Hugo Åkesson: Studying short exact sequences for the sake of topological data analysis

Topological data analysis provides ways of summarizing data sets that otherwise might be too large. One such summary is the barcode, which contains geometrical information of the data. It turns out that we can consider morphisms between barcodes, and even short exact sequences of them:
  Y -> E -> X
The main question of my thesis is: for fixed barcodes Y and X, what can the barcode E look like, in a short exact sequence as above? This is motivated by the recent development of noise systems, a tool for constructing distances between barcodes. The answer to our question turns out to be quite pictorial.