Ricci flow from spaces with edge type conical singularities

里奇流在EPSRC的“几何和拓扑”研究领域的最新突破中已经被视为一种工具：它是证明庞加莱猜想和几何化猜想、可微球定理、证明安德森-契格-柯林-田猜想在三维空间和广义小猜想的基础。在M. Simon和P. Topping在三维空间证明Anderson-Cheeger-Colding-Tian猜想的工作中，使用了Ricci流来平滑Ricci极限空间。在这个方向上，一个进一步开放的问题是里奇流是否可以用来平滑更高维度的正弯曲多面体空间。最近，Bamler-Cabezas-Rivas-Wilking和Gianniotis-Schulze用不同的方法在这个方向上迈出了第一步。Gianniotis-Schulze的工作表明，可以从具有孤立圆锥奇点的紧致流形开始构造里奇流，该流形以正弯曲锥为模型。作为从多面体空间流动的第一步，Lucas Lavoyer de Miranda正致力于将Gianniotis-Schulze的结果扩展到圆锥形奇点沿着封闭曲线出现的情况。有一些新的想法和技术有待发展。这个项目的成功完成将是开发一种从正弯曲多面体空间流动的方法的基础。

Ricci Flow has seen been a tool in recent breakthroughs in the EPSRC research area of 'Geometry and Topology': It was fundamental in the proof of the Poincaré and Geometrization Conjectures, the Differentiable Sphere Theorem, the proof of the Anderson-Cheeger-Colding-Tian conjecture in dimension three, and the Generalized Smale Conjecture. In work of M. Simon and P. Topping on the proof of the Anderson-Cheeger-Colding-Tian conjecture in dimension three, Ricci Flow was used to smooth out Ricci limit spaces. A further open question in this direction is if Ricci Flow can be used to smooth out positively curved polyhedral spaces in higher dimension. First steps in this direction have recently been achieved by Bamler-Cabezas-Rivas-Wilking and Gianniotis-Schulze with different approaches. The work of Gianniotis-Schulze shows that it possible to construct a Ricci Flow starting from a compact manifold with isolated conical singularities which are modelled on positively curved cones. As a first step towards flowing from polyhedral spaces, Lucas Lavoyer de Miranda is working on extending the results of Gianniotis-Schulze to the case that the conical singularities occur along a closed curve. There are several new ideas and techniques to be developed. Successful completion of this project will be fundamental in developing an approach to flowing from positively curved polyhedral spaces.