We propose a measure of shape which is appropriate for the study of a complicated geometric structure, defined using the topology of neighborhoods of the structure. One aspect of this measure gives aâ€¦ (More)

We introduce a fractal dimension for a metric space based on the persistent homology of subsets of that space. We exhibit hypotheses under which this dimension is comparable to the upper boxâ€¦ (More)

Although random cell complexes occur throughout the physical sciences, there does not appear to be a standard way to quantify their statistical similarities and differences. The various proposals inâ€¦ (More)

We study the asymptotic behavior of the persistent homology of i.i.d. samples from a d-Ahlfors regular measure on a metric space â€” one that satisfies uniform bounds of the form 1 c r â‰¤ Î¼ (Br (x)) â‰¤ câ€¦ (More)

We define notions of local topological convergence and local geometric convergence for embedded graphs in R, and study their properties. The former is related to BenjaminiSchramm convergence, and theâ€¦ (More)

The three dimensional structure of DNA in the nucleus (chromatin) plays an important role in many cellular processes. Recent experimental advances have led to high-throughput methods of capturingâ€¦ (More)