Harmonic Analysis on Manifolds

This project considers problems in harmonic analysis related to partial differential equations on curved spaces, with a focus on the Schrödinger equation in quantum mechanics. The PI will investigate the regularity properties of solutions to the linear Schrödinger equation and how these properties depend on the initial data. Central to the project is the question of how the geometry and shape of space influence the behavior of quantum waves governed by the equation. These problems are connected to modern developments in quantum physics, data science, and engineering. The PI also aims to build connections with other fields in mathematics such as number theory, dynamical systems and spectral theory. The project provides training opportunities for graduate students interested in mathematical analysis. More specifically, the PI will develop new mathematical tools to study wave behavior and derive space-time estimates for the linear Schrödinger equation on manifolds under varying geometric assumptions. Examples include hyperbolic manifolds with trapped geodesics, the flat tori, and settings involving constraints from interacting potentials, such as those arising in the many-body Schrödinger equation. The main questions focus on how the geometric and dynamical properties of the underlying space and equation influence the behavior of solutions. These estimates have broad applications in nonlinear partial differential equations arising from different physical contexts and are connected to other areas of mathematics, including number theory and geometry. The analysis employs local harmonic analysis and global analysis on manifolds. The local analysis involves adaptations of variable coefficient arguments developed in the Euclidean setting, along with tools from microlocal analysis, while the global analysis exploits tools from geometry. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.