Locales for Topological Data Analysis

In the past 20 years Topological Data Analysis has been a vibrant area of research a lot due to the developments in applied and computational algebraic topology. Essentially it applies the qualitative methods of topology to problems of machine learning, data mining or computer vision. Under this topic, persistent homology is an area of mathematics interested in identifying a global structure by inferring high-dimensional structure from low-dimensional representations and studying properties of a often continuous space by the analysis of a discrete sample of it, assembling discrete points into global structure. A recent approach to the study of persistent homology using techniques of lattice theory is presented in this talk where we will discuss some interpretations of the achieved complete Heyting algebra and possible applications of the topological information on the correspondent dual space.