Graduate student Alex Cohen has proven a higher-dimensional fractal uncertainty principle, a significant advance in a field that combines quantum mechanics, chaos theory, and fractal geometry. This new principle establishes that waves cannot be confined to fractal structures in higher dimensions, a finding with implications for understanding quantum chaos.
The fractal uncertainty principle, now extended to higher dimensions by graduate student Alex Cohen, states that waves cannot be simultaneously localized in both position and frequency near a fractal set. This mathematical result, which bridges chaos theory and quantum mechanics, has been described as a "foundational result" by mathematicians familiar with the work. Cohen's proof, published in the Annals of Mathematics, addresses a challenge posed by Semyon Dyatlov of MIT, who had previously established the principle in one dimension.
Alex Cohen, a doctoral student at MIT, has successfully proven a higher-dimensional version of the fractal uncertainty principle, a mathematical concept that merges quantum theory, chaos, and fractal geometry. This principle asserts that waves cannot be confined to fractal sets in higher dimensions, a finding that could offer new insights into the behavior of quantum particles in chaotic systems. The work builds upon earlier proofs by Semyon Dyatlov, who established the principle in one dimension, and has been published in the Annals of Mathematics.
Cohen's proof, detailed in the Annals of Mathematics, addresses a gap in the understanding of how quantum waves behave in complex, self-similar structures. The fractal uncertainty principle, first explored in one dimension by Semyon Dyatlov and Jean Bourgain, posits that a function cannot be precisely located in both position and frequency if its support is a fractal set. This has direct implications for quantum chaos, where particles moving in chaotic environments can trace fractal paths. The principle suggests that quantum particles, unlike their classical counterparts, cannot be trapped on such paths.
Dyatlov, a mathematician at MIT, had previously proved the one-dimensional version of the fractal uncertainty principle in 2016. His work demonstrated that waves could not be confined to one-dimensional fractal paths, suggesting a fundamental difference in how quantum particles and classical objects behave in chaotic systems. Cohen's achievement extends this by proving the principle in higher dimensions, a task that proved considerably more challenging. His method involved a novel construction of a "damping function" in two dimensions, a key ingredient needed to extend the proof. Wilhelm Schlag of Yale University, who advised Cohen as an undergraduate, described this construction as "brilliant."
The mathematical framework behind the fractal uncertainty principle involves harmonic analysis, a field that studies functions and their Fourier transforms. The principle essentially states that if a function's Fourier transform is concentrated on a fractal set, the function itself cannot be localized on a fractal set in a complementary way, and vice versa. This is analogous to the more familiar Heisenberg uncertainty principle in quantum mechanics, which states that one cannot simultaneously know both the position and momentum of a particle with arbitrary precision.
Cohen's work has already found applications. Elena Kim, a mathematician at Harvard University, has utilized the fractal uncertainty principle to study chaotic systems in unconventional mathematical spaces. In 2025, Kim, along with Nicholas Miller of the University of Oklahoma, applied a higher-dimensional version of the principle to extend results in hyperbolic spaces. Mathematicians are now exploring further applications, including the potential to prove that waves in chaotic systems not only spread out completely but do so evenly, a result that could have implications for understanding the three-dimensional world.
The fractal uncertainty principle has roots in the study of quantum chaos, specifically the question of whether quantum particles behave differently from classical particles when subjected to chaotic conditions. Classical objects can become trapped on fractal paths in chaotic systems, such as a billiard ball on a table with complex boundaries. However, the fractal uncertainty principle implies that quantum waves cannot be confined in this manner; they would inevitably "leak out." This fundamental difference underscores the unique nature of quantum behavior in chaotic environments.
The development of the fractal uncertainty principle has been an ongoing effort, with contributions from mathematicians like Jean Bourgain, who passed away shortly after contributing key ideas to Dyatlov's initial proof. Cohen's successful extension to higher dimensions represents a significant step forward, providing a more powerful mathematical tool for analyzing complex systems. The research connects disparate areas of mathematics and physics, offering a deeper understanding of the interplay between chaos, fractals, and quantum mechanics.