The Restriction Problem and the Polynomial Method - Lecture I

Explore the fascinating world of Stein's restriction conjecture and the polynomial method in this illuminating lecture. Delve into the intricacies of estimating functions with Fourier transform supported on hypersurfaces, such as spheres in Rn. Discover how these functions can be decomposed into sums over wave packets supported on long thin tubes. Learn about Guth's groundbreaking introduction of the polynomial method in restriction theory, particularly its application in studying tube intersections. Gain a comprehensive understanding of Stein's restriction conjecture and the Kakeya conjecture through a concise introduction. Examine the polynomial method's role in addressing these problems, covering topics such as the uncertainty principle, bump functions, ball multipliers, Kenshin's inequality, and the polynomial partition theorem. Engage with various examples, proofs, and scenarios to deepen your grasp of this complex mathematical subject.

Intro
Uncertainty principle
Bump function
Ball multiplier
Kenshins inequality
Peppermint counter
Spot sprouting
The restriction problem
Examples
Plan
Warmup
Easy estimate
Proof of similarity theorem
Cellular case
Kakaya problem
The polynomial partition theorem
The bad scenarios