alt Hacker NewsIn 2025, Hong Wang won the Fields Medal for proving the Kakeya conjecture in three dimensions. Here, OpenAI is claiming to have solved a stronger problem that proves the Kakeya conjecture in all dimensions!
That's so insane to me. This top mathematician (and colleagues) at her pinnacle spent years proving it in three dims and now OAI proves it in every dimension...
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The question for me is, would OpenAI have been able to solve this without Hong Wang's work?
OpenAI's proof goes down a completely different route than the route taken for Kakeya in three dimensions and also the existing proof of local smoothing in two dimensions.
Hong Wang and Josh Zahl proved something a bit stronger, namely that the Kakeya conjecture holds for families of tubes satisfying the "Convex Wolff Axioms". From the outside this looks like a weird technical detail but if you unpack this you get an exceptional example of 'the journey is worth more than the destination'.
The idea for resolving Kakeya in R3 originates in the work of Jean Bourgain and a connection to the discretized sum-product phenomenon (the idea, in R, that "sum" and "product" are geometrically incompatible). This approach is summarized in Terry Tao's blog post from 12 years ago now [1].
Basically, the efforts to make this approach work resulted in the development of an entirely new and extraordinarily active subfield that one might call "continuous incidence geometry". This led to ideas like restricted projection theory (with spectacular applications in homogeneous dynamics and number theory), decoupling (another one of Thomas Wolff's ideas, and was realized by Bourgain and Demeter), the resolution of the Furstenberg set theorem, etc. There are very far-ranging applications, some of which even appeared on HN, and which a priori look totally unrelated, like the Heilbronn Triangle problem [2]
I know this is out of the context of most here on HN but let me say it in short: the development of the sum-product approach to Kakeya has resulted in a huge improvement of our understanding of the fundamental additive and multiplicative structure of Euclidean space.
OpenAI's proof of local smoothing in Rd shortcuts this entire matter. Don't get me wrong, the proof is phenomenal, and it's amazing that Local Smoothing admits a short, essentially self-contained proof (only uses some facts about elliptic PDEs which I can't comment about unfortunately since it isn't my area).
Hong Wang didn't win the fields medal for proving Kakeya in three dimensions (only). She won because of her incredible contributions to the development of continuous incidence geometry. The proof of local smoothing certainly contains a lot of very interesting insights into the wave equation but, if this proof had appeared in 2014 just like this, I do not think we would have had the same amount of development.
To me, this is the heart of the matter. For mathematicians, only having these incredible new proofs with no "social consequences" would be extraordinary. However, if we are not careful, the idea that math is "solved" will degrade the mathematics tradition in its own right. Where will be in 10 years, in 50 years? Humans, by virtue of being much worse at computations than computers, must build the "correct" abstractions in order to discover new mathematics. If this process is somehow lost, independently of the computer's ability to prove an infinite tower of new results, the mathematics tradition will suffer for it.
[1]: https://terrytao.wordpress.com/2014/05/07/stickiness-grainin...
I mean, the models (starting with Bel clearly and Astra probably--though Astra probably is worse than top humans in some areas) are superhuman at math.
To be honest this doesn't feel as such impressive as the others. It was recently proven in 3 dimensions, so it could be "just" an extension of that. I haven't read the paper anyways so this is pure spectulation.