A 25 year old grad student cracked a quantum uncertainty problem that stumped mathematicians for a decade

[Fraction]

Alex Cohen proved the fractal uncertainty principle in all higher dimensions while still a doctoral student at MIT, extending a result that mathematician Semyon Dyatlov had only managed to prove in one dimension back in 2016 with help from the late Jean Bourgain, and the achievement became Cohen's PhD thesis and landed him an assistant professorship at NYU at just 25

The core idea builds on the ordinary quantum uncertainty principle, the more precisely you know a particle's position, the less precisely you can know its momentum, and this deeper mathematical relationship comes from the Fourier transform, which decomposes any function into a set of simple waves, the fractal version of this principle says that if a function looks fractal, meaning it's riddled with holes at every scale like the Cantor set, its Fourier transform cannot also look fractal, and vice versa

This matters for real physics because fractal like patterns show up naturally when studying chaos, imagine a ball bouncing forever between three bumpers in a pinball machine without ever settling into a repeating path, the collection of points where it hits each bumper forms a fractal shape sometimes called fractal dust, and the fractal uncertainty principle proves that a quantum wave, unlike a classical ball, can never get trapped in that same fractal pattern, it always leaks out and spreads

Cohen's breakthrough required inventing a stricter version of what counts as fractal, since ordinary porous shapes like the Sierpinski carpet can still contain uninterrupted straight lines that break the uncertainty principle, his fix, which he called line porosity, requires that any line drawn through the fractal must still encounter plenty of holes, and that stricter definition is what let his proof actually go through where more than a decade of prior attempts had failed

The breakthrough moment came from unpublished notes that Bourgain had shared with Dyatlov before his death in 2018, notes that Dyatlov later passed on to Cohen, containing a hint rooted in a well known 1960s result called the Beurling-Malliavin theorem that let Cohen build the specialized mathematical tool, called a damping function, that his proof depended on, a nice reminder that even genuinely novel breakthroughs often stand on decades old ideas nobody had connected in quite the right way before

Since Cohen posted the proof in 2023 it's already been used to extend related results to higher dimensional chaotic spaces, and mathematicians see it as a foundational tool likely to ripple through harmonic analysis and signal processing more broadly, with Peter Sarnak of the Institute for Advanced Study calling it a pretty remarkable achievement for someone still working on their thesis

Graduate Student Proves the Fractal Uncertainty Principle | Quanta Magazinewww.quantamagazine.org

[Saka31]

A grad student solving something that stumped a legendary mathematician like Bourgain for years is the kind of story that should get way more mainstream attention

[StayReadyKev91]

Agreed, its inspiring but also a little intimidating, most PhD theses dont end with a professorship offer at 25

[Blue Sasha]

The line porosity fix is such an elegant piece of problem reformulation, sometimes the hardest part of a proof is figuring out the right definition to work with

[Anvil79]

Totally, redefining the problem correctly is often 90 percent of the actual mathematical work even before you start proving anything

[Cheeky Shaun]

The pinball analogy for fractal dust made this way more intuitive than I expected going in, good explainer writing

[PlanckLimit36]

That detail about not knowing it was already known is kind of perfect, sometimes ignorance of the difficulty is exactly what lets you push through

[SpinState]

Agreed, explaining why a quantum wave cant get trapped the way a classical ball could really landed the core insight clearly

[DarkMatter24]

Foundational results in Fourier analysis eventually seep into everything from signal processing to engineering, wouldnt be surprised if this shows up in unexpected places down the line

[CrimsonFury]

Love that Bourgains unpublished notes ended up being the key, and that Cohen later found out the trick wasnt even secret knowledge, just something he hadnt encountered yet

[Dave]

Curious how this eventually connects to the still open Sarnak Rudnick conjecture about waves spreading evenly, not just spreading at all

[ReacherBadger]

That does seem like the natural next target for people building on Cohens result, spreading everywhere versus spreading evenly are genuinely different claims