Abstract:

A family of surfaces moves by mean curvature flow if the velocity at each point is given by the mean curvature vector. Mean curvature flow first arose as a model of evolving interfaces and has been extensively studied over the last 40 years.

In this talk, I will give an introduction and overview for a general mathematical audience. To gain some intuition we will first consider the one-dimensional case of evolving curves. We will then discuss Huisken's classical result that the flow of convex surfaces always converges to a round point. On the other hand, if the initial surface is not convex we will see that the flow typically encounters singularities. Getting a hold of these singularities is crucial for most striking applications in geometry, topology and physics. Specifically, singularities can be either of neck-type or conical-type. We will discuss examples from the 90s, which show, both experimentally and theoretically, that flow through conical singularities is highly nonunique.

Finally, I will discuss recent work with Kyeongsu Choi, Or Hershkovits and Brian White, where we proved that mean curvature flow through neck-singularities is unique. The key for this is a classification result for ancient asymptotically cylindrical flows that describes all possible blowup limits near a neck-singularity. In particular, this confirms the mean-convex neighborhood conjecture. Assuming Ilmanen's multiplicity-one conjecture, we conclude that for embedded two-spheres mean curvature flow through singularities is well-posed.

Speaker: Robert Haslhofer

Robert Haslhofer currently is an associate professor at the University of Toronto. 

Research Interests: Geometric Analysis, Differential Geometry, Partial Differential Equations, Calculus of Variations, Stochastic Analysis, General Relativity

Academic Appointments:

Associate Professor, University of Toronto, 2021 – now

Assistant Professor, University of Toronto, 2015 – 2021

Courant Instructor, Courant Institute of Mathematical Sciences, 2012 – 2015

Teaching Assistant, Department of Mathematics, ETH Z¨urich, 2008 – 2012

Teaching Assistant, Institute for Theoretical Physics, ETH Z¨urich, 2008

Junior Tutor, Department of Mathematics, ETH Z¨urich, 2004 – 2007

Education:

PhD in Mathematics at ETH Z¨urich (diploma with distinction), 2008 – 2012

MSc in Mathematics at ETH Z¨urich (diploma with distinction), 2006 – 2008

BSc in Mathematics at ETH Z¨urich (diploma with distinction), 2003 – 2006

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