Mathematical Sciences: Differential Geometry

首席研究员将研究偏微分方程和微分几何理论中的几个问题。他将构造具有特殊完整度的完备黎曼流形，并研究它们与“超膜”和“超引力”物理理论的关系。在3流形几何和外微分系统理论中出现的退化偏微分方程将是两个进一步研究的主题。近年来，引力和亚原子粒子的物理学，以及微分几何和几何偏微分方程的数学之间的相互作用变得越来越强烈。几何结构已经发展了几十年，现在已经足够强大，可以处理现代物理学的许多新问题。首席研究员将解决一个与最近的“超重力”理论有关的黎曼流形的完整性问题。

The principal investigator will study several problems from the theories of partial differential equations and differential geometry. He will construct complete Riemannian manifolds with special holonomy and study their relationship with the physical theories of "supermembranes" and "supergravity." Degenerate partial differential equations which arise in the geometry of 3- manifolds and the theory of exterior differential systems will be the subjects of two further investigations. The interplay of ideas between the physics of gravitation and subatomic particles, and the mathematics of differential geometry and geometric partial differential equations, has become intense in recent years. Geometric structures which have been under development for many decades are now sufficiently powerful to handle many new problems of modern physics. The principal investigator will solve a problem in holonomy of Riemann manifolds which relates to recent theories of "supergravity."