Larry Guth --- The Kakeya conjecture in $\mathbb{R}^3$, after Hong Wang and Joshua Zahl

samedi 28 mars 2026, 11:30 → 12:30 Europe/Paris

Amphithéâtre Charles Hermite (IHP - Bâtiment Borel)

The Kakeya conjecture is a question about how thin tubes can overlap with each other in Euclidean space.  It connects to problems in Fourier analysis and PDE, such as understanding L^p type estimates for solutions of the wave equation.  This connection has prompted a lot of work on the problem in the harmonic analysis community.

The 2-dimensional case of the Kakeya conjecture has a 1-page proof and has been known since the 1970s.  Recently, Wang and Zahl proved the 3-dimensional case of the Kakeya conjecture.  Their work builds on contributions of many people, including Bourgain, Wolff, Katz, Laba, Tao, Orponen, and Shmerkin.

There are interesting examples related to algebraic geometry which show that some cousins of the Kakeya conjecture are false.  These counterexamples are also important clues for understanding the problem.  In the proof of Kakeya, we imagine a hypothetical counterexample, we prove that it would have to have a great deal of algebraic structure, and we finally show that it cannot exist at all.  A hypothetical counterexample to the Kakeya conjecture is just a set of thin tubes in 3-dimensional space which overlap each other a lot.  We will try to describe how to prove that such a set has algebraic structure.