With his assumptions established, Cohen moved on to the real test. He quickly realized that this was unlike any Fourier-related problem he had worked on before. “I tried to use all my tools to demonstrate the fractal uncertainty principle, and none of them came even remotely close to working,” he said. Feeling stuck, he returned to Dyatlov and Bourgain’s one-dimensional proof of principle and tried to understand exactly how it worked. Dyatlov and Bourgain used an unusual method in their demonstration. It involved isolating one peak of a fractal-like function at a time and showing that the Fourier transform of that peak would be spread. By doing this for all the peaks and considering how the Fourier transforms would be added, they showed that the total Fourier transform could never be equal to zero often enough to form a fractal: there would not be enough holes. Jean Bourgain, seen here in 2012, was the author of more than 500 articles covering much of mathematics. Isolating each peak required building a very specific function that, when multiplied by the original fractal-like function, would extract only the peak and be close to zero for the rest. This is called the damping function and it must be perfectly tuned to work. “Building this is a challenge,” Cohen said. But he knew that if he could do it in higher dimensions, he could unlock the entire test. Cohen consulted Dyatlov about his plan to build this special feature. Before Bourgain died in late 2018, he too struggled with this issue and shared his unpublished notes with Dyatlov. Now Dyatlov shared them with Cohen. “Bourgain was a legendary analyst,” Cohen said. Reading the note was like “receiving this unfinished knowledge from him.” The notes contained exactly the clue Cohen needed. “It just blew my mind,” Cohen said. “It really solved my problem.” Before reading Bourgain’s note, Cohen had some ideas about how to build the damping function, but they were very 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 a complex analysis: the study of functions of imaginary numbers, including the square root of minus 1. This detour allowed Cohen to construct a much more flexible object, which could then be used to construct the damping function indirectly. Semyon Dyatlov, a mathematician at the Massachusetts Institute of Technology, demonstrated the one-dimensional fractal uncertainty principle in 2016. Armed with this idea, Cohen needed to find a way to create just the right version of this flexible object to produce an adequate damping function. “Building something like this that has very specific properties is not trivial. It’s delicate,” said Wilhelm Schlag of Yale University, with whom Cohen studied during his undergraduate degree. “In two dimensions, no one knew how to do that, and Alex came up with a brilliant construction of such a thing.” Cohen surprised the mathematics world when he published the proof online in May 2023. “His paper is very beautiful and made a great impression,” Schlag said. Cohen later discovered that the trick that Bourgain’s note revealed to him was not actually a secret. The method arose from a well-known theorem from the 1960s called the Beurling-Malliavin theorem. “I thought I had this special inside knowledge,” Cohen said. “Later I found out that everyone in the field already knew this strategy.” Had he known that his inside information was no secret, Cohen might have given up too soon. “I think I was very confident because I didn’t know other people had tried it,” he said. Funhouse Chaos Shortly after Cohen shared his result, other mathematicians began using it to discover new evidence about how waves behave in chaotic situations. In nature, chaos appears in systems like turbulent water and weather: situations in which objects that start close together quickly end up in drastically different places. These systems are too complex to be described mathematically. Instead, mathematicians seeking to study chaos often turn to a strange type of space that has chaos built into it, called hyperbolic space. In hyperbolic space, parallel lines diverge dramatically and move away from each other as you follow their paths. (It is the opposite of a sphere, where parallel lines converge). This means that small separations between objects can become huge over time – the telltale sign of chaos.
