# persistence

## merge tree monoid

$$\def\Real{\mathbb{R}}\def\tree{\mathcal{T}}$$ Consider a path metric space $$M$$ and a continuous function $$f:M\to \Real$$. For simplicity, all such pairs here will have sublevel sets of $$f$$ compact (so that the function is bounded from below). Then one can form the so-called merge tree (see, e.g. here, for motivation, and some history), a new topological space (let’s …

## singularities, biparametric persistence and cubical complexes

$$\def\Real{\mathbb{R}}\def\phd{\mathbf{P}H}\def\CAT{\mathtt{CAT}}$$ The goal of this note is to define the biparametric persistence diagrams for smooth generic mappings $$h=(f,g):M\to\Real^2$$ for smooth compact manifold (M). Existing approaches to multivariate persistence are mostly centered on the workaround of absence of reasonable algebraic theories for quiver representations for lattices of rank 2 or higher, or similar artificial obstacles. Singularities …