麻省理工学院-数学

研究计划：几何分析导论

摘要：

本课程旨在向有志学生介绍现代几何分析的基础知识。课程从偏微分方程和微分几何的基本概念和技术入手，最终引导学生阅读该领域中的重要期刊论文。

在为学生提供必要背景知识后，课程的最后阶段将过渡到几何分析中的两个核心研究主题：极小曲面和几何流。极小曲面这一课题已有一个多世纪的历史，它模拟了自然界中出现的几何形状，如肥皂泡和黑洞。另一方面，几何流则是几何形状上的时间相关动力系统。值得注意的例子包括Ricci流、平均曲率流、Yamabe流、Calabi流等。特别是，Ricci流被用于解决庞加莱猜想（由Hamilton和Perelman于2002-03年提出），以及可微球定理（由Brendle和Schoen于2007年提出）。反平均曲率流也被Huisken和Ilmanen用于证明广义相对论中的时间对称Penrose不等式。

Introduction to Geometric Analysis Research Schedule

Abstract:

This course aims at introducing the basics of modern geometric analysis to motivated students. It starts with an introduction to the elementary concepts and techniques in partial differential equations and differential geometry, and eventually guides students to read some important journal papers in this area.

After equipping students with necessary backgrounds, the last phase in the course transitions into the study of two central research topics in geometric analysis: minimal surfaces and geometric flows. Minimal surfaces, a topic with over a century of history, model on geometric shapes appeared in the nature, ranging from soap bubbles and black holes. On the other hand, geometric flows are time-dependent dynamical systems on geometric shapes. Notable examples include the Ricci flow, mean curvature flow, Yamabe flow, Calabi flow, etc. In particular, the Ricci flow was applied to resolve the Poincare Conjecture (by Hamilton and Perelman in 2002- 03), and the Differentiable Sphere Theorem (by Brendle and Schoen in 2007). The inverse mean curvature flow was also used by Huisken and Ilmanen to prove the time-symmetric Penrose in- equality in General Relativity.