Geometric Flows
Project Description
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.
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.
Supervisor
FONG Tsz Ho
Quota
10
Course type
UROP1000
UROP1100
UROP2100
UROP3100
UROP3200
UROP4100
Applicant's Roles
The project is designed for undergraduate students who have plan to (or considering to) continue on graduate studies in differential geometry, or other related areas in pure mathematics, especially partial differential equations, mathematical general relativity, etc.
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 provided that the participant is willing to study basic differential geometry of curves, surfaces, and manifolds while engaging in the project. 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.
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 provided that the participant is willing to study basic differential geometry of curves, surfaces, and manifolds while engaging in the project. 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.
Applicant's Learning Objectives
Through this project, students will:--
1) be guided to read journal papers in geometric analysis under supervision;
2) be introduced the current main stream research in geometric analysis;
3) have chance to try out some entry-level research problems in differential geometry.
1) be guided to read journal papers in geometric analysis under supervision;
2) be introduced the current main stream research in geometric analysis;
3) have chance to try out some entry-level research problems in differential geometry.
Complexity of the project
Moderate