Geometric Analysis plays a central role in mathematics research and has led to many important breakthroughs. The course will first make sure the students familiar with some fundamental material and techniques in Geometric Analysis. It will closely follow the book “Lectures on Differential Geometry” by Schoen and Yau. The material of Chapter 1 Comparison Theorem and Gradient Estimates, Chapter 3 Eigenvalue Problems, and Chapter 4 Heat Kernel on Riemannian Manifolds will be the main focus of the first part of the course. Another reference will be Peter Li’s “Lecture Notes on Geometric Analysis (and Curvature and Function Theory on Riemannian Manifolds)”. Some notes/videos of the lecturer’s 2022 Geometry II course can also be reference for students with less background.

Another focus of the course is to guide students to study related papers and results of some recent research and development in Geometric Analysis. The detailed content of the course will depend on the background of the students.