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Graduate Student Proves a Quantum Uncertainty Principle for Fractals
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chaos theory
Graduate Student Proves a Quantum Uncertainty Principle for Fractals
By
Shalma Wegsman
August 12, 2026
The math, which combines chaos, quantum theory, and infinitely complex fractal structures, has been called a “foundational result.”
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Ada Zejun Shen/Quanta Magazine
Introduction
By Shalma Wegsman
Staff Writer
August 12, 2026
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chaos theory
fractals
harmonic analysis
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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.
Alex Cohen’s proof of the fractal uncertainty principle has already proven influential across math.
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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.
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Mark Belan/Quanta Magazine
The uncertainty principle comes built in. A simple sine wave, which spreads infinitely...