Geometric Analysis and Topology Seminar

Spring 2012


The seminar's usual time is Friday at 11:00am in room 512 of Warren Weaver Hall (Directions). Special times and dates are marked in red. Click on the title of a talk for the abstract (if available).

Mar 2
11am
Tobias Lamm
(Frankfurt)
Rigidity results for conformal immersions in R^n 512 WWH
Mar 8
2pm
Dominic Dotterrer
(Toronto)
The (co)isoperimetric problem in polyhedra 312 WWH
Apr 13
11am
Tom Ilmanen
(ETH)
New Results in Mean Curvature Flow in R^3 512 WWH


Organizers: Sylvain Cappell, Jeff Cheeger, Larry Guth, and Bruce Kleiner.

Previous semesters:

Abstracts:

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.
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.
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)