The Kakeya Conjecture: A Mathematical Problem with a Long History at UBC Math and a 2026 Fields Medal Connection

By Krysten Casumpang and Dr. Pablo Shmerkin

Dr. Hong Wang, Professor of Mathematics at New York University's Courant Institute School of Mathematics, Computing, and Data Science and the Institut des hautes études scientifiques (IHES), has been named one of four recipients of the 2026 Fields Medal. Dr. Wang was awarded this prestigious prize, the highest honour in mathematics, in part because of the advancement she made in solving the Kakeya problem in three dimensions alongside her collaborator, UBC Mathematics Professor Dr. Joshua Zahl (currently on leave at the Chern Institute of Mathematics at Nankai University). 

The Fields Medal recognizes that Dr. Wang has completely reshaped her fields: harmonic analysis and geometric measure theory. Harmonic analysis studies complicated objects by decomposing them into wave patterns, and geometric measure theory investigates the size and shape of rough objects. Both areas have long been areas of excellence within UBC Mathematics through the work of Dr. Izabella Laba, Dr. Malabika Pramanik, Dr. Pablo Shmerkin, and Dr. Joshua Zahl.

The Kakeya Conjecture

In a series of three monumental papers [1], Dr. Wang and Dr. Zahl solved the three-dimensional Kakeya problem, a central problem that concerns how much thin tubes in 3D space can intersect each other. Understanding this problem gives insight into how strongly waves moving in different directions can reinforce each other, as, roughly speaking, the tubes in the Kakeya problem correspond to parts of space where waves are localized. The evolution of waves is described by the wave equation, which has multiple applications in science and engineering.

Although the Kakeya problem is theoretical, applications of mathematical theories often arise unexpectedly and after many years. “The Kakeya problem is a beautiful illustration of the fundamental unity of all of mathematics and beyond,” shares Dr. Pablo Shmerkin. The problem had resisted the efforts of many top mathematicians over decades, including several past Fields medalists. But, as Dr. Shmerkin notes, “The Fields Medal is not just for ticking the box ‘solved’ on this and many other important problems, but for developing ideas and techniques that will shape mathematics for decades to come.”

As a testament to UBC Mathematics’ leadership in the fields of harmonic analysis and geometric measure theory, Dr. Zahl and Dr. Shmerkin were awarded the 2026 Clay Research Award earlier this year.

[1] "The Assouad dimension of Kakeya sets in \(\mathbb{R}^{3}\)" (Published in Inventiones Mathematicae, 2025), "Sticky Kakeya sets and the sticky Kakeya conjecture" (Published in the Journal of the American Mathematical Society, 2026), and "Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions" (arXiv, 2025)