Mathematical Sciences: Twistors, Complex Manifolds, and Differential Geometry

主要研究者将用两种不同的方法研究复维常数标量曲率Kaehler流形的构造。其中第一个涉及到广义的Gibbon-Hawking ansatz，而第二个使用了Banach空间反函数定理，应用于具有正规型点奇点的Kaehler势。他也将研究具有四元数完整的完全非紧黎曼流形的整体变形。最后，他将研究具有自旋结构的黎曼曲面的通用曲线的整体结构。在这个项目中，首席研究员将分析与扭转和弦理论有关的复杂流形。这些数学理论已经被用来解释亚原子粒子的一些行为。流形是数学对象，包括规则的曲线和曲面作为例子。这两个例子都是基于实数的多个副本。在这项研究中，流形是基于复数的副本，而不是实数，将被用来分析几个问题，从物理。

The principal investigator will study construction of constant scalar curvature Kaehler manifolds of complex dimension 2 by two different methods. The first of these involves a generalized Gibbon-Hawking ansatz, while the second uses the Banach space inverse function theorem applied to Kaehler potentials with point singularities of conormal type. He will also study global deformations of complete non-compact Riemannian manifolds with quaternionic holonomy. Lastly, he will investigate the global structure of the universal curve for Riemann surfaces with spin structure. In this project, the principal investigator will analyze complex manifolds which relate to twistor and string theory. These mathematical theories have been used to explain some of the behavior of subatomic particles. Manifolds are mathematical objects which include regular curves and surfaces as examples. Both of these examples are based on multiple copies of the real numbers. In this study, manifolds which are based on copies of the complex numbers, instead of the real numbers, will be used to analyze several questions arising from physics.