53Dxx Symplectic and contact geometry
This subtopic introduces the core ideas in symplectic and contact geometry, 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
53D05 Symplectic manifolds, general
Overview
Symplectic manifolds, general. This topic treats symplectic and contact geometry, focusing on structures underlying Hamiltonian dynamics, invariants, and rigidity/flexibility phenomena.
Related Wikipedia Page
Wikipedia: Symplectic geometry
Useful Links
Key Ideas
- symplectic forms and Hamiltonian flows
- contact structures and Reeb dynamics
- holomorphic-curve and Floer-theoretic techniques
Typical Uses
Used to model conservative dynamics and study geometric invariants in even and odd dimensional settings.
Applications
- Classical mechanics
- Low-dimensional topology
- Mirror symmetry and string-inspired geometry
References
Recommended Textbooks
53D10 Contact manifolds, general
Overview
Contact manifolds, general. This topic treats symplectic and contact geometry, focusing on structures underlying Hamiltonian dynamics, invariants, and rigidity/flexibility phenomena.
Related Wikipedia Page
Wikipedia: Symplectic geometry
Useful Links
Key Ideas
- symplectic forms and Hamiltonian flows
- contact structures and Reeb dynamics
- holomorphic-curve and Floer-theoretic techniques
Typical Uses
Used to model conservative dynamics and study geometric invariants in even and odd dimensional settings.
Applications
- Classical mechanics
- Low-dimensional topology
- Mirror symmetry and string-inspired geometry
References
Recommended Textbooks
53D12 Lagrangian submanifolds; Maslov index
Overview
Lagrangian submanifolds; Maslov index. This topic treats symplectic and contact geometry, focusing on structures underlying Hamiltonian dynamics, invariants, and rigidity/flexibility phenomena.
Related Wikipedia Page
Wikipedia: Symplectic geometry
Useful Links
Key Ideas
- symplectic forms and Hamiltonian flows
- contact structures and Reeb dynamics
- holomorphic-curve and Floer-theoretic techniques
Typical Uses
Used to model conservative dynamics and study geometric invariants in even and odd dimensional settings.
Applications
- Classical mechanics
- Low-dimensional topology
- Mirror symmetry and string-inspired geometry
References
Recommended Textbooks
53D15 Almost contact and almost symplectic manifolds
Overview
Almost contact and almost symplectic manifolds. This topic treats symplectic and contact geometry, focusing on structures underlying Hamiltonian dynamics, invariants, and rigidity/flexibility phenomena.
Related Wikipedia Page
Wikipedia: Symplectic geometry
Useful Links
Key Ideas
- symplectic forms and Hamiltonian flows
- contact structures and Reeb dynamics
- holomorphic-curve and Floer-theoretic techniques
Typical Uses
Used to model conservative dynamics and study geometric invariants in even and odd dimensional settings.
Applications
- Classical mechanics
- Low-dimensional topology
- Mirror symmetry and string-inspired geometry
References
Recommended Textbooks
53D17 Poisson manifolds; Poisson groupoids and algebroids
Overview
Poisson manifolds; Poisson groupoids and algebroids. This topic treats symplectic and contact geometry, focusing on structures underlying Hamiltonian dynamics, invariants, and rigidity/flexibility phenomena.
Related Wikipedia Page
Wikipedia: Symplectic geometry
Useful Links
Key Ideas
- symplectic forms and Hamiltonian flows
- contact structures and Reeb dynamics
- holomorphic-curve and Floer-theoretic techniques
Typical Uses
Used to model conservative dynamics and study geometric invariants in even and odd dimensional settings.
Applications
- Classical mechanics
- Low-dimensional topology
- Mirror symmetry and string-inspired geometry
References
Recommended Textbooks
53D18 Generalized geometries (Ã la Hitchin)
Overview
Generalized geometries (Ã la Hitchin). This topic treats symplectic and contact geometry, focusing on structures underlying Hamiltonian dynamics, invariants, and rigidity/flexibility phenomena.
