Sheaves, Cosheaves and Applications

Loading...
Thumbnail Image
Degree type
Doctor of Philosophy (PhD)
Graduate group
Mathematics
Discipline
Subject
algebraic topology
applied topology
cosheaves
sheaves
stratification theory
Applied Mathematics
Mathematics
Funder
Grant number
License
Copyright date
2015-11-16T20:14:00-08:00
Distributor
Related resources
Contributor
Abstract

This thesis develops the theory of sheaves and cosheaves with an eye towards applications in science and engineering. To provide a theory that is computable, we focus on a combinatorial version of sheaves and cosheaves called cellular sheaves and cosheaves, which are finite families of vector spaces and maps parametrized by a cell complex. We develop cellular (co)sheaves as a new tool for topological data analysis, network coding and sensor networks. We utilize the barcode descriptor from persistent homology to interpret cellular cosheaf homology in terms of Borel-Moore homology of the barcode. We associate barcodes to network coding sheaves and prove a duality theorem there. A new approach to multi-modal sensing is introduced, where sheaves and cosheaves model detection and evasion sets. A foundation for multi-dimensional level-set persistent homology is laid via constructible cosheaves, which are equivalent to representations of MacPherson's entrance path category. By proving a van Kampen theorem, we give a direct proof of this equivalence. A cosheaf version of the i'th derived pushforward of the constant sheaf along a definable map is constructed directly as a representation of this category. We go on to clarify the relationship of cellular sheaves to cosheaves by providing a formula that takes a cellular sheaf and produces a complex of cellular cosheaves. This formula lifts to a derived equivalence, which in turn recovers Verdier duality. Compactly-supported sheaf cohomology is expressed as the coend with the image of the constant sheaf through this equivalence. The equivalence is further used to establish relations between sheaf cohomology and a herein newly introduced theory of cellular sheaf homology. Inspired to provide fast algorithms for persistence, we prove that the derived category of cellular sheaves over a 1D cell complex is equivalent to a category of graded sheaves. Finally, we introduce the interleaving distance as an extended metric on the category of sheaves. We prove that global sections partition the space of sheaves into connected components. We conclude with an investigation into the geometry of the space of constructible sheaves over the real line, which we relate to the bottleneck distance in persistence.

Advisor
Robert W. Ghrist
Date of degree
2014-01-01
Date Range for Data Collection (Start Date)
Date Range for Data Collection (End Date)
Digital Object Identifier
Series name and number
Volume number
Issue number
Publisher
Publisher DOI
Journal Issue
Comments
Recommended citation