In his doctoral thesis, "Sheaves, Cosheaves and Applications," Justin Curry develops the theory of sheaves and cosheaves with a focus on their applications in science and engineering. To make the theory computable, he introduces cellular sheaves and cosheaves, which are finite families of vector spaces and maps parametrized by a cell complex. These cellular (co)sheaves become a new tool for topological data analysis, network coding, and sensor networks. Curry lays the foundation for multi-dimensional level-set persistent homology through constructible cosheaves that are equivalent to representations of MacPherson's entrance path category. He proves a van Kampen theorem to provide a direct proof of this equivalence. Additionally, he constructs a cosheaf version of the i'th derived pushforward of the constant sheaf along a definable map as a representation of this category. Curry clarifies the relationship between cellular sheaves and cosheaves by providing a formula that defines derived equivalence, which recovers Verdier duality. He expresses compactly-supported sheaf cohomology 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 newly introduced cellular sheaf homology. To provide fast algorithms for persistence, Curry proves that the derived category of cellular sheaves over a 1D cell complex is equivalent to a category of graded sheaves. Finally, he introduces an extended pseudo-metric called interleaving distance on the category of sheaves. He proves that global sections partition space into connected components and investigates the geometry of constructible sheaves over the real line related to bottleneck distance in persistence. Overall, Curry's work provides significant contributions to both theoretical mathematics and practical applications in various fields such as computer science, physics, biology, economics etc., offering new insights into how these concepts can be applied in practice.
Justin Curry wrote a big paper about math that helps people in science and engineering. He made some new tools called cellular sheaves and cosheaves to make the math easier to use. These tools can help with things like analyzing data, coding networks, and using sensors. He also found a way to make the math faster by using something called "graded sheaves". Finally, he made up a new way to measure how close two things are in the math world.
- Sheaves: A mathematical tool used in topology and geometry to study spaces.
- Cosheaves: A related tool used for studying functions on spaces.
- Topological data analysis: Using math to analyze shapes and spaces.
- Vector spaces: Spaces where you can add or multiply things together.
- Algorithms: A set of instructions for solving problems.
Sheaves, Cosheaves and Their Applications: A Summary of Justin Curry's Doctoral Thesis
Justin Curry’s doctoral thesis “Sheaves, Cosheaves and Applications” presents a comprehensive overview of the theory of sheaves and cosheaves with a focus on their applications in science and engineering. In this article, we will provide an overview of his work as well as its implications for various fields such as computer science, physics, biology, economics etc.
Introduction to Sheaves and Cosheaves
Curry introduces cellular sheaves and cosheaves which are finite families of vector spaces and maps parametrized by a cell complex. These (co)sheaf structures become powerful tools for topological data analysis, network coding, sensor networks etc. He then proves a van Kampen theorem to provide a direct proof that these cellular (co)sheave structures are equivalent to representations of MacPherson's entrance path category. Additionally he provides a formula that defines derived equivalence between cellular sheaf cohomology and newly introduced cellular sheaf homology which recovers Verdier duality.
Multi-Dimensional Level-Set Persistent Homology
Curry lays the foundation for multi-dimensional level-set persistent homology through constructible cosheaves that are equivalent to representations of MacPherson's entrance path category. He constructs a cosheaf version of the i'th derived pushforward of the constant sheaf along a definable map as representation for this category. Compactly supported sheaf cohomology is expressed as coend with image of constant sheaf through this equivalence while global sections partition space into connected components related to bottleneck distance in persistence.
Fast Algorithms For Persistence
To provide fast algorithms for persistence Curry proves that derived category of cellular sheave over 1D cell complex is equivalent to graded categories which can be used in practice in various fields such as computer science or economics etc.. Finally he introduces extended pseudo metric called interleaving distance on category of sheave which allows further exploration into geometry related topics like bottleneck distance in persistence .
Conclusion
Overall Justin Curry’s work provides significant contributions both theoretical mathematics and practical applications offering new insights into how these concepts can be applied in practice . His research paper has opened up many possibilities for further exploration into topological data analysis , network coding , sensor networks etc., providing valuable insight into how these concepts can be used effectively .
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. 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 defines 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 pseudo-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.