Related Wikipedia Page
Wikipedia: Symplectic geometry
Useful Links
Key Ideas
- symplectic forms and Hamiltonian flows
- contact structures and Reeb dynamics
- holomorphic-curve and Floer-theoretic techniques
Typical Uses
Used to model conservative dynamics and study geometric invariants in even and odd dimensional settings.
Applications
- Classical mechanics
- Low-dimensional topology
- Mirror symmetry and string-inspired geometry
References
Recommended Textbooks
53D20 Momentum maps; symplectic reduction
Overview
Momentum maps; symplectic reduction. This topic treats symplectic and contact geometry, focusing on structures underlying Hamiltonian dynamics, invariants, and rigidity/flexibility phenomena.
Related Wikipedia Page
Wikipedia: Symplectic geometry
Useful Links
Key Ideas
- symplectic forms and Hamiltonian flows
- contact structures and Reeb dynamics
- holomorphic-curve and Floer-theoretic techniques
Typical Uses
Used to model conservative dynamics and study geometric invariants in even and odd dimensional settings.
Applications
- Classical mechanics
- Low-dimensional topology
- Mirror symmetry and string-inspired geometry
References
Recommended Textbooks
53D22 Canonical transformations
Overview
Canonical transformations. This topic treats symplectic and contact geometry, focusing on structures underlying Hamiltonian dynamics, invariants, and rigidity/flexibility phenomena.
Related Wikipedia Page
Wikipedia: Symplectic geometry
Useful Links
Key Ideas
- symplectic forms and Hamiltonian flows
- contact structures and Reeb dynamics
- holomorphic-curve and Floer-theoretic techniques
Typical Uses
Used to model conservative dynamics and study geometric invariants in even and odd dimensional settings.
Applications
- Classical mechanics
- Low-dimensional topology
- Mirror symmetry and string-inspired geometry
References
Recommended Textbooks
53D25 Geodesic flows in symplectic geometry and contact geometry
Overview
Geodesic flows in symplectic geometry and contact geometry. This topic treats symplectic and contact geometry, focusing on structures underlying Hamiltonian dynamics, invariants, and rigidity/flexibility phenomena.
Related Wikipedia Page
Wikipedia: Symplectic geometry
Useful Links
Key Ideas
- symplectic forms and Hamiltonian flows
- contact structures and Reeb dynamics
- holomorphic-curve and Floer-theoretic techniques
Typical Uses
Used to model conservative dynamics and study geometric invariants in even and odd dimensional settings.
Applications
- Classical mechanics
- Low-dimensional topology
- Mirror symmetry and string-inspired geometry
References
Recommended Textbooks
53D30 Symplectic structures of moduli spaces
Overview
Symplectic structures of moduli spaces. This topic treats symplectic and contact geometry, focusing on structures underlying Hamiltonian dynamics, invariants, and rigidity/flexibility phenomena.
Related Wikipedia Page
Wikipedia: Symplectic geometry
Useful Links
Key Ideas
- symplectic forms and Hamiltonian flows
- contact structures and Reeb dynamics
- holomorphic-curve and Floer-theoretic techniques
Typical Uses
Used to model conservative dynamics and study geometric invariants in even and odd dimensional settings.
Applications
- Classical mechanics
- Low-dimensional topology
- Mirror symmetry and string-inspired geometry
References
Recommended Textbooks
53D35 Global theory of symplectic and contact manifolds
Overview
Global theory of symplectic and contact manifolds. This topic treats symplectic and contact geometry, focusing on structures underlying Hamiltonian dynamics, invariants, and rigidity/flexibility phenomena.
Related Wikipedia Page
Wikipedia: Symplectic geometry
Useful Links
Key Ideas
- symplectic forms and Hamiltonian flows
- contact structures and Reeb dynamics
- holomorphic-curve and Floer-theoretic techniques
Typical Uses
Used to model conservative dynamics and study geometric invariants in even and odd dimensional settings.
Applications
- Classical mechanics
- Low-dimensional topology
- Mirror symmetry and string-inspired geometry
References
Recommended Textbooks
53D37 Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category
Overview
Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category. This topic treats symplectic and contact geometry, focusing on structures underlying Hamiltonian dynamics, invariants, and rigidity/flexibility phenomena.
