The Curvature of 4-Manifolds

爱因斯坦广义相对论方程通过约束4维时空的曲率来控制引力场。这个项目涉及这些方程在黎曼环境中的一系列推广，其中没有空间和时间的差异，重点是试图确定哪些4维空间包含这些方程的解。该项目还将通过参与与该项目有关的活动，帮助培训一批研究生进行数学研究。这个项目特别寻求发现和阐明四维空间中曲率和拓扑之间的基本联系。其中一个焦点是紧致流形上的正则度量的构造，主要是借助于Kaehler和共形几何。另一个重点是了解相关的解的模空间，以及相关的气泡现象。这一领域的进展可能会对纯数学以外的领域产生重大影响，特别是因为许多要研究的几何问题与经典和量子引力问题以及理论物理的其他领域有着内在的联系。该项目还包括对学生和初级研究人员的培训和指导。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。

The Einstein equations of general relativity govern the gravitational field by constraining the curvature of a 4-dimensional space-time. This project concerns a family of generalizations of these equations in the Riemannian setting, where there is no difference between space and time, with an emphasis on trying to determine which 4-dimensional spaces carry solutions of these equations. The project will also help train a group of graduate students in mathematical research through their participation in activities related to the project. This project specifically seeks to discover and elucidate fundamental links between curvature and topology in dimension four. One focus is the construction of canonical metrics on compact manifolds, primarily by means of Kaehler and conformal geometry. Another focus is on understanding the relevant moduli spaces of solutions, and associated bubbling phenomena. Progress in this area could have significant repercussions outside of pure mathematics, especially because many of the geometrical questions to be investigated are intrinsically linked to problems in classical and quantum gravity, as well to other areas of theoretical physics. The project also includes training and mentoring of students and junior researchers.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.