Senior Honors Thesis Presentation: Introduction to Fourier Restriction Theory

Abstract: In this talk, we will introduce the basics of Fourier restriction theory. We will describe the general question and motivate the theory with an example from physics. We will then pay particular attention to the setting of the sphere with its surface measure. We will examine endpoint cases and discuss the Tomas-Stein theorem, one of the first major results in this area. We will then demonstrate necessary conditions for restriction inequalities on the sphere, which will motivate the restriction conjecture. Finally, we will discuss the connection between the restriction conjecture and an important problem in geometric measure theory: the Kakeya conjecture. We will give a proof that the restriction conjecture implies the Kakeya conjecture for the Minkowski dimension.

Faculty Advisor: Alan Chang