Topics in Ricci flow, Riemannian geometry, metric geometry and PDE

里奇流、黎曼几何、度量几何和偏微分方程主题

• 批准号: 2104917
• 金额: --
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2104917
• 关键词: Topics; Ricci; flow; Riemannian; geometry

The context and potential impact: Geometric analysis is a booming area of mathematics research, with a very large number of spectacular breakthroughs to its name over recent years. It has an unbeaten record of solving problems in other areas of mathematics, such as topology and differential geometry, and this is expected only to accelerate. Aims and objectives: At the broadest scale, the main aim is to contribute to this great ongoing advance. One area of particular interest is the solution of the Ricci flow on noncompact manifolds in the absence of curvature bounds. As a key ingredient in this programme, Luke is currently investigating a completely new approach to proving integral curvature bounds. This new approach completely avoids the highly technical nature of similar previous results, and offers the hope to prove much stronger results.Alignment to EPSRC strategy: This is fundamental research with great potential. The general area of research has been repeatedly stressed as requiring support from EPSRC by the two major international reviews of mathematics that EPSRC has commissioned. It offers hope of progress across multiple areas that EPSRC strategy supports, including Geometry, Nonlinear Analysis, Mathematical Analysis, etc.

背景和潜在影响：几何分析是数学研究的一个蓬勃发展的领域，近年来有大量引人注目的突破。它在解决其他数学领域的问题方面有着不败的记录，如拓扑学和微分几何，预计这只会加速。宗旨和目标：在最广泛的范围内，主要目标是促进这一正在取得的巨大进步。特别感兴趣的一个领域是解决方案的里奇流的非紧流形的曲率界限的情况下。作为该计划的关键成分，卢克目前正在研究一种全新的方法来证明积分曲率界。这种新方法完全避免了类似的先前结果的高度技术性，并提供了希望证明更强大的结果。一般领域的研究已一再强调，需要支持EPSRC的两个主要的国际审查的数学，EPSRC已委托。它提供了在EPSRC战略支持的多个领域取得进展的希望，包括几何，非线性分析，数学分析等。