Geometric Analysis and Topology Seminar

Fall 2015

The seminar's usual time is Friday at 11:00am in 517 Warren Weaver Hall (Directions). Special times and dates are marked in red. Click on the title of a talk for the abstract (if available).

Upcoming seminars:

Oct 30,  11am
517 WWH
Or Hershkovits
Mean Curvature Flow of Reifenberg Sets.
Nov 20,  11am
517 WWH
Sylvester Eriksson-Bique
Quantitative Bi-Lipschitz embeddings of bounded curvature manifolds and orbifolds
Dec 4,  11am
517 WWH
Jason Behrstock
Curve complexes for cube complexes

Organizers: Sylvain Cappell, Jeff Cheeger, Bruce Kleiner, and Robert Young.

Previous semesters:


Mean Curvature Flow of Reifenberg Sets, Or Hershkovits.  The mean curvature flow, the gradient flow of the area functional, is one of the most natural geometric flows to consider for embedded hypersurfaces in R^{n+1}. Classically, given a sufficiently smooth hypersurface (for which both the area and its gradient are defined), there exists a unique flow starting from it that exists for some positive time. Moreover, the flow smooths the hypersurface instantaneously. In the early 90s it was shown by Ecker and Huisken that the smoothness assumption can be weakened to the class of uniformly locally Lipschitz hypersurfaces (for which the area is defined, but its gradient may not be). When n>1, this is the least regular object for which the flow was known to exist. In this talk, we will discuss the short time existence and uniqueness of smooth mean curvature flow in arbitrary dimension starting from a class of sets which is general enough to include some fractal sets (for which even the area is not defined). Those so-called (ε,R) Reifenberg sets have a weak metric notion of a tangent hyperplane at every point and scale r<R with accuracy determined by ε), but those tangents are allowed to tilt as the scales vary. We show that if X is an (ε,R) Reifenberg set with ε sufficiently small, there exists a unique smooth mean curvature flow emanating from X. When n>1, this provides the first known example of instant smoothing, by mean curvature flow, of sets with Hausdorff dimension larger than n. If time permits, we will discuss the arbitrary co-dimensional case.
Quantitative Bi-Lipschitz embeddings of bounded curvature manifolds and orbifolds, Sylvester Eriksson-Bique.  In this talk we will outline a multi-scale method for constructing bi-Lipschitz embeddings for bounded curvature manifolds and orbifolds. The results heavily rely on ideas from collapsing theory. We outline two proofs; one based solely on collapsing theory and a more algebraic proof. The algebraic approach leads to more quantitative collapsing theory results for certain model spaces, and thus permits us to give some explicit bounds for the bi-Lipschitz constants.
Curve complexes for cube complexes Jason Behrstock.  For CAT(0) cubical groups we develop analogues of tools which have played a key role in the study of the mapping class group, namely, the theory of curve complexes and subsurface projections. We will describe these parallel structures, which we axiomatize as "hierarchical hyperbolic spaces", and some new results that can be proven as a result of this new approach. This is joint work with Mark Hagen and Alessandro Sisto.

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