How to simplify complicated plane curves

Title How to simplify complicated plane curves


SpeakerProfessor Richard Schoen (Stanford University, United States National Academy of Sciences, Academician


Time4:00-5:00 pm; April 25; 2013


PlaceConference Room A404, Department of Mathematical Sciences


Abstract: The curve-shortening flow has been much studied in differential geometry over the past 25 years. In this talk we will introduce the problem, which involves the flow of a plane curve in the normal direction with speed equal to the curvature. It was shown by M. Grayson in 1987 that the flow converts an arbitrary embedded closed curve into a circle! Over the next 10 years Grayson's proof was clarified and simplified by R. Hamilton and G. Huisken. In 2009, B. Andrews and P. Bryan gave a still more direct proof.


We will describe the main ingredient in these simplifications, which involves finding a geometric quantity that is improved under the flow and prevents the formation of certain types of singularities. Although this is a hard theorem, its proof uses only two-variable calculus and a lot of cleverness. The curve-shortening flow is the simplest of several geometric flows whose study is a topic of current research interest in differential geometry. These flows have certain features and methods in common, particularly with regard to the formation of singularities. This talk is intended for undergraduate mathematics students.


ContactYoujin Zhang