# 2017 ECNU Summer School

## of Geometry and Analysis on Manifolds

# 课程简介:

### 课程１

**课程名：Some Connections of Supersymmetric Quantum Field Theories (SUSY QFT) to Index Theory and Applications****主讲人：Han, Fei (National University of Singapore)****简介**： It has been known that de-Rham cohomology is related to SUSY 0d QFT; K-theory and Atiyah-Singer index theorem are related to SUSY 1d QFT; elliptic cohomology and elliptic genera are related to SUSY 2d QFT. In these talks, I will briefly survey this point of view and then show that investigating along this line, one can develop variations and enrichment of the classic theory (like twisted cohomolgy, twisted K theory and etc). We will also introduce some applications of these theories to geometric and topological questions.### 课程2

**课程名: Volume entropy and geometry of negatively curved manifolds****主讲人: Wang, Xiaodong (Michigan State University)****简介**： Mostow’s strong rigidity theorem for locally symmetric spaces is a famous result in geometry. In the real hyperbolic case it asserts that two compact hyperbolic manifolds of dimension at least three with isomorphic fundamental groups must be isometric. In these lectures I will focus on the geometry of negatively curved manifolds. I will discuss compactifications, the volume entropy, Patterson-Sullivan measures and some other important concepts. The final goal is to explain the proof of a deep theorem of Besson-Courtois-Gallot on volume entropy which yields the Mostow strong rigidity in all rank one cases in a unified way and a lot more.### 课程3

**课程名：Introduction to the General Kahler-Ricci Flow****主讲人: Zhang, Zhou (University of Sydney)****简介**： For this series, we start with a concise but mostly self-contained introduction of Kahler geometry for setting up the Kahler-Ricci flow. Then we briefly discuss the relations and differences between several popular Kahler-Ricci flows. The focus will be on the so-called general Kahler-Ricci flow, highlighting the relations and interactions with the original Ricci flow and Minimal Model Program in algebraic geometry. The progress over the past ten years or so features the combination of classic tools in geometric analysis, for example, Maximum Principle, and machinery from pluripotential theory in several complex variables.### 课程4

**课程名: Eigenvalue and Heat Kernel Comparisons****主讲人：Wei, Guofang ( U. C. Santa Barbara & ECNU)****简介**： We will discuss various Laplacian eigenvalue upper and lower bounds, including Cheng's Dirichelet first eigenvalue comparison for balls, Neumann first eigenvalue comparison, gap comparison, and its extension to integral curvature and p-Laplacian. Heat kernel estimate for integral curvature will also be discussed.

# 课程地点：

华东师范大学闵行校区数学楼401报告厅# 时间表:

时间 | 6月26日 | 6月27日 | 6月28日 | 6月29日 | ||

上午 | 9:00-10:15 | Han, Fei | Han, Fei | Han, Fei | Han, Fei | |

10:30-11:45 | Wang, Xiaodong | Wang, Xiaodong | Wang, Xiaodong | Wang, Xiaodong | ||

午休 | ||||||

下午 | 13:30-14:45 | Zhang, Zhou | Zhang, Zhou | Zhang, Zhou | Zhang, Zhou | |

15:15-16:30 | Wei, Guofang | Wei, Guofang | Wei, Guofang | Wei, Guofang |