The Big Idea
Persistent homology can identify significant topological features of data. In this context, data usually means a cloud of points, often representing measurements of some object, and possibly occurring in a high-dimensional space. Topological features include clusters, loops, and voids. Homology is a powerful mathematical tool that identifies topological features.
In the usual setting, data is indexed by a single parameter, such as time. Suppose that any fixed time value determines a particular configuration of data, for which homology can be computed. In this case, persistent homology is able to detect topological features of the data that persist over long intervals of time. Persistence information can be expressed algebraically in a persistence module, or graphically in a persistence diagram or barcode.
My work involves the computation and visualization of multidimensional persistent homology. Often, data is indexed not by a single parameter, but by multiple parameters, such as time and distance. In this case, understanding and visualizing persistent homology is much more complicated than in the one-dimensional setting. Michael Lesnick and I have developed algorithms to efficiently compute multidimensional persistent homology. We are also working on a software program, soon to be released, that will allow interactive visualization of multidimensional persistence. Two papers about our algorithms are also in progress.
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More information about my software is coming soon!