Challenges in Negative and Nonpositive Curvature

这些项目将研究一些关于负曲线和非正曲线度量的基本问题。 例如，关于空间上存在这种几何结构的问题，即如何构造非正或负弯曲的度量。这不仅包括一般的测地线度量，而且还包括更规则的光滑版本，即黎曼度量。主要研究员计划继续他的拓扑空间的负弯曲度量，及其变化：Teichmuller空间的负弯曲度量和模空间的负弯曲度量，连同其推广到非正曲率。他还计划继续研究CAT（0）空间的一些基本问题，曲率小于无穷大的空间中测地线的存在，并继续研究流形拓扑的一般领域及其在几何学中的应用。该奖项反映了NSF的法定使命，并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，被认为值得支持。

These projects will study some fundamental questions about negatively curved and nonpositively curved metrics. For instance, questions about the existence of such geometric structures on spaces, that is, how to construct nonpositively or negatively curved metrics. This includes not just general geodesic metrics but also the more regular smooth versions, i.e, Riemannian metrics.The principal investigator plans to continue his study of the topology of the space of negatively curved metrics, and its variations: the Teichmuller space of negatively curved metrics and the moduli space of negatively curved metrics, together with its generalizations to nonpositive curvature. He also plans to continue his work on some fundamental questions about CAT(0) spaces, the existence of geodesics in spaces of curvature less than infinity, and continue his research on the general area of manifold topology and its applications to geometry.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.