Course of Undergraduate Young Talent Project: 4-manifolds and Ricci flow
Instructor: Bennett Chow (周培能 Zhou, Pei-Neng)
Teaching Language: English
Prerequisites: Basic knowledge of topology and geometry
Schedule: March 26 - June 13, 2018 (12 weeks)
8 am to 10 am on Mondays and 3 pm to 5 pm on Wednesdays
Venue: Room 508, Teaching Building 3 (三教), Peking University

Office hours: Mondays 10:15 am to 11:15 am. Wednesdays 1:30 pm to 2:30 pm.

Teaching objective in syllabus: The main purpose of this course is to introduce students to questions in 4-manifold geometry and topology and a possible tool to solve some questions, especially those related to geometry, Hamilton’s Ricci flow.

Introduction: Ricci flow deforms Riemannian metrics on manifolds by a nonlinear heat equation. In dimension 3 this led to the proof of the Thurston-Hamilton-Perelman geometrization theorem. In dimension 4 it is unknown if Ricci flow can be used to derive topological conclusions. In this course we discuss some of what is currently known in this direction. Our focus will be to present the prerequisite knowledge to understand various conjectures on this topic.

Topics for Part I (4 weeks): The proof of Hamilton's 3-manifolds with positive Ricci curvature theorem.

Collected Notes, Part I: Foundations of Ricci flow These are the combined notes (Notes 1 - 7 plus a few more details) for the first 4 weeks (Monday, March 26 - Wednesday, April 18), edited, rearranged, and with references added. Last revised (version 2): April 15, 9:33 am (Beijing time).

Notes 1: Review of Riemannian geometry Solutions to Exercises in Notes 1
Notes 2: Elements of Ricci flow
Notes 3: Entropy monotonicity and no local collapsing
Notes 4: Hamilton's 3-manifolds with positive Ricci curvature theorem
Notes 5: Compact singularity models
Notes 6: Strong maximum principle
Notes 7: Cheeger--Gromov compactness theorem

Topics for Part II (3 weeks): We discuss gradient Ricci solitons, with an emphasis on shrinkers. The goal is to lead up to the statement of Munteanu and Wang's dichotomy conjecture for 4-dimensional noncompact shrinking gradient Ricci solitons.

Notes 8: Introduction to gradient Ricci solitons
Notes 9: Lower bound for the scalar curvature of GRS
Notes 10: Kazdan-Warner identity (supplement)
Notes 11: Shrinking GRS and quadratic curvature decay
Notes 12: The scalar to Riemann bound for 4-dimensional shrinking GRS
Notes 13: Hamilton's matrix Harnack estimate
Notes 14: Lecture notes on Perelman's kappa-compactness theorem (by Yongjia Zhang)
Notes 15: Introduction to Perelman's L-geometry

REFERENCES:

Part 1: Ricci flow:
“Comparison theorems in Riemannian geometry” Jeff Cheeger and David Ebin, AMS. Chapter 1
Lectures on the Ricci flow by Topping
“Hamilton’s Ricci Flow” by Chow, Lu and Ni, Science Press and AMS (2006). Preview material (chapter 1)
“The Ricci flow: techniques and applications. Parts I--IV.” By Chow, Chu, Glickenstein, Guenther, Isenberg, Ivey, Knopf, Lu, Luo, and Ni, Mathematical Surveys and Monographs, 135, 144, 163, 206, AMS, Providence, RI, 2007, 2010, 2011, 2015.

Part 2: Geometry and topology:
“A basic course in algebraic topology” by Massey, Graduate Texts in Mathematics 127, Springer-Verlag, 1991.
Algebraic topology by Hatcher, Cambridge University Press, 2002.
“Characteristic classes” by Milnor and Stasheff, Princeton University Press.
“Knots and links” by Dale Rolfsen, AMS.
The topology of 4-manifolds by Robion Kirby, Springer-Verlag (1989).
“The wild world of 4-manifolds” by Scorpan, American Mathematical Society, 2005.
“4-manifolds and Kirby calculus” by Robert Gompf and Andras Stipsicz, American Mathematical Society.

Part 3: Advanced topics:
Notes on Perelman’s papers by Kleiner and Lott
Ricci flow and the Poincaré conjecture by Morgan and Tian, Clay Mathematics Monographs, 3, AMS, Providence, RI, 2007
Notes and commentary on Perelman’s Ricci flow papers
Various papers on arXiv.