Selected topics in geometric analysis Ⅰ
Lecturer

Liangdi Zhang (张良迪, Assistant Professor)
Date

2024-02-27 ~ 2024-05-23
Schedule
Weekday    Time    Venue    Online    ID    Password
Tue,Thu    15:20-16:55    YMSC-B534    ZOOM    293 812 9202    BIMSA

Prerequisite

Basic knowledge on Riemannian geometry and PDE.
Introduction

This course is divided into two main topics. The first centres around the introduction of the Ricci flow. The second aims to prove the differentiable sphere theorem.
Syllabus

1. Preliminaries: curvatures on Riemannian manifolds
2. Existence and uniqueness of the Ricci flow.
3. Hamilton’s maximum principle
4. Hamilton’s classic results in dimensions 2, 3 and 4
5. Differentiable sphere theorem
Reference

[1] B. Andrews, C. Hopper, The Ricci flow in Riemannian geometry: a complete proof of the differentiable 1/4-pinching sphere theorem, Lecture Notes in Mathematics, Springer, Heidelberg, (2011).
[2] C. Bohm, B. Wilking, Manifolds with positive curvature operator are space forms, Ann. of Math. 167 (2008) 1079-1097.
[3] S. Brendle, R. M. Schoen, Manifolds with 1/4-pinched curvature are space forms, J. Amer. Math. Soc. 22(1) (2009) 287-307.
[4] S. Brendle, Ricci flow and the sphere theorem, Graduate Studies in Mathematics, AMS Press, (2010).
[5] D. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Differ. Geom. 18 (1983) 157-162.
[6] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differ. Geom. 17 (1982) 255-306.
[7] R. S. Hamilton, Four-manifolds with positive curvature operator, J. Differ. Geom. 24 (1986) 153-179.
[8] R. S. Hamilton, The Ricci flow on surfaces, Contemp. Math. 71 (1988) 237-261.
[9] W. X. Shi, Deforming the metric on complete Riemannian manifolds, J. Differ. Geom. 30 (1989) 223-301.
Video Public

Yes
Notes Public

Yes
Audience

Advanced Undergraduate, Graduate
Language

Chinese
Lecturer Intro

Liangdi Zhang received his Ph.D. degree from Zhejiang University in June 2021. He worked as a postdoc at Beijing Institute of Mathematical Sciences and Applications (BIMSA) and Tsinghua University from Augest 2021 to Augest 2023. He is currently an assistant professor at BIMSA. His research interests include differential geometry and geometric analysis.
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