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Rigidity results for conformal immersions in R^n, Tobias Lamm.
By a classical result of Codazzi every closed, totally
umbilic surface in R^n is a round sphere. De Lellis and Muller proved a
rigidity statement
corresponding to this result. More precisely, they showed that for
every closed surface in R^3, whose traceless second fundamental
form is "small" in L^2, there exists a conformal parametrization
whose distance to a standard parametrization of a round sphere is
small in W^{2,2}.
In a recent joint work with H. Nguyen (Warwick) we were able to extend
this result to arbitrary codimensions. Moreover, we obtained related
rigidity results for inversions of the catenoid and Enneper`s minimal
surface.
In my talk I will review the analytic preliminaries (i.e. the results
of Muller-Sverak and Kuwert-Li) and I will sketch the proof of the
above mentioned results.
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The (co)isoperimetric problem in polyhedra, Dominic Dotterrer.
In discrete settings, many natural combinatorial optimization problems can be reformulated as an isoperimetric-type problem. In the last decade, a number of (surprising) applications have leveraged solutions to these problems. The talk will be in three parts. I will begin by explaining some of these geometro-topological applications. Then I will describe a technique for filling cycles in a hypercube. I will finish by describing an interesting family of cubical cellular cycles which turn out to be isoperimetric minimizers in the cube.
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New Results in Mean Curvature Flow in R^3, Tom Ilmanen.
I will report on numerous new results for the evolution of
surfaces by mean curvature
flow in R^3, particularly when the density ratios are less than two. It
includes a structure theorem for self-similar shrinking solutions,
a monotonicity formula for self-expanding solutions, isolation
of the cylinder, positive mean curvature in the neighborhood of a neckpinch,
and progress toward the genericity conjecture for positive mean curvature
singularities.
(joint with Colding, Minicozzi, White)
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