Mean Curvature Flow

Date: 26 - 30 September 2016

Location: Mathematical Institute, University of Oxford

Mean curvature flow (MCF) is the negative gradient flow of volume, so any hypersurface flows through hypersurfaces in the direction of steepest descent for volume and eventually becomes extinct in finite time.  It has been studied in material science to model things such as cell, grain, and bubble growth.  It also plays an important role in image processing and has been widely studied numerically.   The flow is well-defined classically until the first singularity.  The level set method was invented to study the flow, even past singularities, and it is now widely used in a variety of settings.

Before the flow becomes extinct, changes occur as it goes through singularities.   To understand the flow, it is crucial to understand these.  This can be done completely for embedded curves, but there are infinitely many possible types of singularities in all higher dimensions.  Recent years have seen major progress on some of the most natural questions:  Is it possible to bring the hypersurface into general position so that there is a smaller set of “generic” singularities?  What does the flow look like near these singularities?  What is the structure of the singular set itself? 

This workshop will explore recent developments in MCF, focusing on which singularities can occur, which of these are generic, what the flow looks like near these singularities, and the structure of the singular set.

Speakers: Sigurd Angenent (Wisconsin), Jacob Bernstein (Johns Hopkins), Yoshikazu Giga (Tokyo), Gerhard Huisken (Tübingen), Bruce Kleiner (NYU), Bob Kohn (NYU), Felix Otto (MPIM Leipzig), Felix Schulze (UCL), James Sethian (Berkeley), Peter Topping (Warwick), Lu Wang (Wisconsin), Mu-Tao Wang (Columbia), Brian White (Stanford)

Some of the topics that will be covered in the workshop are described in the paper Level set flow  by Colding and Minicozzi.

Image:  R. I. Saye and J. A. Sethian, Lawrence Berkeley National Laboratory and University of California, Berkeley