In this dissertation, we investigate three separate problems in harmonic analysis, partial differential equations, and numerical analysis.
The first concerns an important family of inequalities in analysis known as the Brascamp–Lieb inequalities, which subsume several inequalities significant in their own right, including Hölder’s inequality, the Loomis–Whitney inequality, and Young’s inequality. Several variants and extensions of these inequalities have been developed, some of which have proved to be very useful in Fourier restriction theory. We formulate Brascamp–Lieb inequalities for algebraic objects known as quiver representations and establish necessary and sufficient conditions for such inequalities. We also show that, unlike in classical Brascamp–Lieb inequalities, Gaussians do not saturate certain quiver Brascamp–Lieb inequalities.
The second considers equations modelling wave-like behaviour whose solutions exhibit a phenomenon known as scattering. The problem of deducing properties of these equations from how their solutions scatter has been extensively studied and can be conceived of as the determination of the properties of a physical object from the way it interacts with waves. We study the scattering behaviour of a semilinear wave equation and employ the method of deconvolution developed by Killip, Murphy, and Vișan to determine more general nonlinearities under significantly weaker conditions than those previously assumed by Sá Barreto, Uhlmann, and Wang.
The third addresses the alternating least squares (ALS/AltLS) method, a widely used algorithm for computing the CP decomposition of a tensor. This decomposition expresses a tensor as a sum of simpler tensors and is applied in numerous sciences to reveal patterns and features of multidimensional data. We analyze the theoretical convergence of the CP-AltLS method for so-called orthogonally and incoherently decomposable tensors and establish quantitative, explicit, and precise local convergence theorems for these tensors. In contrast to existing convergence results, our arguments are constructive, more direct, and less technical; and extend to non-orthogonal decompositions while remaining applicable to tensors of arbitrary rank. We also perform numerical experiments confirming our theorized rates of convergence.