
The VietorisRips Complex of the Circle.
[Abstract,
Slides,
Poster]
Given a metric space and a positive connectivity parameter, the VietorisRips simplicial complex has a vertex for each point in the metric space, and contains a set of vertices as a simplex if its diameter is less than the connectivity parameter. A theorem of JeanClaude Hausmann states that if the metric space is a Riemannian manifold and the connectivity parameter is sufficiently small, then the VietorisRips complex is homotopy equivalent to the original manifold. What happens for larger connectivity parameters? We show that as the connectivity parameter increases, the VietorisRips complex of the circle obtains the homotopy type of the circle, the 3sphere, the 5sphere, the 7sphere, ..., until finally it is contractible. The same progression of homotopy types occurs for the ambient Čech complex of the circle, i.e. a nerve complex of circular arcs, as the length of the arcs increases. Joint with Michał Adamaszek.

Evasion Paths in Mobile Sensor Networks.
[Abstract,
Slides,
Poster,
Multimedia]
Suppose that ballshaped sensors wander in a bounded domain. A sensor doesn't know its location but does know when it overlaps a nearby sensor. We say that an evasion path exists in this sensor network if a moving intruder can avoid detection. In "Coordinatefree coverage in sensor networks with controlled boundaries via homology", Vin de Silva and Robert Ghrist give a necessary condition, depending only on the timevarying connectivity data of the sensors, for an evasion path to exist. Using zigzag persistent homology, we provide an equivalent condition that moreover can be computed in a streaming fashion. However, no method with timevarying connectivity data as input can give necessary and sufficient conditions for the existence of an evasion path. Indeed, we show that the existence of an evasion path depends not only on the fibrewise homotopy type of the region covered by sensors but also on its embedding in spacetime. For planar sensors that also measure weak rotation and distance information, we provide necessary and sufficient conditions for the existence of an evasion path. Joint with Gunnar Carlsson.

IMA Workshop on Topological Systems: Communication, Sensing, and Actuation, Mar 2014. [Video]

Rocky Mountain Algebraic Combinatorics Seminar, Nov 2013.

SIAM Conference on Applied Algebraic Geometry, Aug 2013.

Applied Topology in Będlewo, July 2013.

MSRI Workshop on Algebraic Topology, June 2013. [Video]

Ayasdi Topology Day, June 2013.

Stanford CompTop Seminar, May 2013.

Special Session on Applied and Computational Topology at MAA MathFest, Aug 2012.

Algebraic Topology: Applications and New Directions, July 2012.

Minisymposium on Applied Algebraic Topology at SIAM Annual Meetings, July 2012. [Video]

Schloss Dagstuhl Seminar on Applications of Combinatorial Topology to Computer Science, Mar 2012.

AMS Special Session on Computational and Applied Topology, Joint Meetings, Jan 2012.

SIAM Conference on Applied Algebraic Geometry, Oct 2011.

Nudged Elastic Band in Topological Data Analysis.
[Abstract]
We use the nudged elastic band method from computational chemistry to analyze highdimensional data. Our approach is inspired by Morse theory, and as output we produce an increasing sequence of small cell complexes modeling the dense regions of the data. We test the method on data sets arising in social networks and in image processing. Furthermore, we apply the method to identify new topological structure in a data set of optical flow patches. Joint with Atanas Atanasov and Gunnar Carlsson.

Topological Data Analysis: Understanding Optical Flow. IMA Short Course on Applied Algebraic Topology, June 2009.
[Slides]