Statement of Current Research

I am interested in computational topology and geometry, combinatorial topology, and topology applied to data analysis and to sensor networks. My current research: §1. Advances the theory of Vietoris–Rips simplicial complexes. Given a set of points X sampled from a metric space M , what information can one recover about M? One approach is to build a Vietoris–Rips simplicial complex, which depends on the choice of a scale parameter r, on top of vertex set X. If M is a Riemannian manifold, X is sufficiently dense, and r is sufficiently small, then the Vietoris–Rips complex of X at scale r is homotopy equivalent to M . In practice (e.g. when trying to estimate the shape of a data set X) one does not know how to choose scale r, and so the philosophy of persistent homology is to vary the scale from small to large and to trust those topological features which persist. However, the theory of Vietoris–Rips complexes is very poorly understood as the scale r increases, even though such complexes arise naturally in applications of persistent homology. My research addresses the following question: how do Vietoris–Rips complexes of manifolds behave as we increase the scale parameter? Surprising answers arise: as a first example, the Vietoris–Rips complexes of the circle obtain the homotopy types of the circle, 3-sphere, 5-sphere, 7-sphere, . . . , as scale r increases. §2. Applies topology to data analysis and sensor networks. I work on provably-stable methods for combining persistent homology with machine learning techniques. I also study a Morse-theoretic approach, using the nudged elastic band method from computational chemistry, to build coarse cell complex models for data. These methods have been applied to data arising from social networks, image processing, computer vision, microarray analysis, discrete dynamical systems, and PDEs. In minimal sensor network problems, one is given only local data measured by many weak sensors but tries to answer a global question. For example, if sensors are scattered in a domain and each cannot measure its location but instead only the identities of its neighboring sensors, can we determine if the entire domain is covered? Topological tools (including Vietoris–Rips complexes) are useful for this passage from local to global. Together these topics form a unified program in applied topology, with persistent homology and Vietoris–Rips complexes as recurring tools.