The study of geometric flows is a central topic in modern differential geometry. They are time-dependent dynamical systems on geometric shapes (which are called "manifolds" by geometers and topologists). Notable examples include the Ricci flow, mean curvature flow, Yamabe flow, Calabi flow, etc. In particular, the Ricci flow was applied to resolve the Poincare Conjecture (by Hamilton and Perelman in 2002-03), and the Differentiable Sphere Theorem (by Brendle and Schoen in 2007). The inverse mean curvature flow was also used by Huisken and Ilmanen to prove the time-symmetric Penrose inequality in General Relativity.
This project aims at introducing the basics of geometric flows to motivated undergraduate students, through reading some important journal papers in this area. Students who have excellent performance in UROP 1100 will be invited to continue on the project for more original research.
In this project, students will get a taste of geometric flows by reading some important journal papers in this area under the guidance of the supervisor. Students who take part in this project should already have a solid foundation on elementary analysis, multivariable calculus and linear algebra. Prior knowledge in differential geometry will be a bonus, but is not required. At the end of the project, students in UROP 1100 are expected to write an expository article on the journal paper they have read. For students in UROP 2100 or higher, some original research works are expected.
Through this project, students will:--
1) be guided to read journal papers in pure mathematics under supervision;
2) strengthen their background knowledge on Multivariable Calculus and Differential Equations through exploring some geometric flows;
3) improve critical thinking skills through reading journal papers;
4) learn how to search and locate good journal papers in pure mathematics;
5) build up a solid foundation on differential geometry for future research in pure/applied mathematics as well as in other related areas including physics, computer science, etc.