General Information
| Student: | Scott Harman |
|---|---|
| Office: | 444 |
| School: | Rutgers University |
| E-mail: | scottkh1995@gmail.com |
| Project: | Geometric Analysis |
Project Description
The theory of curve-shortening flow essentially models the evolution of a curve in the plane. The evolution is provided that, at each point, the curve moves along the normal vector proportionally to the curvature. Usually, these curves shrink to a point provided they satisfy the correct conditions. I will impose other conditions on the curve and plane, such as different metrics and halting conditions, and analyze the results.Weekly Log
- Week 1:
- Week 2:
- Week 3:
- Week 4:
I was introduced to people in the program and my mentor Professor Woodward. I was introduced to the idea of curve-shortening flow and (a little bit of) Ricci flow. I began studying the curve-shortening flow from several papers and attempted to understand the basics of the project.
After going through the papers I was provided as reference, I was startled at how difficult they could be and needed further reading and references in order to understand the concepts behind flows and geometric analysis. I spent the week reading through several textbooks, such as Topology and Differential Geometry, in order to familiarize myself more with what was going on. I tackled the concepts of manifolds, curvature, and more abstract surfaces in greater detail. I also met with my mentor for the second time and he advised me on which literature I should consult for the project and several potential models for the curve-shortening flow that I could introduce.
I familiarized myself with differential geometry and began to study surfaces and manifolds, and in particular, how curves behave when embedded onto manifolds. I studied one surface, a hyperboloid of one sheet, in detail in order to understand the mechanisms of curvature and flow with an example, and began to understand notions of geodesic curvature and Riemmanian metric.
This week I was encountering a lot of road-blocks. I was having a lot of trouble understanding some of the more advanced math concepts being presented to me, such as the PDE the flow obeys on surfaces and how metric and the surface the curve is embedded on change the properties of the flow. It was not a great week but towards the end of it I felt that I was beginning to crack the math a bit more and understand it. Most of the week was just spent reading my differential geometry textbook or searching the internet for answers.
Presentations
- First Presentation
Additional Information
- My Mentors