Applications of Riemannian Geometry and Ricci flow in physics

黎曼几何和里奇流在物理学中的应用

• 批准号: 203614-2007
• 负责人: Woolgar, Eric
• 金额: $ 2.25万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2010
• 资助国家: 加拿大
• 起止时间: 2010-01-01 至 2011-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=453407
• 关键词: Applications; Riemannian; Geometry; Ricci; flow

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.

在过去的三十年或更长时间里，偏微分方程和黎曼几何的数学领域结合在一起，创建了称为几何分析的领域。这在一定程度上是受到广义相对论的推动，从一开始这种综合就取得了相当大的进展。近年来，几何分析的某些领域取得了惊人的数学进展。一个例子是里奇流，现在人们对它的理解已经达到了百年前的庞加莱猜想和更广泛的瑟斯顿猜想的主张，描述了所有光滑的 3 维致密物体（“流形”）的性质。该项目的目的是将其中一些进展应用于物理学问题。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。相关流程对于理解爱因斯坦广义相对论的时间无关解非常重要。该项目旨在探索这些联系。