Hong Wang (MIT PhD 2019, now Associate Professor at NYU Courant & Permanent Professor at IHES) together with Joshua Zahl has achieved a major breakthrough in geometric measure theory.
In February 2025, they posted a 127-page preprint proving the long-standing three-dimensional Kakeya set conjecture.
A Kakeya set in ℝ³ is a compact set that contains a unit line segment in every possible direction. For over a century, mathematicians wondered how “small” such a set could be. Wang and Zahl proved that these sets must have full Minkowski and Hausdorff dimension 3; they cannot hide in lower-dimensional space, even though they can have zero volume.
This result settles a fundamental question with deep connections to Fourier analysis, the restriction problem, PDEs, and incidence geometry. Their work builds on years of incremental progress (including their own earlier papers from 2022 onward) and introduces powerful new volume estimates for unions of tubes and convex sets.