Computational and applied algebraic topology research

I'm interested in problems arising from efforts to use the invariants of algebraic topology to analyze real data sets. The basic point of departure is the idea of studying finite metric spaces ("point clouds") by constructing associated simplicial complexes and looking at the resulting topological invariants.

Of course, constructing the simplicial complexes typically involves a choice of feature scale. A very nice recent idea is to develop homological invariants which encode information across all feature scales at once (more precisely, which track changes in homological invariants as the feature scale changes). This is usually denoted as "persistence" or "persistent topological invariants". See the Stanford computational topology page for a comprehensive overview (and a lot of interesting work in this direction).

My work in this area has three basic directions, carried out with I. Gal, M. Mandell, and M. Pancia.

Foundations for persistent homotopy theory

A basic principle of modern homotopy theory is that homotopical information is captured by topologized mapping spaces. In the simplicial context, the simplicial approximation theorem tells us we must subdivide the source complex in order to realize all homotopy classes of maps. By studying maps out of small models of test complexes (e.g., spheres), subdivision provides a notion of scale. We develop a version of the simplicial approximation theorem in the setting of simplicial complexes, adapted for computation of persistent homotopy groups.

Quantitative homotopy theory in topological data analysis
joint with Michael A. Mandell
To appear in Foundations of Computational Mathematics.

Statistical methods in computational algebraic topology

To interpret the results of computing the persistent homology of a data set, it would be very useful to have access to the tools and language of statistical hypothesis testing. Moreover, a focus on the data set as a sample from an underlying distribution gives us a perspective from which to quantify robust behavior in the face of outlying data points. To this end, we have studied the persistent homology of a metric measure space in terms of a distribution of barcodes associated to samples of a fixed size. The resulting invariants are robust in the sense of being relatively insensitive to large changes in small (in measure) parts of the space. In addition, we can approximate these invariants and construct confidence intervals for understanding the dependence of the results on sample variation using resampling methods.

Persistent homology for metric measure spaces, and robust statistics for hypothesis testing and confidence intervals
joint with Itamar Gal, Michael Mandell, and Mathew Pancia.
To appear in Foundations of Computational Mathematics.
Resampling methods for computing the persistent homology of metric measure spaces
joint with Michael Mandell

Practical computation of persistent homology

Computing persistent homology efficiently for large data sets is a fundamental question. The number of simplices in Rips and Cech complexes can grow exponentially in the number of vertices, and witness complexes suffer the same growth as a function of the number of witnesses. One approach to resolving this is via simplicial collapse methods. We have developed a version of this which is compatible with persistent homology.

Simplicial collapse for efficient computation of persistent homology
joint with Itamar Gal