Nonlinear Heat Equations in Geometry, Mechanics and Imaging

PI旨在研究黎曼几何中热方程奇异解的渐近性，如Ricci流和平均曲率流，以及涉及四阶方程的自由边界问题。 PI感兴趣的进一步问题涉及Monge-Kantorovich质量传递问题的公式化，特别是可能出现在医学成像问题中的变化。微分几何和偏微分方程（PDE）在现代科学和工程中起着核心作用。 PI有兴趣从“纯数学”的角度研究这些主题，并与工程师和科学家合作，寻找几何和PDE在实际问题中的新应用。 PI感兴趣的特定应用领域涉及医学科学中生成的图像的自动处理（脑扫描，虚拟结肠镜检查等）。

The PI intends to study the asymptotics of singular solutions to heat equations which occur in Riemannian geometry, such as Ricci flow and mean curvature flow, and also free boundary problems involving fourth order equations. Further problems of interest to the PI are related to the formulation of the Monge-Kantorovich mass-transfer problem, and in particular variations thereof which may appear in medical imaging problems.Differential Geometry and partial differential equations (PDE) play a central role in modern science and engineering. The PI is interested both in studying these subjects from the "pure-mathematics" point of view, and in collaborating with engineers and scientists to find new applications of geometry and PDE to practical problems. Particular applied areas of interest to the PI involve the automated processing of images (brain scans, virtual colonoscopy, etc) generated in the medical sciences.