几何分析中的基本概念与原理

摘要

The present article is based on a talk that I gave at the Chinese Academy of Sciencesin Beijing on October 9, 2001, on the occasion of the inaugration of the Partner Group on GeometricAnalysis for the Max Plank Institute for Mathematics in the Sciences in Leipzig at the CAS. It aims atproviding a perspective on the present state and future research lines of geometric analysis through asurvey of the basic principles and concepts. These are based on ideas from both Riemannian geometryand quantum field theory, and in many cases, they lead to variational problems for minimizing aphysical action or optimizing a geometric quantity.%本文是根据作者在中国科学院"中德几何分析伙伴小组"成立仪式上的演讲整理而成.主要目的在于通过回顾几何分析中的基本概念及基本定理,了解现阶段的研究情况和将来的研究线路.主要描述黎曼几何与量子场论中的思想,特别特别是极化物理量与几何量产生的变分问题.