Moduli Spaces and Geometric Microlocal Analysis

Northeastern University, Boston MA

Investigators

Abstract

Metrics with singularities are important objects in differential geometry and arise naturally in algebraic geometry, mathematical physics, number theory, representation theory etc. The PI in particular studies constant curvature metrics with singularities, which naturally appear for instance in the degeneration of smooth metrics and geometric flows. These objects also have interesting applications in physics as they can be viewed as solitons in gauge theory, and are related to Higgs bundles and magnetic vortices. Such metrics solve partial differential equations, called curvature equations, where singularities can be viewed as boundary data. Recently there have been many developments in the interplay of analysis and geometry in this field, featuring new and surprising phenomena such as stratification and bubbling. The goal of the project is to develop new techniques that will help discover new features of such objects, and give insight into problems related to singular metrics, including singular uniformization, moduli spaces, and solutions of partial differential equations on manifolds with singular geometry. 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. 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.