Microlocal analysis and singularities

Degenerations and singularity formations play important roles in the study of differential geometry and arise naturally in multiple other areas of mathematics, including, algebraic geometry, mathematical physics, number theory, and representation theory. This project concerns the geometry of singularities and has interesting connections with an array of disciplines including modeling of electromagnetic systems, gauge theory and string theory. Building on her track record, the PI plans to use effective techniques to solve a wide variety of problems and discover new and sharper analytic results. Alongside her research, the PI will engage in various outreach activities, with a focus on fostering mentoring networks for undergraduate and graduate students. This project involves studying singular metrics using geometric microlocal analysis. The PI will address problems related to metric degenerations, including the perturbation theory, singular uniformization problems, parametrization of moduli spaces and related geometric properties on singular metrics. Tools include geometric resolutions or compactifications which resolve the singularities and provide more detailed analytic properties, in addition to interactions with algebraic and differential geometry which in turn suggest which PDE techniques need to be developed. The PI will use these methods to study problems such as the stratification and singularities of the moduli space of constant curvature conical metrics, hyperbolic metrics with cusps and their spectral gaps, and gauge-theoretic singular metrics such as gravitational instantons and their degenerations. 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.