Mathematical Sciences: Complex Geometry and Mathematical Physics

这两位研究人员将研究起源于数学物理和复杂几何的各种几何结构。其中包括射影代数流形、复流形、四元数Kahler流形和超流形。这项研究将包括与四元数Kahler约化有关的动量映射、自对偶流形以及四元数Kahler结构的扭量表示。用计算机研究二维Yang-Mills-Higgs方程静态自对偶解的动力学。这两位研究人员将继续研究新的“超几何”，它是规范理论的一部分，目的是了解宇宙学和亚原子物理之间的关系。其中一人将在奈杰尔·希钦的框架内继续他关于旋风几何的工作。第二部分将利用“四元数商”进一步得到具有不寻常自旋完整的完备非紧致流形的知识。这是叶理的一种性质，衡量了它们的剪切倾向。

The two investigators will study a variety of geometric structures which have their origins in mathematical physics as well as complex geometry. These include projective algebraic manifolds, complex manifolds, quaternionic Kahler manifolds, and supermanifolds. The investigation will include a study of the momentum mapping in connection with quaternionic Kahler reduction, self-dual manifolds, and the twistor representation of quaternionic Kahler structures. The dynamics of the static self- dual solutions of the two-dimensional Yang-Mills-Higgs equations will be studied with the use of a computer. The two investigators will continue their studies of the new "supergeometry" which was created as part of gauge theory to understand the relationship between cosmology and subatomic physics. One will continue his work on the twistor geometries in Nigel Hitchin's framework. The second will use "quaternionic quotients" to derive further knowledge of complete noncompact manifolds with unusual spin holonomy. This last is a property of foliations and measures their shearing tendencies.