Counting functions on the interval and the foundations of TDA

Topology offers a set of descriptors---trees, persistence diagrams, and sheaves---for the analysis of data where shape, broadly speaking, is important. I will present a new technique called "chiral merge trees" especially suited to the study of time series. Counting the number of chiral merge trees that realize a given persistence diagram refines Arnold's Calculus of Snakes and has a suggestive entropic interpretation. Since the space of trees is CAT(0), the existence of unique Fréchet means overcomes certain statistical challenges in persistence. Finally, I will discuss how constructible cosheaves provide a unifying data structure for the study of multi-variate data, where questions of numerical approximation and convergence are leading to the study of analysis on these categorically defined structures.