Topics in Ricci flow, Riemannian geometry, metric geometry and PDE
Lead Research Organisation:
University of Warwick
Department Name: Mathematics
Abstract
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.
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.
Organisations
People |
ORCID iD |
| Peter Topping (Primary Supervisor) | |
| Luke Peachey (Student) |
| Description | An original aim of my project was to investigate Ricci flow on noncompact manifolds in the absence of curvature bounds. One of the major outcomes of my research has been a new rigidity result for noncompact spacetimes that admit complete Ricci flows. Rigidity results are much sort after in geometric analysis as they provide additional structure, that was a priori unknown, to useful objects of study. My work in this area has led to a number of interesting new research questions, including the question of if there are similar rigidity results in higher dimensions. As well as studying Ricci flow, my research project also involved the study of mean curvature flow. Much like Ricci flow, mean curvature flow is another central area of study in geometric analysis, with applications in geometry, topology, general relativity, material science and image processing. Another major outcome of my research has been a new non-uniqueness result for noncompact solutions to mean curvature flow. Given a geometric flow and a class of solutions, a uniqueness statement shows that there is a canonical evolution within the class of solutions from the original initial data under the flow. As such, a non-uniqueness result highlights important intricacies previously unknown in the evolution equation. This work has also opened up new research avenues, including finding the correct geometric hypotheses to impose on noncompact solutions, in a way analogous to completeness of solutions in Ricci flow, to ensure the evolution under the flow is canonical. |
| Exploitation Route | I hope to build upon my outcomes and pursue some of the research questions detailed in the above section in future research projects. I am currently unaware of other ongoing research building upon my results, although this may changes in the future. |
| Sectors | Other |