Some course notes.

All the mistakes in the notes are blamed on me, and please let me know if you figure out any mistakes.

1. Morse theory perspective of mean curvature flow: dynamics, and interpolation method - Ao Sun
   This is the lecture note of Masterclass: Recent Progress on Singularity Analysis and Applications of the Mean Curvature Flow at Copenhagen Centre for Geometry & Topology (GeoTop), University of Copenhagen, from April 29 - May 3, 2024. I tried to give an overview of the Morse theory perspective of mean curvature flow.

2. Lefschetz Fibrations - Benson Farb, Danny Calegari, and Eduard Looijenga
   This is the course note of MATH 34600 Topics in Geometry and Topology taught by Professors Benson Farb, Danny Calegari, and Eduard Looijenga in 2023 Spring at UChicago. The main purpose of this course is to discuss what Lefschetz fibrations are, how they are used, and basic open questions about them that remain. Because I am not familiar with algebraic geometry, I may introduce many mistakes, but I really enjoy this course.

3. Dynamical System - Amie Wilkinson
   This is the course note of MATH 35600 Topics in Smooth Dynamics and Ergodic Theory taught by Professor Amie Wilkinson in 2021 Fall at UChicago. The note was taken together with Adrian Chun Pong Chu. The main purpose of this course is to prove two theorems in dynamical systems: the marked length spectrum rigidity theorem proved by Otal and Croke, and the metric on a torus with no conjugate points is flat proved by Burago-Ivanov. Many backgrounds on geodesic flows, ergodic theory, and entropy are discussed in the course.

4. Hyperbolic 3-Manifolds - Danny Calegari
   This is the course note of MATH 37101 Hyperbolic 3-Manifolds taught by Professor Danny Calegari in 2021 Spring at UChicago. The course is mainly about Thurston's hyperbolization theorem. In particular, this course discussed two important papers by Thurston (Hyperbolic Structures on 3-manifolds, I: Deformation of acylindrical manifolds, Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle). You can find 100 figures in this note!

5. Ricci Flow - Richard Bamler
   This is the course note of Topics Class on Ricci flow (Math 277) taught by Professor Richard Bamler in 2020 Fall at UC Berkeley. The first half of the course is about some fundamental results in Ricci flow, and the second half of the course is about three papers by Richard Bamler posted in 2020.

6. Topics in Elliptic PDEs - David Jerison
   This is the course note of 18.158 at MIT taught by Professor David Jerison in 2020 Spring. The note was taken together with Shengwen Gan, Feng Gui, and Chunxia Tao. The course covers the topic of minimal surfaces, free boundary problem, optimal transport, and level set equations. Since David still wants to modify these notes carefully, at this moment I would not share them here but can share them with you upon request. This note is different from the topic on free boundary problem notes.

7. Free Boundary Problem - David Jerison
   This is the course note of 18.158 at MIT taught by Professor David Jerison in 2015 Fall. At that time I was a fresh PhD student and did not know much about elliptic PDEs and Latex. However, I realize that the material in that course was so important, and I finally decide to type down the notes after four years since I took the course. Unfortunately, the notes only cover half of the class. Since David will teach this course in 2020 Spring again, I would no longer update these notes, and I will try to get a better version this time.

8. Comparison Geometry and Ricci Spaces - Tristan Collins
   This is the course note of 18.966 at MIT taught by Professor Tristan Collins in 2019 Spring. The note was taken together with Feng Gui. This note contains basic materials of comparison geometry and Ricci Spaces.

9. Decoupling Theory - Larry Guth
   This is the course note of 18.118 at MIT taught by Professor Larry Guth in 2017 Fall. Because I am lazy, this note actually contains only the content from the first half of the semester. You can go to Larry's homepage to see the complete lecture notes taken by the students of this class.

10. Analytic and Geometric Estimates for Complex Monge-Ampere Equation - Jian Song
    This note is based on the talk given by Professor Jian Song in Thematic Program on K\"ahler Geometry on June 19-23, 2017 at University of Notre Dame. This note contains an introduction to complex Monge-Ampere equation.

11. Minimal Surface - William Minicozzi
    This is the course note of 18.137 at MIT taught by Professor Bill Minicozzi in 2016 Spring. The content which has already appeared in this note is basically the first several chapters of the book "A Course in Minimal Surfaces". The rest part of the course is extremely hard to take note. If Bill teaches this again in the future I promise I will take a better note.

2021 Uchicago REU Lecture notes Heat Equation and Geometric Flows. 

I will update the notes after each lecture.