Biparametric persistence for smooth filtrations

TL;DR

This paper introduces biparametric persistence diagrams for smooth mappings on manifolds, using Whitney theory to address the lack of algebraic frameworks in multivariate persistence.

Contribution

It proposes a novel approach to biparametric persistence based on Whitney theory, bypassing algebraic limitations of existing methods.

Findings

Defines biparametric persistence diagrams for smooth maps

Connects persistence with Whitney theory and Morse theory

Provides a new perspective on multivariate persistence

Abstract

The goal of this note is to define biparametric persistence diagrams for smooth generic mappings $h=(f,g):M\to V\cong \mathbb{R}^2$ 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. We approach the problem from the Whitney theory perspective, similar to how single parameter persistence can be viewed through the lens of Morse theory.