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

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，近年来它的名字取得了大量惊人的突破。它在解决其他数学领域的问题方面有着不败的记录，例如拓扑学和微分几何，预计这一速度只会加快。宗旨和目标：在最广泛的范围内，主要目标是为这一正在进行的巨大进步做出贡献。一个特别有趣的领域是非紧流形上的Ricci流在没有曲率界的情况下的解。作为这一计划的一个关键因素，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以前类似结果的高度技术性，并提供了证明更强大结果的希望。与EPSRC战略一致：这是具有巨大潜力的基础研究。EPSRC委托进行的两次主要的国际数学审查一再强调，一般研究领域需要EPSRC的支持。它提供了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。

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.