Comment by bamb008
3 hours ago
Doesn't this make it seem like Wang had the wrong intuition, if her meyhod does not generalize? She proved Kakeya only in dimension 3, and also not prove the Hausdorff 3D Kakeya. In retrospect it seems humans were asking the wrong question, if a couple dozen pages long manuscript can subsume so many other efforts.
AI has only been able to advance the math frontier for a couple of months, fully grasping the applications, connections, and possible generalizations takes longer.
They did have the right intuition; it's pretty uncontroversial to believe that local smoothing in Rd would have been resolved within the next decade without AI input. For example, Hong, Josh and coauthors were pretty far along the way to finishing the R4 case when they got scooped in the previous batch. The timeframes are too short, of course.
My original comment was perhaps badly written: my point was that the OpenAI proof does not subsume the previous efforts because they are proving something stronger than Kakeya using very geometric methods based on the discretized sum-product theory in the real line. This approach has basically revolutionized modern analysis. Kakeya is just one very publicly visible problem. All of the other byproducts are in total of much greater interest than the resolution of Kakeya itself.
It's like Fermat's last theorem: who cares about solutions to x^n + y^n = z^n? It's the underlying theory-building in algebraic geometry that is the actual interesting product in its own right. If someone finds an extraordinarily clever 1 page proof of Fermat's last theorem it does not mean that the progress on algebraic geometry was useless.
I want to point out as well that it's quite common for humans to find very short clever proofs of difficult old results. Those proofs have value of course but rarely do they generalize. One of the goals of mathematics is to understand the fundamental mechanism underlying why things are true; proofs (of any origin) that are too slick is antithetical to this.