Geometric aspects of partial differential equations

阿尔琼项目的主要目的是调查一系列关于偏微分方程(PDE)行为的试探性猜想，这些猜想具体出现在几何学中。这类方程的性质似乎往往比一般的偏微分方程组理论所期望的要强得多。这项研究的一个关键目标是用非常粗略的初始数据来理解平均曲率流的适定性。一种全新的方法论正在尽可能低的维度中进行，包括为经历曲线缩短流动的曲线长度建立急剧的衰减结果。近年来在Ricci流中的类似结果在微分几何的研究中有着深远的意义。该项目符合EPSRC的“数学分析”和“几何和拓扑学”研究领域。

The key aim of Arjun's project is to investigate a selection of tentative conjectures concerning the behaviour of partial differential equations (PDE) that arise specifically in geometry. Such equations appear often to have properties that are much stronger than one would expect from generic PDE theory. One key objective within this research is to understand the well-posedness of mean curvature flow with very rough initial data. A completely novel methodology is being pursued in the lowest possible dimension, involving establishing sharp decay results for the length of curves undergoing curve shortening flow. Analogous results in Ricci flow over recent years have had profound implications in the study of differential geometry. The project fits into both the `Mathematical Analysis' and `Geometry and Topology' EPSRC research areas.