Formal Sciences & Matter / Mathematics

Kakeya Conjecture in 3D

Hong Wang and Joshua Zahl prove the three-dimensional Kakeya conjecture in February 2025; Wang receives a Fields Medal in 2026.

The question: how small can a set be that contains a straight line segment pointing in every direction? Besicovitch showed that in the plane the area can be arbitrarily small, but the conjecture says the set must still have full dimension (3 in 3D space). Wang and Zahl posted the proof on 24 February 2025, settling a problem open since the 1970s; Nets Katz called it a once-in-a-century result. It matters for Fourier analysis and partial differential equations.