lot more room for creativity and beauty in mathematics than people sometimes think — the progress we’ve made with harnessing PDEs to explain the natural and physical world is a clear example of that creativity and beauty in action.