Along with his assumptions in place, Cohen moved on to the precise proof. He shortly realized that this was in contrast to any Fourier-related downside he had labored on earlier than. “I attempted utilizing all my instruments to show the fractal uncertainty precept, and none of them even got here remotely near working,” he mentioned.
Feeling caught, he went again to Dyatlov and Bourgain’s proof of the precept in a single dimension and sought to know precisely the way it labored.
Dyatlov and Bourgain used an unusual technique of their proof. It concerned isolating one peak from a fractal-like operate at a time and exhibiting that the Fourier remodel of that peak would unfold out. Doing this for all peaks, and contemplating how the Fourier transforms would add collectively, they proved that the overall Fourier remodel may by no means equal zero usually sufficient to type a fractal — there wouldn’t be sufficient holes.
Isolating every peak required setting up a really particular operate that, when multiplied by the unique fractal-like operate, would pull out simply the height and be near zero in all places else. That is referred to as a damping operate, and it must be completely tailored to work. “It is a difficult factor to assemble,” Cohen mentioned. However he knew that if he may do it in greater dimensions, he may unlock your complete proof.
Cohen consulted Dyatlov about his plan to assemble this particular operate. Earlier than Bourgain died in late 2018, he too struggled with this downside, and he shared his unpublished notes with Dyatlov. Now, Dyatlov shared them with Cohen. “Bourgain was a legendary analyst,” Cohen mentioned. Studying the be aware felt like “receiving this unfinished information from him.”
The notes contained precisely the trace Cohen wanted. “It simply blew my thoughts,” Cohen mentioned. “It actually unlocked the issue for me.”
Earlier than studying Bourgain’s be aware, Cohen had just a few concepts for assemble the damping operate, however they had been extremely difficult and exact, just like the designs for constructing a home brick by brick. The be aware revealed an sudden solution to do it. It concerned taking a detour into complicated evaluation — the examine of features of imaginary numbers, which embody the sq. root of detrimental 1. This detour allowed Cohen to construct a way more versatile object, which may then be used to assemble the damping operate not directly.
Armed with this perception, Cohen then wanted to discover a solution to create simply the fitting model of this versatile object to provide a correct damping operate. “To assemble one thing like this that has very particular properties is very nontrivial. It’s delicate,” mentioned Wilhelm Schlag of Yale College, with whom Cohen studied as an undergraduate. “In two dimensions, no one knew how to try this, and Alex got here up with a superb building of such a factor.”
Cohen shocked the mathematics world when he posted the proof online in Could 2023.
“His paper may be very stunning, and it made an enormous impression,” Schlag mentioned.
Later, Cohen came upon that the trick revealed to him in Bourgain’s be aware wasn’t truly a secret. The strategy got here from a widely known theorem from the Sixties referred to as the Beurling-Malliavin theorem. “I assumed that I had this particular inside information,” Cohen mentioned. “I came upon later that everybody within the subject already knew about this technique.”
Had he recognized that his insider tip was no secret, Cohen might need given up too quickly. “I feel I had plenty of confidence as a result of I didn’t know different individuals had tried it,” he mentioned.
Funhouse Chaos
Quickly after Cohen shared his end result, different mathematicians began utilizing it to unlock new proofs about how waves behave in chaotic conditions.
In nature, chaos seems in programs like turbulent water and the climate — conditions the place objects that begin shut collectively shortly find yourself in drastically completely different locations. These programs are too complicated to explain mathematically. As a substitute, mathematicians in search of to review chaos usually flip to an odd type of house that has chaos in-built, referred to as hyperbolic house.
In hyperbolic house, parallel strains diverge dramatically, getting farther from one another as you comply with their paths. (It’s the other of a sphere, the place parallel strains converge.) Which means that small separations between objects can turn into big down the road — the telltale signal of chaos.
