Microlocal analysis and singularities

退化和奇点形成在微分几何的研究中扮演着重要的角色，并自然地出现在数学的其他多个领域，包括代数几何、数学物理、数论和表示论。这个项目涉及奇点的几何学，并与一系列学科有有趣的联系，包括电磁系统建模、规范理论和弦理论。在她过往记录的基础上，PI计划使用有效的技术来解决各种问题，并发现新的、更尖锐的分析结果。除了她的研究，PI还将参与各种外展活动，重点是培养本科生和研究生的指导网络。这个项目涉及使用几何微局域分析来研究奇异度量。PI将解决与度量退化有关的问题，包括扰动理论、奇异均匀化问题、模空间的参数化以及与奇异度量相关的几何性质。工具包括几何分辨率或紧凑化，它们解决奇点并提供更详细的分析性质，此外还与代数和微分几何交互，这反过来又建议需要开发哪些PDE技术。PI将使用这些方法来研究常曲率圆锥度量的模空间的分层和奇异性、带尖点的双曲度量及其谱间隙、以及规范理论的奇异性度量，例如引力瞬子及其简并。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.