Mathematical Sciences: Twistor, Complex Manifolds, and Differential Geometry

克劳德·勒布伦将研究黎曼几何中或与黎曼几何有关的一些问题，在这些问题中，复分析技术可以通过扭曲对应的方式来应用--即通过将黎曼流形的点视为某些相关复流形的紧致复子流形。其中几个问题，特别是关于爱因斯坦方程和杨-米尔斯方程的解的问题，都是由物理驱动的。另一些是几何学和分析的内部问题，涉及具有几何性质的特定偏微分方程解的存在或不存在。这项研究将在三个主要方向上进行。首先，他将寻找四元数流形的大类例子。第二个方向是关于Witten对Yang-Mills场的解释在嵌入的三阶无穷小邻域方面的进一步扩展。LeBrun已经在自对偶流形的情况下进行了这样的推广。该计划的最后部分涉及定义CR结构概念的四元数模拟。

Claude LeBrun will study a number of problems in or related to Riemannian geometry in which complex-analytic techniques may be brought to bear by means of a twistor correspondence - that is, by thinking of the points of a Riemannian manifold as compact complex submanifolds of some associated complex manifold. Several of these problems, notably questions concerning solutions to Einstein's equations and the Yang-Mills equations, are motivated by physics. Others are problems internal to geometry and analysis concerning the existence or non-existence of solutions to particular partial differential equations of a geometric character. There are three principal directions in which the research will proceed. First he will seek large categories of examples of quaternionic manifolds. The second direction concerns further extensions of Witten's interpretation of Yang-Mills fields in terms of third order infinitesimal neighborhoods of embeddings. Such extensions have already been carried out by LeBrun in the case of self dual manifolds. The final part of the program involves defining a quaternionic analog of the notion of CR structure.