题目：Collapsed Riemannian Manifolds with  Curvature Bound    

报告人：Prof. Xiaochun Rong

摘要：In this mini-course, we give a quick introduction to the study of geometric and topological structures on an -collapsed Riemannian n-manifold M, i.e., every unit ball on M has the volume ɛ (a small constant depending on n), while a suitable bound on curvature is imposed preventing a scaling of the metric.

 This mini-course consists of two parts. In the first part, we will present new proofs for the Gromov’s theorem on almost flat manifolds, and the nilpotent fibration theorem of Cheeger-Fukaya-Gromov on collapsed n-manifolds with sectional curvature bounded in absolute value, and applications. In the second part, we will review basic Cheeger-Colding-Naber theory on a Ricci limit space, i.e., the Gromov-Hausdorff limit of a sequence of Riemannian manifolds with Ricci curvature bounded below. We will focus on the class of collapsing with local bounded covering geometry, and present partial nilpotent structures on such a collapsed manifold. We also discuss recent applications. 

CONTENTS

Part I. Collapsing with bounded sectional curvature

Lecture 1. Overview on collapsed Riemannian manifolds with local bounded covering geometry

Lecture 2. Basic properties of Gromov-Hausdorff convergence

Lecture 3. Fiber bundle construction via Harmonic coordinates

Lecture 4. A new proof of Gromov’s almost flat manifolds

Lecture 5. Applications

Part II. Collapsing with Ricci curvature bounded below

Lecture 6. Ricci-limit spaces, and the subclass of local bounded covering geometry

Lecture 7. Nilpotent structures on collapsed manifolds with local Ricci bounded covering geometry

Lecture 8. Quantitative maximal rigidity of Ricci curvature bounded below

Lecture 9. Compact Ricci limit spaces of solvable manifolds

报告时间：每周一三五上午9:30-11:30，1月6日开始 

地点：校本部教二楼 627教室 腾讯会议：353-4740-8528