Workshop on Uniqueness in Analysis and Geometry

Schedule

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 diffusion-reaction structure that makes them prone to finite-time 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 S³ and the Lawson conjecture
	Abstract: We present our proof that the Clifford torus is the only embedded minimal surface in S³ 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 (10-340) - By Invitation Only

Thursday - December 11, 2014
9:00	Felix Schulze	A local regularity theorem for mean curvature flow with triple edges
	Abstract: We consider the evolution by mean curvature flow of surface clusters, where along triple edges three surfaces are allowed to meet under an equal angle condition. We show that any such smooth flow, which is weakly close to the static flow consisting of three half-planes meeting along the common boundary, is smoothly close with estimates. Furthermore, we show how this can be used to prove a smooth short-time existence result. This is joint work with B. White.
10:15	Lu Wang	A Sharp Lower Bound for the Entropy of Closed Hypersurfaces up to Dimension Six
	Abstract: In this talk, we prove the entropy of closed hypersurfaces of dimension less than 7 is uniquely (modulo scalings and translations) minimized by the round sphere. This is joint work with Jacob Bernstein.
11:30	Leon Simon	Cylindrical tangent cones
	Abstract: Continuation of the first talk, including some recent progress in the analysis of one special class of examples of cylindrical cones.
2:00	Brian Krummel	Higher regularity of the branch set of harmonic functions and minimal submanifolds
	Abstract: I will discuss work on the fine structure and higher regularity of the branch set of two-valued solutions to the Laplace's equation and the minimal surface system. It is known from joint work of myself with Neshan Wickramasekera that the branch set is countably $(n-2)$-rectifiable. This follows from the monotonicity formula for frequency functions due to F. J. Almgren and asymptotics near branch points that are established using a modification of a blow-up method, which was originally applied by Leon Simon to multiplicity one classes of minimal submanifolds. In recent independent work, I show that the branch set is locally real analytic on a relatively open dense subset of the branch set using the blow-up method to establish asymptotics along the branch set and using a regularity argument for differential equations involving partial Legendre-type transformations.
3:15	Brian White	Open Problems
	Abstract: I will discuss some open problems and partial results about uniqueness in minimal surface theory and mean curvature flow.