Alex Cohen Extends Fractal Uncertainty Principle to All Higher Dimensions at 25
Updated
Updated · Quanta Magazine · Aug 12
Alex Cohen Extends Fractal Uncertainty Principle to All Higher Dimensions at 25
1 articles · Updated · Quanta Magazine · Aug 12
Summary
A 2025 Annals of Mathematics paper by Alex Cohen proved the fractal uncertainty principle in all higher dimensions, solving a problem that had resisted mathematicians since a 2016 one-dimensional version.
The result shows a wave and its Fourier transform cannot both concentrate on porous fractal sets, sharpening how quantum waves differ from classical trajectories in chaotic systems.
Cohen’s key step was defining stricter “line porosity” to rule out higher-dimensional counterexamples and building a delicate damping function using ideas linked to Bourgain’s unpublished notes and complex analysis.
The proof has already been used: in 2025, Elena Kim and Nicholas Miller applied it to higher-dimensional hyperbolic spaces, extending results that waves in chaotic settings cannot stay trapped.
Beyond quantum chaos, mathematicians see the theorem as a foundational Fourier-analysis tool with possible applications ranging from pure math to signal processing.
If quantum waves refuse to be trapped in chaotic fractal mazes, what hidden paths do they take instead?
Could a mathematical rule preventing waves from getting stuck revolutionize how we process complex signals?
How did a hidden clue in unpublished notes help unlock a quantum mystery that baffled mathematicians for years?
Cohen’s 2025 Solution to the Fractal Uncertainty Principle in Higher Dimensions: Mathematical Impact and Future Directions
Overview
In 2025, Alex Cohen of MIT published a landmark paper in the Annals of Mathematics, introducing a higher-dimensional fractal uncertainty principle (FUP) that revolutionized how mathematicians understand wave behavior in complex environments. By formulating a new version of the Beurling–Malliavin theorem and introducing the concept of line porosity, Cohen proved that fractal sets avoiding straight lines must satisfy the FUP. This breakthrough quickly energized the mathematical community, leading to expository guides, major conference sessions, and new research on quantum chaos and signal processing. Cohen’s work not only opened new frontiers in mathematical physics but also highlighted practical challenges, such as detecting line obstructions in high-dimensional data.