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: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=362604
• 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30年或更长的时间里，偏微分方程和黎曼几何的数学领域已经走到一起，创造了称为几何分析的领域。这部分是由广义相对论推动的，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步。一个例子是利玛窦流，它现在被理解到它已经产生了百年历史的庞加莱猜想和更广泛的瑟斯顿猜想的主张，描述了所有光滑的三维紧致物体（“流形”）的性质。这个项目的目的是将这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。一个相关的流动承诺是重要的理解爱因斯坦的广义相对论的时间无关的解决方案。本项目旨在探索这些联系。