Xuwen Zhu

Sponsor: NSF

Moduli Spaces and Geometric Microlocal Analysis

Metrics with singularities are important objects in differential geometry and arise naturally in algebraic geometry, mathematical physics, number theory, representation theory, etc. This project involves studying singular metrics using geometric microlocal analysis. The central idea is to introduce new objects, called compactifications or resolutions, to resolve the singularities. These resolutions will in turn suggest which analytic techniques need to be developed. The PI intends to use this method to study problems such as the moduli space construction of constant curvature conical metrics and its relation to vortices, hyperbolic metrics with cusps and asymptotic geometry of the compactified Riemann moduli space, and gauge-theoretic partial differential equations with singular metric background.