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Introduction to the proof of the Kakeya conjecture

Larry Guth

Abstract

Recently, Hong Wang and Joshua Zahl announced a proof of the 3-dimensional Kakeya conjecture. This is a survey article on the proof of Kakeya. We introduce the problem, discuss previous work and some of the difficulties of the problem, and describe the new ideas in the recent work.

Introduction to the proof of the Kakeya conjecture

Abstract

Recently, Hong Wang and Joshua Zahl announced a proof of the 3-dimensional Kakeya conjecture. This is a survey article on the proof of Kakeya. We introduce the problem, discuss previous work and some of the difficulties of the problem, and describe the new ideas in the recent work.
Paper Structure (16 sections, 4 theorems, 62 equations)

This paper contains 16 sections, 4 theorems, 62 equations.

Table of Contents

  1. The Kakeya problem
  2. A key obstacle: the Heisenberg group
  3. The idea of stickiness
  4. Sticky vs. not sticky
  5. The $L^2$ method
  6. New ideas for the not sticky case
  7. Some ideas that don't work
  8. Looking at $\mathbb{T}$ inside a small ball
  9. A surprising induction
  10. Density of the Kakeya set in small balls
  11. When $\mathbb{T}_B$ clusters into slabs
  12. How do we reduce to the not-sticky-at-all-scales case?
  13. The high density lemma
  14. What about the Katz-Zahl example?
  15. What about higher dimensions?
  16. ...and 1 more sections

Key Result

Theorem 1.2

(Wang-Zahl, WZ) If $\mathbb{T}$ is a set of $\delta$-tubes in $\mathbb{R}^3$, and $\Delta_{max}(\mathbb{T}) \lessapprox 1$, then

Theorems & Definitions (5)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 3.1
  • Lemma 6.1
  • Lemma