Moduli Space of Canonical Metrics on Four-Manifolds

世纪以来，专门研究微分几何的数学家一直在研究什么是最好的几何（即，正如一个沙滩球可以被塑造成一个完美的球体，在所有点和所有方向上具有相同的曲率，所有二维空间都可以被塑造成高度对称和均匀的几何形状，同样，所有三维空间都可以被切割成带有这种几何形状的碎片。但相应的问题对于四维空间是完全开放的。 此外，这个维度在其他维度中脱颖而出，因为某些四维空间可以支持无限多个不同的光滑结构，这些结构可以容纳具有根本不同性质的几何形状。这些事实使得四维几何在许多方面不可预测，并成为跨学科研究的肥沃领域，因为该学科已经在许多方向发展并分支到数学和物理的其他领域。 这个研究项目是基于包括几何，拓扑和分析领域之间的相互作用。 研究人员将深入分析一类无界的4维空间，从广义相对论的角度来看，这是很重要的，携带一个几何模型的复数，其几何是温和的统一。在另一个方向上，研究者将研究曲率类似均匀的封闭四维空间。 更详细地说，研究人员将对渐近局部欧几里得凯勒流形进行全面分析。这是一类特殊的非紧四维流形，它是理论物理中引力瞬子的推广。该项目在理解闭常数量曲率Kähler或几乎Kähler流形的全局结构方面有着深远的应用。第二组项目集中于研究4-流形的黎曼性质，并结合来自Seiberg-Witten理论的技术。特别是，研究者将分析爱因斯坦度量或常数标量曲率度量的模空间及其在Gromov-Hausdorff拓扑中的紧化。所采用的技术来自规范理论，全球分析，代数几何和微分拓扑。

For more than a century, mathematicians specializing in the subject of differential geometry have worked on the question of what is the best geometry (i.e., shape) that a given space can have: Just as a beachball can be shaped into a perfect sphere, with the same amount of curvature at all points and in all directions, all 2-dimensional spaces can be shaped into highly symmetric and uniform geometries, and similarly all 3-dimensional spaces can be cut into pieces that carry such geometries. But the corresponding question is completely open for 4-dimensional spaces. Moreover, this dimension stands out among others, since certain 4-dimensional spaces can support infinitely many different smooth structures, which may admit geometries with radically different properties. These facts make 4-dimensional geometry in many ways unpredictable, and a fertile field for interdisciplinary research, as the subject has grown in many directions and branched into other areas of mathematics and physics. This research project is based on interactions between fields including geometry, topology, and analysis. The investigator will undertake an in-depth analysis of a class of unbounded 4-dimensional spaces that are important from the perspective of general relativity, that carry a geometry modeled on the complex numbers, and whose geometry is mildly uniform. In another direction, the investigator will examine closed 4-dimensional spaces whose curvature is similarly uniform. In more detail, the investigator will undertake a comprehensive analysis of the asymptotically locally Euclidean Kähler manifolds. This is a special class of non-compact 4-dimensional manifolds that generalizes the gravitational instantons from theoretical physics. The project has far-reaching applications in understanding the global structure of closed constant scalar curvature Kähler or almost-Kähler manifolds. A second group of projects focuses on the study of the Riemannian properties of 4-manifolds in conjunction with techniques coming from Seiberg-Witten theory. In particular, the investigator will analyze the moduli space of Einstein metrics or constant scalar curvature metrics and their compactifications in Gromov-Hausdorff topology. The techniques employed come from gauge theory, global analysis, algebraic geometry, and differential topology.