‘Once in a Century’ Proof Settles Math’s Kakeya Conjecture

Hong Wang, Member (2019–21) in the School of Mathematics, now at New York University’s Courant Institute, has proven the three-dimensional Kakeya conjecture. She resolved the problem, which was open for more than a century, alongside her colleague Joshua Zahl from The University of British Columbia.

Originating in 1917, the Kakeya conjecture is a deceptively simple mathematical problem that can be visualized as follows: “Hold a pencil in midair, then point it in every direction while minimizing the volume of space it moves through.” 

Wang and Zahl’s work has “established an absolute limit to how small such a pattern of movements can be.” Despite its straightforward appearance, the problem has, until now, “eluded some of the greatest living mathematicians.” 

Their work has built on that of other IAS scholars. Jean Bourgain, Professor (1994–2018) in the School of Mathematics, made substantial contributions to the Kakeya problem, as did Larry Guth, Member (2010–11) in the School. Their insights were crucial for Wang and Zahl.

In addition to this, Wang and Zahl employed a technique pioneered by Terence Tao, Member (2005, 2023) in the School of Mathematics, and his colleague Nets Katz. In 2014, Tao and Katz “examined a pesky class of Kakeya sets. Their proof showed that every set in that particular class had a dimension of three.” This provided a “road map” for Wang and Zahl’s initial work.

There are important consequences for Wang and Zahl’s new proof, which resolves a landmark open problem, but further work remains. The conjecture about Kakeya sets has been made for all dimensions, and in higher dimensions the geometry becomes more complicated. The Kakeya conjecture is also closely connected to important restriction conjectures about Fourier transforms that are still open. Thus, this breakthrough will lead to much more exciting research.

Read the full article at Quanta Magazine.