With his assumptions in place, Cohen moved on to the actual proof. He quickly realized that this was unlike any Fourier-related problem he had worked on before. “I tried using all my tools to prove the fractal uncertainty principle, and none of them even came remotely close to working,” he said.
Feeling stuck, he went back to Dyatlov and Bourgain’s proof of the principle in one dimension and sought to understand exactly how it worked.
Dyatlov and Bourgain used an uncommon method in their proof. It involved isolating one peak from a fractal-like function at a time and showing that the Fourier transform of that peak would spread out. Doing this for all peaks, and considering how the Fourier transforms would add together, they proved that the total Fourier transform could never equal zero often enough to form a fractal — there wouldn’t be enough holes.
Isolating each peak required constructing a very specific function that, when multiplied by the original fractal-like function, would pull out just the peak and be close to zero everywhere else. This is called a damping function, and it needs to be perfectly tailor-made to work. “This is a challenging thing to construct,” Cohen said. But he knew that if he could do it in higher dimensions, he could unlock the entire proof.
Cohen consulted Dyatlov about his plan to construct this special function. Before Bourgain died in late 2018, he too struggled with this problem, and he shared his unpublished notes with Dyatlov. Now, Dyatlov shared them with Cohen. “Bourgain was a legendary analyst,” Cohen said. Reading the note felt like “receiving this unfinished knowledge from him.”
The notes contained exactly the hint Cohen needed. “It just blew my mind,” Cohen said. “It really unlocked the problem for me.”
Before reading Bourgain’s note, Cohen had a few ideas for how to construct the damping function, but they were highly complicated and precise, like the designs for building a house brick by brick. The note revealed an unexpected way to do it. It involved taking a detour into complex analysis — the study of functions of imaginary numbers, which include the square root of negative 1. This detour allowed Cohen to build a much more flexible object, which could then be used to construct the damping function indirectly.
Armed with this insight, Cohen then needed to find a way to create just the right version of this flexible object to produce a proper damping function. “To construct something like this that has very specific properties is highly nontrivial. It’s delicate,” said Wilhelm Schlag of Yale University, with whom Cohen studied as an undergraduate. “In two dimensions, nobody knew how to do that, and Alex came up with a brilliant construction of such a thing.”
Cohen stunned the math world when he posted the proof online in May 2023.
“His paper is very beautiful, and it made a huge impression,” Schlag said.
Later, Cohen found out that the trick revealed to him in Bourgain’s note wasn’t actually a secret. The method came from a well-known theorem from the 1960s called the Beurling-Malliavin theorem. “I thought that I had this special inside knowledge,” Cohen said. “I found out later that everyone in the field already knew about this strategy.”
Had he known that his insider tip was no secret, Cohen might have given up too soon. “I think I had a lot of confidence because I didn’t know other people had tried it,” he said.
Soon after Cohen shared his result, other mathematicians started using it to unlock new proofs about how waves behave in chaotic situations.
In nature, chaos appears in systems like turbulent water and the weather — situations where objects that start close together quickly end up in drastically different places. These systems are too complex to describe mathematically. Instead, mathematicians seeking to study chaos often turn to an odd kind of space that has chaos built in, called hyperbolic space.
In hyperbolic space, parallel lines diverge dramatically, getting farther from each other as you follow their paths. (It’s the opposite of a sphere, where parallel lines converge.) This means that small separations between objects can become huge down the line — the telltale sign of chaos.