The original version of this story appeared in Quanta Magazine.
The best perk of Alberto Maspero’s job, he says, is the view from his window. Situated on a hill above the ancient port city of Trieste, Italy, his office at the International School for Advanced Studies overlooks a broad bay at the northern tip of the Adriatic Sea. “It’s very inspiring,” the mathematician said. “For sure the most beautiful view I’ve ever had.”
Italians call Trieste la città della bora, after its famed “bora” wind, which blows erratically down off the Alps and over the city. When the bora is strong enough, it drives the waves into reverse. Instead of breaking against the docks, they stream away from the city, back toward the open sea.
But they never actually get there. Watching from his window on these gusty days, Maspero can see the retreating waves slowly disperse as they exit the port, eventually giving way to a calm, still surface.
The equations that mathematicians use to study the flow of water and other fluids—which Leonhard Euler first wrote down nearly 300 years ago—look simple enough. If you know the location and velocity of each droplet of water, and simplify the math by assuming there’s no internal friction, or viscosity, then solving Euler’s equations will allow you to predict how the water will evolve over any time period. The rich menagerie of phenomena we see in the world’s oceans—tsunamis, whirlpools, riptides—are all solutions to Euler’s equations.
But the equations are usually impossible to solve. Even one of the simplest and most common kinds of solutions—one that describes a steady train of gently rolling waves—is a mathematical nightmare to extract from Euler’s equations. Until about 30 years ago, the bulk of what we knew about these waves came only from a mix of real-world observations and guesswork. For the most part, proofs seemed like a fantasy.
“Before starting math, I thought water waves were something very understood—not a problem at all,” said Paolo Ventura, a postdoctoral fellow at the Swiss Federal Institute of Technology Lausanne and Maspero’s former graduate student. “But in reality, they are just strange.”