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.