# 2019 ECNU Summer School

## of Geometry and Analysis on Manifolds

# 课程简介:

### 课程１

**课程名: eta-Invariants in Differential Topology****主讲人: Goette, Sebastian (University of Freiburg)****简介**： To study the topology of closed manifolds, one can use tools from classical algebraic topology like fundamental group, (co-) homology, K-theory etc. For even dimensional manifolds, one can sometimes describe their cobordism classes by characteristic numbers. For odd-dimensional manifolds M, one may consider so-called boundary defect invariants instead. These are typically defined using relative characteristic numbers on a compact manifold W with boundary ∂W = M. While abstract bordism theory tells us that such W exist, it is not always easy to find a concrete W on which to evaluate the relevant invariants. In the lecture series, we will study an intrinsic approach to certain boundary defect invariants. I will begin by explaining the index theorems by Atiyah-Singer for closed even-dimensional manifolds, and by Atiyah-Patodi-Singer for even dimensional compact manifolds with boundary. The relevant boundary contribution is the so-called η-invariant. Then I will sketch intrinsic descriptions of certain boundary defect invariants in terms of η-invariants. Under additional geometric assumptions like positive scalar curvature or special holonomy, some of these invariants possess natural refinements. In the last part of the series we will consider several examples. For each example, we first describe some particular construction of manifolds, then define appropriate boundary defect invariants, represent them in terms of η-invariants, and finally sketch the computational tools necessary to evaluate them. If time permits, we will see metrics of non-negative sectional curvature on all exotic 7-spheres, and closed 7-manifolds admitting several deformation families of metrics with holonomy G2.### 课程2

**课程名: Quantitative estimates for singular set of Ricci limit space****主讲人: Jiang, Wenshuai (Zhejiang University)****简介**: In these four lectures, we will discuss some recent results about manifold with lower Ricci curvature bound. In the first two lectures, we will briefly recall some results of Cheeger, Colding, Tian and Naber, such as almost metric cone theorem, almost splitting theorem and Cheeger-Naber's quantitative estimate of singular set and applications. In the last two lectures, we will discuss quantitative estimates in Jiang-Naber and Cheeger-Jiang-Naber. We will introduce the neck regions and neck region decomposition theorem for manifold with lower Ricci curvature.### 课程3

**课程名: The Gromov’s theorem on almost flat manifolds****主讲人: Rong, Xiaochun (Capital Normal University and Rutgers University)****简介**: In the collapsing theory of Cheeger-Fukaya-Gromov on collapsed manifolds with bounded sectional curvature, the Gromov’s theorem on almost flat manifold has been a corner stone. We will present a detailed proof for the Gromov’s theorem, and we will discuss its a generalization and applications.Lecture 1. Nilpotent manifolds, equivariant Gromov-Hausdorff convergence.

Lecture 2. Successive blow-ups, Fibration theorems.

Lecture 3. A new proof of Gromov’s theorem on almost flat manifolds.

Lecture 4. Manifolds of almost non-negative Ricci curvature whose Riemannian universal cover is not collapsed.

### 课程4

**课程名: Semiclassical methods in spectral geometry****主讲人: Wang, Zuoqin ( University of Science and Technology of China)****简介**: In this series of lectures, I will give an introduction to semiclassical analysis which plays a role as a bridge between classical mechanics (i.e. symplectic geometry) and quantum mechanics (i.e. spectral theory). I will start with a quick introduction to the necessary symplectic geometry background. Then I will explain Weyl quantization that connect the classical and quantum theory. Finally I will focus on some classical theorems in spectral geometry that are proven using this classical-quantum correspondence point of view.### 课程5

**课程名: Geometric problems related to scalar curvature****主讲人: Shi, Yuguang ( Peking University)****简介**: For manifolds with more than three dimensions, scalar curvature is the weakest curvature invariant. But differential manifolds with nonnegative scalar curvature will have a greater limitation on the topology of manifolds. On the other hand, scalar curvature is explained as energy density in general relativity. So the geometric problems with respect to scalar curvature is closely related to the energy problems in general relativity. In four lectures, I will report on geometric problems related to this, including the proof of the positive mass theorem on Schoen-Yau three-dimensional manifolds.

# 课程地点：

华东师范大学闵行校区数学楼102报告厅# 时间表:

Session | Time | July 1 | July 2 | July 3 | July 4 | July 5 |

Morning Session | 9:00-10:15 | Goette, Sebastian | Goette, Sebastian | Goette, Sebastian | Jiang, Wenshuai | Shi, Yuguang |

10:30-11:45 | Jiang, Wenshuai | Rong, Xiaochun | Rong, Xiaochun | Rong, Xiaochun | Rong, Xiaochun | |

Lunch and Snap | ||||||

Afternoon Session | 14:30-15:45 | Shi, Yuguang | Shi, Yuguang | Jiang, Wenshuai | Shi, Yuguang | Jiang, Wenshuai |

16:00-15:15 | Goette, Sebastian | Wang, Zuoqin | Wang, Zuoqin | Wang, Zuoqin | Wang, Zuoqin |