Study Plan: Curve Shortening Flow

For a while now I have wanted to dig into the details of Gage-Grayson-Hamilton work on the "curve shortening flow." I have several reasons for this: 

1. If I am to call myself a differential geometer, I will really have to learn some more about geometric analysis fundamentals. CSF seems like the most accessible version of a large class of geometric evolution equations which are very important. (For example, Perelman's work on the Poincaré conjecture is essentially a control over the singularities of the Ricci flow introduced by Hamilton.)
2. There is this big project I never finished that would likely go easier if I knew some more about these kinds of things. Maybe this would be a good way to get back into it.
3. Discretized versions of the problem seem to be a place for getting students involved in mathematics. I have tried lots of different angles for getting students involved in research, and I have learned the hard way that I need a class of problems that satisfy these requirements:
1. easy to get started (low floor)
2. possibility of exciting new developments (high ceiling)
3. I have to know something about it.
4. I am teaching differential geometry this term, and I like to take the opportunity when teaching a course to deepen my knowledge of the material in some significant way.
5. I am giving a talk at an undergraduate conference pretty soon, and I want to talk about these ideas. (I gave them an abstract that would push me to learn more.)
6. The stuff is just cool.

So, I have started reading the primary literature already. I will write about what I am learning in small pieces, so this is an opportunity for me to organize the ideas into a sequence of blog posts. 

The list below is limited a bit by my current state of knowledge. I know more details about the beginning of the list (items 1-7), and less about the end which will likely expand quite a bit. 

Here is what I think will happen:

1. Fundamentals on smooth plane curves: adaptations of the Frenet-Serret apparatus to 2d; some basic formulae for computation
2. convex plane curves, in particular the idea of a support function; new expressions for length and area
3. some isoperimetric inequalities (following a paper by Harley Flanders). Possibly several posts.
4. the idea of curve shortening flow
5. Gage's isoperimetric inequality
6. The evolution of the isoperimetric profile under CSF.
7. The Hausdorff metric on closed sets
8. CSF shrinking convex curves to "round points." Likely several posts. 
9. About parabolic PDEs and their solutions.
10. Grayson's whisker lemma and the case of embedded non-convex curves.
11. Curves on surfaces.

This project is mainly for my own benefit. I want to get this stuff into my head, and writing it really helps me. If you find it interesting, too, come along.