Mathematics Colloquium

If a function has Fourier transform supported on a sphere, what can we say about this function?

Given a collection of long thin tubes pointing in different directions, how much do they overlap?

These two questions are closely related. In this talk, we will discuss how understanding the second question leads to progress on the first one. More precisely, we will discuss Stein's restriction conjecture and Sogge's local smoothing conjecture for the wave equation.