The Gromov-Hausdorff metric relies on discrete approximations to determine how metrically similar two spaces are, and recently has been applied to recognition problems in computer imaging, among many important uses. Under the right circumstances, this "coarse" metric nonetheless can capture topological information such as the fundamental group or even homeomorphism type.
On the other hand, discrete homotopies, recently developed by Berestovskii-Plaut and extended in joint work with Jay Wilkins, can be used to not only determine the fundamental group of many compact spaces, it
also adds geometric information in the form of the Homotopy Critical Spectrum, which is closely linked to the Length and Covering Spectra. We will give a basic introduction to these two concepts that is accessible to anyone who knows what a metric space is, and will conclude with a finiteness theorem linking these two
concepts, which generalizes finiteness theorems of Anderson and Shen-Wei.