55Uxx Applied algebraic topology
This subtopic introduces the core ideas in applied algebraic topology, including foundational concepts, standard methods, and the main questions used to organize the area. Typical uses include building mathematical background, framing related research problems, and supporting applications in neighboring fields where these concepts provide useful structure.
Specific topics
55U05 Abstract complexes in algebraic topology
Overview
Abstract complexes in algebraic topology. This topic covers applied algebraic topology, where homological and homotopical invariants are used in data, networks, and computational models.
Related Wikipedia Page
Wikipedia: Applied topology
Useful Links
Key Ideas
- persistent and computational invariants
- algebraic-topological descriptors of data
- stability and robustness of summaries
Typical Uses
Used to extract shape information from complex data and to compare structures beyond metric summaries.
Applications
- Topological data analysis
- Sensor/network analysis
- Shape and image understanding
References
Recommended Textbooks
55U10 Simplicial sets and complexes in algebraic topology
Overview
Simplicial sets and complexes in algebraic topology. This topic covers applied algebraic topology, where homological and homotopical invariants are used in data, networks, and computational models.
Related Wikipedia Page
Wikipedia: Applied topology
Useful Links
Key Ideas
- persistent and computational invariants
- algebraic-topological descriptors of data
- stability and robustness of summaries
Typical Uses
Used to extract shape information from complex data and to compare structures beyond metric summaries.
Applications
- Topological data analysis
- Sensor/network analysis
- Shape and image understanding
References
Recommended Textbooks
55U15 Chain complexes in algebraic topology
Overview
Chain complexes in algebraic topology. This topic covers applied algebraic topology, where homological and homotopical invariants are used in data, networks, and computational models.
Related Wikipedia Page
Wikipedia: Applied topology
Useful Links
Key Ideas
- persistent and computational invariants
- algebraic-topological descriptors of data
- stability and robustness of summaries
Typical Uses
Used to extract shape information from complex data and to compare structures beyond metric summaries.
Applications
- Topological data analysis
- Sensor/network analysis
- Shape and image understanding
References
Recommended Textbooks
55U20 Universal coefficient theorems, Bockstein operator
Overview
Universal coefficient theorems, Bockstein operator. This topic covers applied algebraic topology, where homological and homotopical invariants are used in data, networks, and computational models.
Related Wikipedia Page
Wikipedia: Applied topology
Useful Links
Key Ideas
- persistent and computational invariants
- algebraic-topological descriptors of data
- stability and robustness of summaries
Typical Uses
Used to extract shape information from complex data and to compare structures beyond metric summaries.
Applications
- Topological data analysis
- Sensor/network analysis
- Shape and image understanding
References
Recommended Textbooks
55U25 Homology of a product, Künneth formula
Overview
Homology of a product, Künneth formula. This topic covers applied algebraic topology, where homological and homotopical invariants are used in data, networks, and computational models.
Related Wikipedia Page
Wikipedia: Applied topology
Useful Links
Key Ideas
- persistent and computational invariants
- algebraic-topological descriptors of data
- stability and robustness of summaries
Typical Uses
Used to extract shape information from complex data and to compare structures beyond metric summaries.
Applications
- Topological data analysis
- Sensor/network analysis
- Shape and image understanding
References
Recommended Textbooks
55U30 Duality in applied homological algebra and category theory in algebraic topology
Overview
Duality in applied homological algebra and category theory in algebraic topology. This topic covers applied algebraic topology, where homological and homotopical invariants are used in data, networks, and computational models.
Related Wikipedia Page
Wikipedia: Applied topology
Useful Links
Key Ideas
- persistent and computational invariants
- algebraic-topological descriptors of data
- stability and robustness of summaries
Typical Uses
Used to extract shape information from complex data and to compare structures beyond metric summaries.
Applications
- Topological data analysis
- Sensor/network analysis
- Shape and image understanding
References
Recommended Textbooks
55U35 Abstract and axiomatic homotopy theory in algebraic topology
Overview
Abstract and axiomatic homotopy theory in algebraic topology. This topic covers applied algebraic topology, where homological and homotopical invariants are used in data, networks, and computational models.
Related Wikipedia Page
Wikipedia: Applied topology
Useful Links
Key Ideas
- persistent and computational invariants
- algebraic-topological descriptors of data
- stability and robustness of summaries
Typical Uses
Used to extract shape information from complex data and to compare structures beyond metric summaries.
Applications
- Topological data analysis
- Sensor/network analysis
- Shape and image understanding
References
Recommended Textbooks
55U40 Topological categories, foundations of homotopy theory
Overview
Topological categories, foundations of homotopy theory. This topic covers applied algebraic topology, where homological and homotopical invariants are used in data, networks, and computational models.
Related Wikipedia Page
Wikipedia: Applied topology
Useful Links
Key Ideas
- persistent and computational invariants
- algebraic-topological descriptors of data
- stability and robustness of summaries
Typical Uses
Used to extract shape information from complex data and to compare structures beyond metric summaries.
Applications
- Topological data analysis
- Sensor/network analysis
- Shape and image understanding
References
Recommended Textbooks
55U99 None of the above
Overview
None of the above. This topic covers applied algebraic topology, where homological and homotopical invariants are used in data, networks, and computational models.
Related Wikipedia Page
Wikipedia: Applied topology
Useful Links
Key Ideas
- persistent and computational invariants
- algebraic-topological descriptors of data
- stability and robustness of summaries
Typical Uses
Used to extract shape information from complex data and to compare structures beyond metric summaries.
Applications
- Topological data analysis
- Sensor/network analysis
- Shape and image understanding
References
Recommended Textbooks