58Dxx Spaces of mappings
This subtopic introduces the core ideas in spaces of mappings, 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
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
Overview
Groups of diffeomorphisms and homeomorphisms as manifolds. This topic studies spaces of mappings between manifolds and spaces, including functional-topological structure and moduli-type behavior.
Related Wikipedia Page
Wikipedia: Mapping space
Useful Links
Key Ideas
- topologies on function spaces
- smooth mapping manifolds
- moduli and deformation perspectives
Typical Uses
Used to treat families of maps as geometric objects in their own right for analysis and topology.
Applications
- Gauge and field theory
- Shape spaces
- Diffeomorphism and embedding space studies
References
Recommended Textbooks
58D07 Groups and semigroups of nonlinear operators
Overview
Groups and semigroups of nonlinear operators. This topic studies spaces of mappings between manifolds and spaces, including functional-topological structure and moduli-type behavior.
Related Wikipedia Page
Wikipedia: Mapping space
Useful Links
Key Ideas
- topologies on function spaces
- smooth mapping manifolds
- moduli and deformation perspectives
Typical Uses
Used to treat families of maps as geometric objects in their own right for analysis and topology.
Applications
- Gauge and field theory
- Shape spaces
- Diffeomorphism and embedding space studies
References
Recommended Textbooks
58D10 Spaces of imbeddings and immersions
Overview
Spaces of imbeddings and immersions. This topic studies spaces of mappings between manifolds and spaces, including functional-topological structure and moduli-type behavior.
Related Wikipedia Page
Wikipedia: Mapping space
Useful Links
Key Ideas
- topologies on function spaces
- smooth mapping manifolds
- moduli and deformation perspectives
Typical Uses
Used to treat families of maps as geometric objects in their own right for analysis and topology.
Applications
- Gauge and field theory
- Shape spaces
- Diffeomorphism and embedding space studies
References
Recommended Textbooks
58D15 Manifolds of mappings
Overview
Manifolds of mappings. This topic studies spaces of mappings between manifolds and spaces, including functional-topological structure and moduli-type behavior.
Related Wikipedia Page
Wikipedia: Mapping space
Useful Links
Key Ideas
- topologies on function spaces
- smooth mapping manifolds
- moduli and deformation perspectives
Typical Uses
Used to treat families of maps as geometric objects in their own right for analysis and topology.
Applications
- Gauge and field theory
- Shape spaces
- Diffeomorphism and embedding space studies
References
Recommended Textbooks
58D17 Manifolds of metrics (esp. Riemannian)
Overview
Manifolds of metrics (esp. Riemannian). This topic studies spaces of mappings between manifolds and spaces, including functional-topological structure and moduli-type behavior.
Related Wikipedia Page
Wikipedia: Mapping space
Useful Links
Key Ideas
- topologies on function spaces
- smooth mapping manifolds
- moduli and deformation perspectives
Typical Uses
Used to treat families of maps as geometric objects in their own right for analysis and topology.
Applications
- Gauge and field theory
- Shape spaces
- Diffeomorphism and embedding space studies
References
Recommended Textbooks
58D19 Group actions and symmetry properties of spaces of mappings
Overview
Group actions and symmetry properties of spaces of mappings. This topic studies spaces of mappings between manifolds and spaces, including functional-topological structure and moduli-type behavior.
Related Wikipedia Page
Wikipedia: Mapping space
Useful Links
Key Ideas
- topologies on function spaces
- smooth mapping manifolds
- moduli and deformation perspectives
Typical Uses
Used to treat families of maps as geometric objects in their own right for analysis and topology.
Applications
- Gauge and field theory
- Shape spaces
- Diffeomorphism and embedding space studies
References
Recommended Textbooks
58D20 Measures (Gaussian, cylindrical, etc.) on manifolds of maps
Overview
Measures (Gaussian, cylindrical, etc.) on manifolds of maps. This topic studies spaces of mappings between manifolds and spaces, including functional-topological structure and moduli-type behavior.
Related Wikipedia Page
Wikipedia: Mapping space
Useful Links
Key Ideas
- topologies on function spaces
- smooth mapping manifolds
- moduli and deformation perspectives
Typical Uses
Used to treat families of maps as geometric objects in their own right for analysis and topology.
Applications
- Gauge and field theory
- Shape spaces
- Diffeomorphism and embedding space studies
References
Recommended Textbooks
58D25 Equations in function spaces; evolution equations
Overview
Equations in function spaces; evolution equations. This topic studies spaces of mappings between manifolds and spaces, including functional-topological structure and moduli-type behavior.
Related Wikipedia Page
Wikipedia: Mapping space
Useful Links
Key Ideas
- topologies on function spaces
- smooth mapping manifolds
- moduli and deformation perspectives
Typical Uses
Used to treat families of maps as geometric objects in their own right for analysis and topology.
Applications
- Gauge and field theory
- Shape spaces
- Diffeomorphism and embedding space studies
References
Recommended Textbooks
58D27 Moduli problems for differential geometric structures
Overview
Moduli problems for differential geometric structures. This topic studies spaces of mappings between manifolds and spaces, including functional-topological structure and moduli-type behavior.
Related Wikipedia Page
Wikipedia: Mapping space
Useful Links
Key Ideas
- topologies on function spaces
- smooth mapping manifolds
- moduli and deformation perspectives
Typical Uses
Used to treat families of maps as geometric objects in their own right for analysis and topology.
Applications
- Gauge and field theory
- Shape spaces
- Diffeomorphism and embedding space studies
References
Recommended Textbooks
58D29 Moduli problems for topological structures
Overview
Moduli problems for topological structures. This topic studies spaces of mappings between manifolds and spaces, including functional-topological structure and moduli-type behavior.
Related Wikipedia Page
Wikipedia: Mapping space
Useful Links
Key Ideas
- topologies on function spaces
- smooth mapping manifolds
- moduli and deformation perspectives
Typical Uses
Used to treat families of maps as geometric objects in their own right for analysis and topology.
Applications
- Gauge and field theory
- Shape spaces
- Diffeomorphism and embedding space studies
References
Recommended Textbooks
58D30 Applications of manifolds of mappings to the sciences
Overview
Applications of manifolds of mappings to the sciences. This topic studies spaces of mappings between manifolds and spaces, including functional-topological structure and moduli-type behavior.
Related Wikipedia Page
Wikipedia: Mapping space
Useful Links
Key Ideas
- topologies on function spaces
- smooth mapping manifolds
- moduli and deformation perspectives
Typical Uses
Used to treat families of maps as geometric objects in their own right for analysis and topology.
Applications
- Gauge and field theory
- Shape spaces
- Diffeomorphism and embedding space studies
References
Recommended Textbooks