15 - 25 July 2013, entitled Geometric Flows of Curves. The course is open to postgraduate students, honours students and researchers who have access to Access Grid facilities.
If you are interested in participating, please register by sending an email to firstname.lastname@example.org by 15 June with the following information:
- your full name
- University affiliation, and
- postal address.
University of Wollongong
Geometric flows of curves are families of curves varying smoothly with time according to a given law. One example of this is the curve shortening flow, a geometric evolution equation for curves where each point moves with velocity given by the curvature vector at that point. This evolution equation is the main focus of our study in this course. Our main goal is to give in detail the original argument of Gage-Hamilton establishing that convex initial curves shrink to smooth round points. This argument has by now been replaced by much more efficient methods. The original argument contains many standard tools, and so it is hoped that a presentation of this in detail will introduce students to valuable techniques and methods with application far beyond the realm of curve flows.
A secondary goal is to improve Gage-Hamilton┐s Theorem to Grayson┐s Theorem, the statement that all embedded initial curves shrink to points. The original proof for this will be discussed and summarised. Time permitting, we will also detail how some modern methods due to Huisken and Andrews-Baker have shortened and replaced many of the steps from the original proofs of these theorems.
Exercises are given throughout and form a crucial part of the learning process. Topics to be covered:
- Elementary geometry of curves
- Short-time existence for general curve flows
- The proof of Gage-Hamilton that convex initial curves shrink to round points under curve shortening flow
- A discussion of Grayson's original proof
- A discussion of the alternative proofs of Grayson┐s theorem due to Huisken and Gage-Hamilton.
The course does not have any formal prerequisites apart from calculus and differential equations. Some knowledge of calculus of variations, the geometry of curves, and PDEs, would be helpful, but is not required. The course is targeted at Honours students, since some mathematical maturity is required and a little bit of discipline to engage in the exercises given during lectures. That said, students from any year are welcome to participate.
Please note: as these short courses are delivered over the Access Grid, AMSI will not make provisions for travel or accommodation. This information can also be viewed on the AMSI web pages.
If you would like to attend this seminar in our access grid room then please check to see if the grid is already booked at this time and send an email to email@example.com to let the CSOs know that you would like to attend.