Differential topology, and specifically Morse theory, provide a suitable setting for formalizing
and solving several problems related to shape analysis. The fundamental idea behind …

The continued and dramatic rise in the size of data sets has meant that new methods are
required to model and analyze them. This timely account introduces topological data …

Detecting meaningful structure in neural activity and connectivity data is challenging in the
presence of hidden nonlinearities, where traditional eigenvalue-based methods may be …

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We study the homology of a filtered d-dimensional simplicial complex K as a single
algebraic entity and establish a correspondence that provides a simple description over …

Persistent homology captures the topology of a filtration-a one-parameter family of
increasing spaces-in terms of a complete discrete invariant. This invariant is a multiset of …

The tree structure is currently the accepted paradigm to represent evolutionary relationships
between organisms, species or other taxa. However, horizontal, or reticulate, genomic …

In this paper, we initiate a study of shape description and classification via the application of
persistent homology to two tangential constructions on geometric objects. Our techniques …

The Vietoris-Rips complex characterizes the topology of a point set. This complex is popular
in topological data analysis as its construction extends easily to higher dimensions. We …

The ever-growing amount of data stored in digital databases raises the question of how to
organize and extract useful knowledge. This paper outlines some current developments in …