Sheaves as Computable and Stable Topological Invariants for Datasets

Explore the connections between sheaf theory and persistent homology in this 23-minute conference talk from the Applied Algebraic Topology Network. Delve into the world of constructible sheaves over normed vector spaces and their similarities to persistence modules. Learn about the convolution distance introduced by Kashiwara-Schapira and its stability properties. Discover the explicit connections between level-sets persistence and derived sheaf theory over the real line, including the construction of functors between 2-parameter persistence modules and sheaves. Examine the concept of Mayer-Vietoris systems and their classification, barcode decomposition, and stability theorems. Understand how these results establish a pseudo-isometric equivalence of categories between derived constructible sheaves and strictly pointwise finite-dimensional Mayer-Vietoris systems. Gain insights into using constructible sheaves as a continuous generalization of level-sets persistence in a computer-friendly way, providing a functorial equivalence between level-sets persistence and derived pushforward of sheaves for continuous real-valued functions.

Introduction
Definitions
Persistence and shifts
What is the derived category
Convolution forces
Shifts
Decomposition theorem
Dirac category
Isometry theorem
Levelset
Persistence Barcode
References
Questions
Matchings