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，近年来有很多引人注目的突破。在解决拓扑学和微分几何等其他数学领域的问题方面，它有着无与伦比的记录，而且预计这种记录只会加速。宗旨和目标：在最广泛的范围内，主要目标是为这一伟大的持续进步作出贡献。一个特别有趣的领域是在没有曲率边界的非紧化流形上的Ricci流的解。作为这个项目的关键组成部分，卢克目前正在研究一种全新的方法来证明积分曲率边界。这种新方法完全避免了先前类似结果的高度技术性，并提供了证明更强有力的结果的希望。与EPSRC战略保持一致：这是具有巨大潜力的基础研究。EPSRC委托进行的两项主要的国际数学评论一再强调，一般的研究领域需要EPSRC的支持。它为EPSRC战略支持的多个领域带来了进步的希望，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.