Mathematical Sciences: Twistor, Complex Manifolds, and Differential Geometry

Claude LeBrun将研究许多黎曼几何中的问题或与之相关的问题，在这些问题中，复解析技术可能通过扭转或对应的方式来发挥作用-也就是说，通过将黎曼流形的点视为一些相关复流形的紧复子流形。其中一些问题，特别是关于爱因斯坦方程和杨-米尔斯方程的解的问题，是由物理学引起的。另一些是几何内部的问题和分析，涉及几何特征的特定偏微分方程解的存在或不存在。这项研究将主要朝三个方向进行。首先，他将寻找大量四元数流形的例子。第二个方向是进一步扩展Witten对Yang-Mills场的解释，根据嵌入的三阶无穷小邻域。这样的扩展已经由LeBrun在自对偶流形的情况下进行了。程序的最后一部分涉及定义CR结构概念的四元数类比。

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.