Distinguishing classes of surfaces in $\mathbb{R}^n$ is a task which arises in many situations. There are many characteristics we can use to solve this classification problem. The Persistent Homology Transform allows us to look at shapes in $\mathbb{R}^n$ from $S^{n-1}$ directions simultaneously, and is a useful tool for surface classification. Using the Julia package DiscretePersistentHomologyTransform, we will look at some example curves in $\mathbb{R}^2$ and examine distinguishing features. 

The TMD algorithm (Kanari et al. 2018) computes the barcode of a neuron (tree) with respect to the radial or path distance from the soma (root). We are interested in the inverse problem: how to understand the space of trees that are represented by the same barcode. Our tool to study this spaces is the stochastic TNS algorithm (Kanari et al. 2020) which generates trees from a given barcode in a biologically meaningful way. 

I will present some theoretical results on the space of trees that have the same barcode, as well as the effect of adding noise to the barcode. In addition, I will provide a more combinatorial perspective on the space of barcodes, expressed in terms of the symmetric group. I will illustrate these results with experiments based on the TNS.

This is joint work with L. Kanari and K. Hess.