
VietorisRips complexes of circles, ellipses, and higherdimensional spheres.
Topology, Geometry, and Data Analysis seminar at The Ohio State University, Feb 2017.
[Abstract]
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. Janko Latschev proves an analogous theorem for sufficiently dense samplings. Little is known about the behavior of VietorisRips complexes for larger values of r, even though these complexes arise naturally in applied topology and persistent homology. We describe how 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. These homotopy types are connected to cyclic and centrally symmetric polytopes and orbitopes. Interestingly, Latschev's result fails for the ellipse with larger r values, and certain VietorisRips complexes of ellipses contain ephemeral summands in their persistent homology modules. We argue that infinite VietorisRips complexes should be equipped with a different topology: an optimal transport or Wasserstein metric thickening the metric on X. Using this new metric, we describe the first change in homotopy type (as r increases) of VietorisRips complexes of higherdimensional spheres. Joint work with Michał Adamaszek, Florian Frick, and Samadwara Reddy.

Metric reconstruction via VietorisRips complexes and optimal transport.
Florida International University Winter Conference on Geometry, Topology, and Applications, Jan 2017.
[Abstract,
Notes]

An introduction to applied and computational topology.
Florida International University Winter Conference on Geometry, Topology, and Applications, Jan 2017.
[Abstract,
Slides]

Cyclic polytopes and nerve complexes.
Rocky Mountain Algebraic Combinatorics Seminar, Oct 2016.
[Abstract,
Notes]

An introduction to computational topology.
CSU Computer Science Colloquium, Oct 2016.
[Abstract,
Slides,
Video]

The theory of VietorisRips complexes: What is known and what is open?
Minisymposium on Applied and Computational Topology at the SIAM Central States Section Meeting, Oct 2016.
[Slides]

What is topology, and how is it applied to data analysis?
Front Range Computational & Systems Biology Symposium, July 2016.

VietorisRips complexes of circles and ellipses.
Joint Mathematics Meetings, AMS Special Session on Applied and Computational Topology, Jan 2016.
[Abstract,
Slides]

Random cyclic dynamical systems.
Rocky Mountain Algebraic Combinatorics Seminar, Sept 2015.
[Abstract]

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.

University of Rochester Data Science Colloquium, April 2015.

Applied Algebraic Topology Research Network, Online Seminar Series, Mar 2015. [Video]

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.

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.

CSU Pattern Analysis Lab, Oct 2015.

Duke Graduate & Faculty Seminar, Feb 2015.

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]