Transformation Groups in Conformal and Projective Geometry

首席研究员将研究刚性的微分几何方面：紧致空间上几何结构的对称性是什么？假设高度对称，空间的几何和拓扑性质是什么？PI的工作集中在伪黎曼几何、共形几何和射影几何的背景下研究这些问题。广义相对论中出现了伪黎曼度量，特别是洛伦兹度量；这里紧化时空是不现实的，但非紧化时空的保形或射影紧化是相当重要的。在物理学中，对称性对应于守恒定律。在几何学中，它们为分类提供了一个框架。PI正在指导两名研究生对该计划的各个方面进行研究，他们预计将得到该奖项的支持。在她的最新进展，有限基本群实解析流形的Lorentzian Lichnerowicz猜想的证明的基础上，PI将研究具有无限基本群但低维的流形的猜想，以及具有基本共形群的平面共形结构的变形刚度。PI将致力于将她的技术应用于射影设置，其中由伪黎曼度量产生的射影结构的类似猜想在高签名中尚未解决。私家侦探将继续调查转型小组和拖拉机解决方案之间的关系。在光滑双曲动力学的背景下，PI将建立在非均匀收缩环的不变微分几何结构的构造上，在合适的假设下找到更精细的不变结构，并将这些结构应用于刚性结果。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。

The Principal Investigator will study differential-geometric aspects of rigidity: what can be the symmetries of a geometric structure on a compact space? Assuming a high degree of symmetry, what can be the geometric and topological properties of the space? The PI's work focuses on these questions in the setting of pseudo-Riemannian, conformal, and projective geometry. Pseudo-Riemannian and, in particular, Lorentzian metrics, arise in General Relativity; here compact spacetimes are not realistic, but conformal or projective compactifications of noncompact spacetimes are quite important. In physics, symmetries correspond to conservation laws. In geometry, they provide a framework for classification. The PI is mentoring two graduate students in research on aspects of this program, who are expected to be supported by the award.Building on her recent advance, the proof of the Lorentzian Lichnerowicz Conjecture for real-analytic manifolds with finite fundamental group, the PI will investigate the conjecture for manifolds with infinite fundamental group but low dimension, and deformation rigidity of flat conformal structures with essential conformal group. The PI will work to apply her techniques in the projective setting, where an analogous conjecture for projective structures arising from pseudo-Riemannian metrics remains unsolved in higher signature. The PI will continue her investigation of relations between transformation groups and tractor solutions. In the context of smooth hyperbolic dynamics, the PI will build on the construction of invariant differential-geometric structures for nonuniformly contracting cocycles, to find more refined invariant structures under suitable hypotheses, and applications of these to rigidity results.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.