Transformation Groups in Conformal and Projective Geometry

主要研究员将研究刚性的微分几何方面：什么可以是紧凑空间上的几何结构的对称性？ 假设高度对称，空间的几何和拓扑性质是什么？ PI的工作集中在这些问题的设置伪黎曼，共形，和射影几何。 伪黎曼度规，特别是洛伦兹度规，产生于广义相对论;在这里紧致时空是不现实的，但非紧致时空的共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就该项目的各个方面进行研究，预计他们将获得该奖项的支持。PI将在她最近的进展基础上，证明具有有限基本群的实解析流形的Lorentzian Lichnerowicz猜想，研究具有无限基本群但低维的流形的猜想，以及具有本质共形群的平坦共形结构的变形刚度。PI将致力于在投影设置中应用她的技术，其中由伪黎曼度量产生的投影结构的类似猜想在更高的签名中仍未解决。PI将继续研究变换群和拖拉机解之间的关系。在光滑双曲动力学的背景下，PI将建立在非均匀收缩cocycles的不变微分几何结构的建设上，在合适的假设下找到更精细的不变结构，并将其应用于刚性results.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.