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The Kakeya conjecture, after Wang and Zahl

Larry Guth

Abstract

This is a Seminaire Bourbaki survey of the proof of the Kakeya conjecture in three dimensions. The survey is written for a broad mathematical audience. We sketch all the ideas in the proof, with many pictures.

The Kakeya conjecture, after Wang and Zahl

Abstract

This is a Seminaire Bourbaki survey of the proof of the Kakeya conjecture in three dimensions. The survey is written for a broad mathematical audience. We sketch all the ideas in the proof, with many pictures.

Paper Structure

This paper contains 33 sections, 17 theorems, 70 equations, 9 figures.

Table of Contents

  1. Statement of main results
  2. Notation
  3. The hero: multiscale analysis
  4. A key obstacle: the Heisenberg group
  5. Key structures and outline of the proof
  6. Grain structure
  7. Complex conjugation structure
  8. Stickiness
  9. Outline of the proof
  10. Avoiding technicalities
  11. Stickiness leads to grain structure
  12. Perfect overlap
  13. Grain structure
  14. Fractal structure of $A$
  15. Grain structure leads to complex conjugation structure
  16. ...and 18 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 $\blacktriangleleft$$\blacktriangleleft$

Figures (9)

  • Figure 1: Intersecting tubes
  • Figure 2: Grain structure
  • Figure 3: When is Lemma \ref{['lemmultmu']} sharp?
  • Figure 4: Illustration of Lemma \ref{['lempartubes']}
  • Figure 5: Perfect overlap of tubes in a grain
  • ...and 4 more figures

Theorems & Definitions (26)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 6.1
  • Lemma 6.2
  • proof : Proof of Lemma \ref{['lemmultmu']}
  • Lemma 6.3
  • Lemma 6.4
  • Lemma 6.5
  • proof : Proof sketch
  • Lemma 6.6
  • ...and 16 more