The concept of a Heisenberg uniqueness pair (Γ,Λ), defined for a curve Γ and set Λ in the plane, was first introduced by Hedenmalm and Motes-Rodríguez (2011) as a variant of the uncertainty principle for the Fourier transform. 

In this talk, we will introduce the work of Hedenmalm et al. on Heisenberg uniqueness pairs and some recent progress on this topic. Additionally, we will discuss closely related Fourier uniqueness pairs as well as Viazovska's theory of Fourier interpolation.