Topology and Data

There is an increasing demand for the analysis of large amounts of data of many different kinds. Often the data is
equipped with a notion of distance, reflecting notions of similarity of the data. Further, it is often the case that the notion of similarity
is not backed up by theoretical constructions, but simply reflects an investigator’s intuitive notion of similarity. This means that
in studying the data set, one should perform analyses which have some degree of robustness to small changes in the metric, or
which have the property that one can track the change in analysis over changes in metrics or related parameters. Topological
methods can be adapted to these situation, and naturally suggest themselves since topology is by its definition robust to changes
in metrics. I will discuss how this point of view plays out in practice, with examples.