Most of my work involves linear algebra in some form. A common theme in this work is structure, whether that structure comes from the geometry of a physical problem or of a manifold of matrices, special algebraic properties like complex symmetry or low-rank structure, or analytic properties involving perturbations from some geometric or algebraic structure.
I am particularly interested in eigenvalue problems and related topics. Eigenvalue-based analysis methods are surprisingly powerful for several reasons. Eigenvalue problems arise naturally in the analysis of differential equations by transform methods. But eigenvalue problems have much broader applications, too, largely because they among the few nonlinear equations for which we have global solution methods, and symmetric eigenvalue problems can be formulated as one of the few non-convex optimization problems for which we have good general-purpose methods. In my work, I have explored eigenvalue problems to understand everything from the difficulty of problems in computer vision, to how people form opinions, to how to cluster and rank in complex networks, to the impact of fabrication errors on novel gyroscope designs.
Since my days as a PhD student, I have worked on structured eigenvalue problems associated with certain PDE discretizations, spectral methods for model reduction, and methods of eigenvalue analysis for parameterized families of matrices. In more recent work, I have focused on eigenvalue problems that arise in network-structured problems, eigenvalue problems for systems with underlying (approximate) symmetry groups, and analysis techniques for nonlinear eigenvalue problems.