Mathematical Sciences: "Studying the Topology of Riemannian Manifolds through Ricci Deformation of the Metric"

首席研究员将研究利玛窦变形的 黎曼流形上的度量这项技术是由 汉密尔顿在1982年，并已成为一个有用的和众所周知的 微分几何中的方法研究的目标是支持 这个奖项是一个更好地了解拓扑结构的 满足曲率和Ricci限制的流形 特别是曲率。抛物偏微分理论 微分方程将被用来获得估计的 解的Ricci变形，然后应导致 附加拓扑信息。 曲率的抛物线变形的一个物理例子是 拉伸的橡皮筋会经历的变形， 浸在粘性液体如油中。只能以 带将渐近变形为圆，如果 封闭区域被重新归一化为1。该方法可用于 研究一个曲面或流形的拓扑结构， situations.首席研究员将尝试获得精确的 估计变形的大小并使用该信息 以获得拓扑信息。

The principal investigator will study Ricci deformation of the metric on Riemannian manifolds. This technique was introduced by Hamilton in 1982 and has since become a useful and well known method in differential geometry. The goal of the research supported by this award is a better understanding of the topology of manifolds which satisfy restrictions on curvature and Ricci curvature in particular. The theory of parabolic partial differential equations will be used to obtain estimates on the solution of the Ricci deformation which should then lead to additional topological information. A physical example of parabolic deformation of curvature is the deformation a stretched rubber band would experience if it were immersed in a viscous fluid such as oil. It is possible to show that the band would asymptotically deform to a circle if the enclosed area were renormalized to one. This method can be used to study the topology of a surface or manifold in more complicated situations. The principal investigator will try to obtain precise estimates for the size of the deformations and use that information to obtain topological information.