Applications of Riemannian Geometry and Ricci flow in physics

在过去的三十多年里，偏微分方程和黎曼几何的数学领域走到了一起，创造了一个叫做几何分析的领域。这在一定程度上是受广义相对论的推动，从一开始，这种综合就取得了相当大的进展。近年来，在几何分析的某些领域有了惊人的数学进展。一个例子是里奇流，现在人们对它的理解已经达到了一个地步，它已经提出了一百年前的庞加莱猜想和更广泛的瑟斯顿猜想，描述了所有光滑的三维致密物体（“流形”）的性质。这个项目的目的是将这些进步中的一些应用于物理问题。例如，里奇流也被认为至少近似地描述了弦理论中的某些动力学行为。在理解爱因斯坦广义相对论的不依赖于时间的解时，一个相关的流是很重要的。这个项目旨在探索这些联系。

In the past thirty years or more, the mathematical fields of partial differential equations and Riemannian geometry have come together to create the field called geometric analysis. This was partly motivated by General Relativity, where from the very beginning this synthesis has yielded considerable progress. In recent years, there have been spectacular mathematical advances in certain areas of geometric analysis. An example is the Ricci flow, which is now understood to the point where it has yielded a claim of the hundred year old Poincare conjecture and the much broader Thurston conjecture, describing the nature of all smooth 3-dimensional compact objects ("manifolds"). The purpose of this project is to apply some of these advances to problems in physics. For example, the Ricci flow is also believed to describe, at least approximately, certain dynamical behaviour in string theory. A related flow promises to be important in the understanding of time-independent solutions of Einstein's general relativity theory. This project is aimed at exploring these connections.