The Kakeya Conjecture: where does it come from and why is it important?

Abstract: Roughly speaking, the Kakeya Conjecture asks to what extent lines which point in different directions can be packed together in a small space. In $\R^2$, the problem is relatively straightforward and was settled in the 1970s. In $\R^3$ it is much more difficult and was only recently resolved in a monumental and groundbreaking work of Hong Wang and Joshua Zahl. This note describes the origins of the Kakeya Conjecture, with a particular focus on its classical connections to Fourier analysis, and concludes with a discussion of elements of the Wang--Zahl proof. The goal is to give a sense of why the problem is considered so central to mathematical analysis, and thereby underscore the importance of the Wang--Zahl result.