Wednesday  December 10, 2014 
9:00 
Leon Simon 
Uniqueness of tangent cones 

Abstract: A brief survey of what we know and don't know about uniqueness
of tangent cones of singular minimal submanifolds, including the
significance of uniqueness in analyzing the structure of the singular
set of a minimal submanifold, and discussion of some of the analytic
techniques relevant to the study of such uniqueness questions.

10:15 
Bob Hardt 
One Dimensional Rectifiable Varifolds and Some Applications 

Abstract: Varifolds were originally introduced by Almgren and Allard to describe interesting limits arising from various 2 dimensional minimal surfaces or other soap film models. A varifold is stationary in a region U if the first variation of its mass is zero under deformations supported in U. With a suitable lower density bound, a stationary one dimensional rectifiable varifold enjoys a regularity property due to F.Almgren and W.Allard (1976). One dimensional varifolds with cost functionals depending nonlinearly in the densities, models ramified transport paths in work of Q.Xia and enjoy a similar regularity. Here we discuss a natural application of varifolds to Michel trusses, which are cost optimal 1 dimensional balanced structures consisting of bars and cables. Introduced in a 1904 paper of an economist A.G. Michel, they have been treated in the Mechanical Engineering literature, in interesting papers by R.Kohn and G. Strang (1983), and by G.Bouchitte, W.Gangbo, and P.Sepulcher (2008). There are many basic open questions about the location, structure, and rigidity of Michel trusses.

11:30 
Dan Knopf 
Universal behaviors in geometric heat flows 

Abstract: Geometric heat flows like Ricci flow and mean curvature flow have proven to be remarkably successful tools to investigate the geometry and topology of manifolds. They are nonlinear PDE with a diffusionreaction structure that makes them prone to finitetime singularities. Perhaps counterintuitively, these singularities are aids, not obstacles, to these flows’ applications — because regions of high curvature tend to be very special. In this talk, we will survey these phenomena and present evidence in favor of the conjectures that (1) solutions asymptotically acquire extra symmetries as they become singular, and (2) generic solutions may be constrained to a small catalog of universal asymptotic profiles. 
2:30 
Simon Brendle 
Embedded minimal tori in S^{3} and the Lawson conjecture 

Abstract: We present our proof that the Clifford torus is the only embedded minimal surface in S^{3} of genus 1. 
4:00 
Natasa Sesum 
Ancient solutions to the mean curvature flow 

Abstract: We will discuss some recent work on ancient solutions to the mean curvature flow. We describe some geometric properties of those and give their more precise asymptotics. This is a joint work with Angenent and Daskalopoulos.

Banquet  Emma Rogers Room at MIT (10340)  By Invitation Only 