Distinguished Geometric Structures with Symmetry in Four Dimensions

里奇流是一个几何演化方程，在几何、拓扑学、物理学、生物学和图像处理等领域都有重要的应用。本项目旨在提供一个完整的描述和分类的高度对称，自相似的解决方案，在四维里奇流。这样的分类对于理解流的行为是必不可少的，但迄今为止还没有被发现。这个项目打算结合纯数学和计算技术来完成这个分类。这样的结果将大大提高对四维流形几何的理解，潜在地导致在几个科学领域的应用以及图像处理。

The Ricci flow is a geometric evolution equation having significant applications in geometry, topology, as well as in physics, biology and image processing. This project aims to provide a complete description and classification of highly symmetric, self-similar solutions to the Ricci Flow in four dimensions. Such a classification is essential to understanding the behaviour of the flow, but has so far evaded discovery. This project intends to combine techniques from pure mathematics with computational techniques to complete this classification. Such an outcome would greatly improve the understanding of the geometry of four-dimensional manifolds, potentially leading to applications in several areas of science as well as image processing.