Abstract: Many persistent topological constructions can be regarded as a sheaf or a cosheaf. For example, given a surface with a height function, its Reeb graph is equivalent to the display space of its cosheaf of components. Alternatively, one can consider the space of all surfaces with height functions and study how this space fibers over the space of Reeb graphs or its downstream space of persistence diagrams. In this talk, I will present some recent progress on classifying embedded spheres with the same persistence, which leverages a recent calculation of the homotopy type of embedded circles in the plane by myself, Gelnett, and Zaremsky. Finally, I’ll present some recent interpolation work for the persistent homology transform (PHT) via Kan extensions. This provides a first theoretical baseline for regression for the PHT, which I hope neural networks will refine in the near future.