This is the homepage of my Part III course called Restriction and Kakeya Phenomena. The course was lectured in 2002 and I have made absolutely no effort to keep these notes up to date. In particular there is no mention of the proof of the finite field Kakeya Conjecture due to Zeev Dvir.

Notes

1. Besicovitch sets
2. The Kakeya problem 1
3. The Kakeya problem 2
4. The circle 1
5. The circle 2
6. Discrete functional analysis
7. Riesz-Thorin interpolation
8. The discrete paraboloid 1
9. The discrete paraboloid 2
10. Montgomery's large-values conjecture and Kakeya
11. Introduction to $\Lambda(p)$ sets
12. Beckner's inequality
13. The influence of boolean functions
14. Sumsets in $\mathbb{F}_2^n$

Non-examinable handouts

Spheres

Example Sheets

Examples 1
Examples 2
Examples 3

A list of errata for the notes is now available here, and will be kept updated.

Internet Resources

 A visit to the webpage of Terry Tao is highly recommended. He has various papers, both technical and expository, on the Kakeya problem. A rather entertaining interactive guide to Besicovitch's construction is available too.

Isabella Laba also has a very useful page giving an introduction to the Kakeya Problem and connections with Harmonic analysis. Her page also summarises some of what is known in the area, and provides further links.