Related Wikipedia Page
Wikipedia: Symplectic geometry
Useful Links
Key Ideas
- symplectic forms and Hamiltonian flows
- contact structures and Reeb dynamics
- holomorphic-curve and Floer-theoretic techniques
Typical Uses
Used to model conservative dynamics and study geometric invariants in even and odd dimensional settings.
Applications
- Classical mechanics
- Low-dimensional topology
- Mirror symmetry and string-inspired geometry
References
Recommended Textbooks
53D40 Symplectic aspects of Floer homology and cohomology
Overview
Symplectic aspects of Floer homology and cohomology. This topic treats symplectic and contact geometry, focusing on structures underlying Hamiltonian dynamics, invariants, and rigidity/flexibility phenomena.
Related Wikipedia Page
Wikipedia: Symplectic geometry
Useful Links
Key Ideas
- symplectic forms and Hamiltonian flows
- contact structures and Reeb dynamics
- holomorphic-curve and Floer-theoretic techniques
Typical Uses
Used to model conservative dynamics and study geometric invariants in even and odd dimensional settings.
Applications
- Classical mechanics
- Low-dimensional topology
- Mirror symmetry and string-inspired geometry
References
Recommended Textbooks
53D42 Symplectic field theory; contact homology
Overview
Symplectic field theory; contact homology. This topic treats symplectic and contact geometry, focusing on structures underlying Hamiltonian dynamics, invariants, and rigidity/flexibility phenomena.
Related Wikipedia Page
Wikipedia: Symplectic geometry
Useful Links
Key Ideas
- symplectic forms and Hamiltonian flows
- contact structures and Reeb dynamics
- holomorphic-curve and Floer-theoretic techniques
Typical Uses
Used to model conservative dynamics and study geometric invariants in even and odd dimensional settings.
Applications
- Classical mechanics
- Low-dimensional topology
- Mirror symmetry and string-inspired geometry
References
Recommended Textbooks
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
Overview
Gromov-Witten invariants, quantum cohomology, Frobenius manifolds. This topic treats symplectic and contact geometry, focusing on structures underlying Hamiltonian dynamics, invariants, and rigidity/flexibility phenomena.
Related Wikipedia Page
Wikipedia: Symplectic geometry
Useful Links
Key Ideas
- symplectic forms and Hamiltonian flows
- contact structures and Reeb dynamics
- holomorphic-curve and Floer-theoretic techniques
Typical Uses
Used to model conservative dynamics and study geometric invariants in even and odd dimensional settings.
Applications
- Classical mechanics
- Low-dimensional topology
- Mirror symmetry and string-inspired geometry
References
Recommended Textbooks
53D50 Geometric quantization
Overview
Geometric quantization. This topic treats symplectic and contact geometry, focusing on structures underlying Hamiltonian dynamics, invariants, and rigidity/flexibility phenomena.
Related Wikipedia Page
Wikipedia: Symplectic geometry
Useful Links
Key Ideas
- symplectic forms and Hamiltonian flows
- contact structures and Reeb dynamics
- holomorphic-curve and Floer-theoretic techniques
Typical Uses
Used to model conservative dynamics and study geometric invariants in even and odd dimensional settings.
Applications
- Classical mechanics
- Low-dimensional topology
- Mirror symmetry and string-inspired geometry
References
Recommended Textbooks
53D55 Deformation quantization, star products
Overview
Deformation quantization, star products. This topic treats symplectic and contact geometry, focusing on structures underlying Hamiltonian dynamics, invariants, and rigidity/flexibility phenomena.
Related Wikipedia Page
Wikipedia: Symplectic geometry
Useful Links
Key Ideas
- symplectic forms and Hamiltonian flows
- contact structures and Reeb dynamics
- holomorphic-curve and Floer-theoretic techniques
Typical Uses
Used to model conservative dynamics and study geometric invariants in even and odd dimensional settings.
Applications
- Classical mechanics
- Low-dimensional topology
- Mirror symmetry and string-inspired geometry
References
Recommended Textbooks