Graduate Student Proves the Fractal Uncertainty Principle | Quanta Magazine

Graduate Student Proves a Quantum Uncertainty Principle for Fractals The math, which combines chaos, quantum theory, and infinitely complex fractal structures, has been called a “foundational result.” Ada Zejun Shen/Quanta Magazine Introduction At the quantum scale, tiny particles behave in bizarre ways. One reason for this is the uncertainty principle, which says that the more you know about where a quantum particle is, the less you can know about how fast it’s moving, and vice versa. Recently, this rule got a rare upgrade. The new uncertainty principle relates to fractals, shapes that remain equally complex no matter how much you zoom in on them. Around a decade ago, Semyon Dyatlov, a mathematician at the Massachusetts Institute of Technology, was studying whether quantum particles behave differently than ordinary particles when put into the same chaotic situations. Sometimes, an object moving chaotically can become trapped into following a fractal-like path forever. Could quantum particles do the same? Quantum particles tend to spread out like waves, which blurs their exact location. To figure out whether quantum particles blur too much to take on these intricate trapped paths, Dyatlov needed a new uncertainty principle — one that could tackle fractals. In 2016, with key ideas from Jean Bourgain — a renowned mathematician who died shortly after this work — Dyatlov proved the fractal uncertainty principle for one-dimensional fractals, which look like jagged lines. These lines can represent the paths taken by objects moving in two dimensions, like balls traveling around a billiard table. That fall, Dyatlov and Bourgain gathered mathematicians from around the world in New Jersey for a workshop, hoping to extend the proof to higher dimensions. An extended proof could be used to study the three-dimensional world and would become a universal mathematical tool in its own right. But the task proved too difficult. By the end of the workshop, “nobody really believed that it could be done,” said one of the attendees, Frédéric Naud, a mathematician from Sorbonne University. It wasn’t until years later that Alex Cohen, while a doctoral student at MIT, finally made a breakthrough. In a paper published in 2025 in the Annals of Mathematics, widely considered to be the field’s top journal, he extended the fractal uncertainty principle to all higher dimensions. The result became Cohen’s thesis and earned him an assistant professorship at New York University at the age of 25. The fractal uncertainty principle is “a foundational result,” said Peter Sarnak of the Institute for Advanced Study — “a pretty remarkable achievement for a guy in his thesis.” Already, this principle has revealed a new deep way that quantum particles differ from classical ones. Pinball Wizard Though the uncertainty principle may seem strange in the context of particles, it can also crop up in less mysterious forms. The briefer a sound, the less sure you can be about the tones that make it up. A short radar pulse can accurately locate a submarine, but it takes a longer signal to determine where it’s moving. All these uncertainty principles, including the quantum one, arise from the same mathematical source. This deeper mathematical uncertainty principle applies broadly to any function — or any curve, roughly speaking, no matter how bumpy and wild it looks. It comes from an equation invented in the 19th century called the Fourier transform. Named for the Frenchman Joseph Fourier, the Fourier transform decomposes any function into a set of simple waves, each with a different frequency, or tone. Add those simple waves together, and you’ll get back your original function. Mark Belan/Quanta Magazine The uncertainty principle comes built in. A simple sine wave, which spreads infinitely throughout space, has one definite frequency, so its Fourier transform is a single peak. But the snap of a snare drum looks like a pulse as it travels through the air. Its Fourier transform has a wide spread of frequencies — a sound with no discernible pitch. In general, the narrower a function, the more spread out its Fourier transform must be, and vice versa. Or, in other words, the more certain you are about where a function peaks in space, the less certain you can be about its frequency, or how it’s changing in time. In quantum mechanics, a particle is described mathematically as a wave, with peaks in places where it’s most likely to be found. The Fourier transform of that wave describes the particle’s motion. This is why a quantum particle’s position and momentum can’t both be precisely known at once: They are related by a Fourier transform. But what if you take the Fourier transform of a function that looks like a fractal? To make sense of the question, it helps to imagine a simple kind of fractal: Start with a line and cut it into three segments. Then delete the center segment. Repeat the process: Cut each remaining segment into three, remove the center, and so on. The result is a fractal set called the Cantor set. The set has the feature of being what mathematicians call porous — it’s full of holes, like a sponge, at every scale. Now imagine that this Cantor set lives on the x-axis, and at each point on this set is a peak representing a frequency. The fractal uncertainty principle says that if you add together waves of that fractal-like set of frequencies, the resulting curve cannot also look like a fractal — it cannot be porous. The reverse also holds: If you take the Fourier transform of a fractal-like function, the result cannot be a fractal-like set of frequencies. Such fractal-like functions might sound hard to come by. But they pop up when mathematicians study what happens to quantum particles — or waves more generally — in chaotic situations. Like a ball bouncing around a pinball machine, an object experiencing chaos will often travel erratically around the entire space. “If you look at your path, it’s going to look like a random scribble,” said Elena Kim, a mathematician at Harvard University. But in rare instances, an object can find stability in the chaos. In a flat pinball machine, a ball could stay trapped bouncing between three bumpers forever. The ball wouldn’t repeat the exact same path, but it would stay confined between the bumpers, bouncing off a slightly different spot each time. And if you marked each spot where the ball hits each bumper, you’d find that the marks make up something like the Cantor set. This collection of marks is also called fractal dust. Most balls that hit the bumpers will fly off. “This is the dust that’s left,” said Maciej Zworski of the University of California, Berkeley. Elena Kim, a mathematician at Harvard University, has used the fractal uncertainty principle to understand chaotic systems in unconventional spaces. Steph Stevens Unlike a pinball, however, a wave cannot be confined to a fractal-like path, according to the fractal uncertainty principle. If you tried to trap a wave between three bumpers, it would leak out and escape. “So there’s something different about quantum and classical [chaos],” Dyatlov said. “And one ingredient that you can try to use for that would be uncertainty principles.” Unfinished Knowledge Cohen arrived at MIT in 2021 a self-described “young, energetic harmonic analyst” — harmonic analysis being a field of mathematics dedicated to studying functions and their Fourier transforms. As a doctoral student, he sat in Dyatlov’s talks about the fractal uncertainty principle. “He would end every talk being like, ‘We don’t have a higher-dimensional fractal uncertainty principle. I wish we had that!’” Cohen said. “I was like, OK, this seems like a fun thing to work on.” His enthusiasm was soon checked. For one thing, mathematicians already knew of many situations where the fractal uncertainty principle would fail in higher dimensions. Cohen’s first challenge was to come up with a clean way to avoid these cases. The typical requirement of being a fractal is to be porous — meaning that there are holes everywhere you look. The two-dimensional fractal called the Sierpiński carpet is an example; it’s built by dividing a square into a grid of smaller squares and removing the middle square, repeatedly. But while this shape has many holes, it’s also possible to draw a straight line that is fully contained within it. Lines like these spell trouble for the fractal uncertainty principle. The Fourier transform of a vertical line returns a horizontal line, and vice versa. In two or more dimensions, both of these lines count as fractals — that’s because a line takes up no area and leaves most of the surrounding space empty, which satisfies the condition of having many holes. So any fractal that contains uninterrupted lines breaks the fractal uncertainty principle. Cohen needed a new, more stringent kind of porosity. He came up with what he called “line porosity” — any line you draw on the fractal should encounter many holes. His proof excludes any fractals that don’t have this condition, including the typical Sierpiński carpet, but a modified Sierpiński carpet with much more empty space satisfies the rule. 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.