Mathematicians Move the Needle on Decades-Old Problem

Mathematicians from New York University and the University of British Columbia have resolved a decades-old geometric problem, the Kakeya conjecture in 3D, which studies the shape left behind by a needle moving in multiple directions.

The Kakeya conjecture was inspired by a problem asked in 1917 by Japanese mathematician Sōichi Kakeya: What is the region of smallest possible area in which it is possible to rotate a needle 180 degrees in the plane? Such regions are called Kakeya needle sets.

Hong Wang, an associate professor at NYU's Courant Institute of Mathematical Sciences, and Joshua Zahl, an associate professor in UBC's Department of Mathematics, in an article recently posted to the preprint server arXiv, which hosts research before it is peer-reviewed and published in a journal, have shown that Kakeya sets, which are closely related to Kakeya needle sets, cannot be "too small"-namely, while it is possible for these sets to have zero three-dimensional volume, they must nonetheless be three-dimensional.

"There has been some spectacular progress in geometric measure theory: Hong Wang and Joshua Zahl have just released a preprint that resolves the three-dimensional case of the infamous Kakeya set conjecture!" wrote UCLA mathematics professor Terence Tao, who won the 2006 Fields Medal, which is awarded every four years to a mathematician under the age of 40.

"It stands as one of the top mathematical achievements of the 21st century," says Eyal Lubetzky, the chair of the Mathematics department at the Courant Institute.