Executive Summary : | The Heisenberg uncertainty principle states that accurate information about a particle's position and momentum cannot be obtained simultaneously. This principle is applied to the uniqueness of the Fourier transform, a long-standing problem in harmonic analysis. The uniqueness of the Fourier transform becomes difficult when the measure is supported on a curve. This problem gained attention in harmonic analysis due to Hedenmalm and Montes-Rodriguez's work on measure supported on a curve whose Fourier transform vanishes on a thin-set. The Heisenberg uniqueness pair (HUP) is a version of the uncertainty principle for the Fourier transform and has significant similarities to mutually annihilating pairs of Borel measurable sets with positive measures. This research proposal aims to study the Heisenberg uniqueness pair, which is equivalent to finding the uniqueness of solutions of partial differential equations induced by algebraic curves in the plane. The study aims to characterize the Heisenberg uniqueness pairs corresponding to the cross-section in the plane, generalize this result for the union of finitely many cross-sections in the plane, and characterize the Heisenberg uniqueness pairs corresponding to the spiral in the plane. The proposal also aims to generalize the concept of the Heisenberg uniqueness pair to noncommutative settings, such as the Heisenberg group. Additionally, it will study Nazarov's uncertainty principle for the Weyl transform and the group Fourier transform on the Heisenberg group. |