2nd degree (Master)
Coordinator Office Hours:
Prof Jake Solomon
Given a submanifold, it is natural to look for an isotopic submanifold with minimal volume. Minimal submanifolds play an important role in fields ranging from the study of soap bubbles to boundary conditions in quantum gravity. Mean curvature flow is an evolution equation for submanifolds that reduces volume as quickly as possible. In good cases, it converges to a volume minimizing submanifold. However, in general, singularities develop that prevent the continuation of the flow. In this this course, we will learn about the classification of singularities.
Classification of singularities in mean curvature flow.
Learning outcomes - On successful completion of this module, students should be able to:
To understand the basic techniques for classifying singularities of mean curvature flow. To prove long term existence in dimension 2 and for convex hypersurfaces.
While there is no formal attendance requirement, students are expected to learn the content of lectures, which may not be available from any of the course reading. Moreover, students are expected to be aware of any announcements made in lecture.
Teaching arrangement and method of instruction:
Lecture and exercises.
Introduction to mean curvature flow, blow-up arguments, type 1 singularties, type 2 singularities, long term existence for convex hypersurfaces, long term existence for curves in the plane.
Mategazza, Carlo, “Lecture Notes on Mean Curvature flow”
Additional Reading Material:
Ecker, Klaus, “Regularity Theory for Mean Curvature Flow”
Zhu, Xi-Ping, “Lectures on Mean Curvature Flows”
End of year written/oral examination 0 %
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Assignments 100 %
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