Spectral Theory and Applications for Models with Localized or Boundary Defects

The project consists of three main research areas in applied mathematics: (i) properties of model operators in condensed matter and topological physics, (ii) performance of numerical algorithms for studying how waves propagate (either electromagnetic waves or fluid waves), and (iii) a study on how the geometry of a domain or graph impacts methods for “partitioning” it into smaller useful pieces for identifying key communities or labels (such as identifying cancer cells from certain genetic markers). Each of these might sound completely unrelated, but the main idea behind the project is that similar methods from harmonic analysis and optimization can be applied to each of these problems when viewed through the right lens. A substantial part of the project will be performed in collaboration with undergraduate and graduate students and postdocs, towards developing theoretical and computational tools, while at the same time maintaining and creating new collaborations with physicists, especially in terms of applications. Special attention will be paid to working with numerical analysts at the forefront of simulating problems in each application domain.Algorithms for solving complicated problems in physics, fluid mechanics and data analysis will be considered, with a special focus on quantifiable error estimates and rigorous constructions of various fundamental modes or resonances for physically important examples. For instance, the principal investigator will extend work that was performed with collaborators on solving the Helmholtz problem using domain decompositions to give further quantitative bounds on numerical schemes for obstacle scattering, in similar way to the cases of scattering by inhomogeneous media. This will give enhanced error estimates for numerical solutions to the Helmholtz equation, which play a major role in various applications for inverse problems in medical imaging, sonar detection, and more. In particular, although numerical simulations are by nature constrained to certain bounded domains, one can understand how to provide damping of a fluid at certain points to allow for a numerical simulation of the way a wave propagates in the open ocean without getting effects from the boundary. This is done by adding a “damping term” that could perturb the system in a significant way. However, upon performing a certain transformation, the essence of this damping lies in computing properties of operators that arise in quantum mechanics with complex potentials. The complex and rich tools that have been recently developed in microlocal analysis, optimization and elliptic theory of partial differential equations allow us to give strong insights and new means of establishing quantitative bounds. These apply to the efficacy of damped fluid models, as well as to questions related to the behavior of a light wave in a crystalline structure with defects. These methods apply to various fields, including but not limited to imaging, lasing and community detection.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.