| Abstract: |
| Static persistence, that is the persistent homology computed for a single filtration function $f:K \to \mathbf{R}$, has been highly utilized throughout topological data analysis. The first work looking at the persistent homology for a parameterized family of functions, $f_{p}:K \to \mathbf{R}$, came in the form of vineyards by Morozov et al, where the parameterization was over some interval, i.e. $p \in [a,b]$. More recently, there has been a growing interest in parameterized persistence where $p \in B$ for some more complex base space. For example, Turner et al introduced the Persistent Homology Transform (PHT), which studies the persistence of a family of functions on an embedded space $|K| \subset \mathbf{R}^n$ parameterized over $p \in \mathbf{S}^{n-1}$. The most general form of this came with the introduction of persistent homology bundles by Hickok, for arbitrary base space $B$. This parameterized input data holds interesting structure; of note is the discovery of monodromy in these bundles, a point in the persistence diagram might not come back to the same place while tracing out a closed loop in the base space. In this talk, we will introduce the concept of a \textit{canopy}: a topological space with some additional structural data reminiscent of a bundle which stores the representatives of persistence classes over the parameterizing base space $B$. Canopies are particularly useful since they are well defined, even in the presence of monodromy. We show that for nice enough parameterized families of functions, we can construct such a canopy and give structure theorems for the types of issues that can give rise to monodromy. |
|