Serkan Salik Rose Hills
Mean Curvature Flow and the PDE-ODI Principle
Mean curvature flow is a way of evolving some initial surface in 3-dimensional space according to a rule that makes its geometry more homogeneous. Unfortunately, you can’t always do this for all time, and the flow frequently incurs “singularities”, where a region of the surface collapses and we are no longer working with a surface, so can’t keep flowing. To understand this singularity formation, we look at the surface at a very small time before it collapses and zoom in really close to the region that collapses. We get a “model” for the singularity in this way, which is a way of understanding what structure in the surface actually caused the singularity to form in the first place. Understanding models of singularities better helps us understand the large-scale structure of the flow, and is one of the central problems in the field. Recently, Bamler-Lai have developed a technique called the PDE-ODI principle, which allows one to understand these singularity models in a more quantitative manner. That will be the focus of this project: we will compute the so-called asymptotics of standard singularity models, which are (surprisingly) not available in the literature.
Message To Sponsor
Thank you so much for your support. I’ve been wanting to do something original for so long, as opposed to just reading more and more of the literature, so I’m very excited for this summer.