Distinguished Geometric Structures with Symmetry in Four Dimensions

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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.