UBC Mathematics Associate Professor Josh Zahl, in collaboration with Hong Wang, an Associate Professor at the Courant Institute of Mathematical Sciences at New York University (NYU), has proven the Kakeya set conjecture in three dimensions.

The Kakeya Conjecture is a long-standing problem in mathematics that deals with the geometry of sets in space. In simple terms, it asks how much space is needed to rotate a needle in all directions within a given area. The conjecture suggests that in three-dimensional space, the minimal space required to rotate a needle in every direction is surprisingly small. Until now, this problem remained unsolved in three dimensions, though it had been proven in lower dimensions.

This breakthrough is significant because it not only solves a key puzzle in mathematics but also opens doors for advancing our understanding of other complex problems in the fields of partial differential equations (which deal with how quantities change in space and time) and Fourier analysis (a method for analyzing complex waveforms). Essentially, the Kakeya Conjecture's solution will help mathematicians tackle other related problems by providing new insights and tools to work with.

Zahl and Wang's work builds upon previous groundbreaking work by Izabella Laba, Nets Katz, Terence Tao [1], Pablo Shmerkin [2], Malabika Pramanik, and Tongou Yang [3].

RELATED PRESS:

'Once in a Centry' Proof Settles Math's Kakeya Conjecture by Joseph Howlett, Quanta Magazine (March 13, 2025)

Mathematicians Move the Needle on Decades-Old Problem - New York University Press Release (March 11, 2025)

[1] Katz, Laba, Tao. An improved bound on the Minkowski dimension of Besicovitch sets in R^3.  Annals of Math (2000)
[2] Orponen, Shmerkin, Wang. Kaufman and Falconer estimates for radial projections and a continuum version of Beck’s Theorem. GAFA (2024)
[3]  Pramanik, Yang, Zahl. A Furstenberg-type problem for circles, and a Kaufman-type restricted projection theorem in R^3. (2022).