Moduli Spaces and Geometric Microlocal Analysis

具有奇点的度量是微分几何中的重要对象，在代数几何、数学物理、数论、表示论等学科中自然出现。PI特别研究具有奇点的常曲率度量，奇点自然出现在光滑度量和几何流的退化中。这些物体在物理学中也有有趣的应用，因为它们可以被视为规范理论中的孤子，并且与希格斯束和磁漩涡有关。这样的度量解偏微分方程，称为曲率方程，其中奇点可以被视为边界数据。近年来，在这一领域中，分析学和几何学的相互作用有了许多发展，出现了分层和冒泡等新的令人惊讶的现象。该项目的目标是开发新技术，以帮助发现这些对象的新特征，并深入了解与奇异度量相关的问题，包括奇异均匀化，模空间和奇异几何流形上的偏微分方程的解。该项目涉及使用几何微局部分析研究奇异度量。中心思想是引入新的对象，称为紧化或分辨率，来解决奇点。这些决议将反过来建议需要发展哪些分析技术。PI计划利用该方法研究常曲率圆锥度量的模空间构造及其与涡旋的关系、带尖的双曲度量与紧化黎曼模空间的渐近几何、具有奇异度量背景的规范理论偏微分方程等问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.