The VietorisRips Complexes of a Circle.
[Abstract,
Slides,
Poster]
Given a metric space X and a distance threshold r > 0, the VietorisRips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of JeanClaude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the VietorisRips complex is homotopy equivalent to the original manifold. Little is known about the behavior of VietorisRips complexes for larger values of r, even though these complexes arise naturally in applications using persistent homology. We show that as r increases, the VietorisRips complex of the circle obtains the homotopy types of the circle, the 3sphere, the 5sphere, the 7sphere, ..., until finally it is contractible. As our main tool we introduce a directed graph invariant, the winding fraction, which in some sense is dual to the circular chromatic number. Using the winding fraction we classify the homotopy types of the VietorisRips complex of an arbitrary (possibly infinite) subset of the circle, and we study the expected homotopy type of the VietorisRips complex of a uniformly random sample from the circle. Moreover, we show that as the distance parameter increases, the ambient Čech complex of the circle also obtains the homotopy types of the circle, the 3sphere, the 5sphere, the 7sphere, ..., until finally it is contractible. Joint with Michał Adamaszek.

Applied Algebraic Topology Research Network, Online Seminar Series, March 3, 2015.

Colloquium at Colorado State University, Jan 2015.

Geometry and Topology Seminar at Tulane University, Nov 2014.

Applied Topology Seminar at the University of Pennsylvania, Nov 2014.

Colloquium at UNC Greensboro, Oct 2014.

Geometry and Topology Seminar at North Carolina State University, Sept 2014.
 IMA Postdoc Seminar, May 2014.
 IMA Postdoc Seminar, Oct 2013.