**Title**: Lecture 3: Group-invariant persistent homology and its use for topological data analysis.

**Speaker**: Patrizio Frosini (University of Bolohna)

**Abstract**: Persistent homology has proven itself efficient in the topological and qualitative comparison of filtered topological spaces, when invariance with respect to every homeomorphism is required. However, we can make the following two observations about the use of persistent homology for application purposes. On the one hand, more restricted kinds of invariance are sometimes preferable (e.g., in shape comparison). On the other hand, in several practical situations filtering functions are not just auxiliary technical tools that can be exploited to study a given topological space, but instead the main aim of our analysis. Indeed, most of the data is usually produced by measurements, whose results are quite often functions defined on a topological space. In fact, in many applications the dataset of interest is seen as a collection Phi of real-valued functions defined on a given topological space X, instead of a family of topological spaces. As a natural consequence, in these cases observers can be seen as collections of suitable operators on Phi. Starting from these remarks, in this lecture we describe a way to combine persistent homology with the use of G-invariant non-expansive operators defined on Phi, where G is a group of self-homeomorphisms of X. This approach gives us a method to study Phi in a way that is invariant with respect to G.

**Seminar URL**: http://www.tgda.osu.edu/mini-course-frosini.html