Symplectic Geometry §

Group: 4 #group-4

Relations §

• Integrable Systems: Integrable systems are a class of dynamical systems with a rich algebraic and geometric structure, and have close connections to symplectic geometry.
• Canonical Transformations: Canonical transformations are a type of transformation that preserves the symplectic structure, and are important in symplectic geometry.
• Gromov’s Non-Squeezing Theorem: Gromov’s non-squeezing theorem is a fundamental result in symplectic geometry, demonstrating the rigidity of symplectic embeddings.
• Fiber Bundles: Symplectic geometry, which studies manifolds with a symplectic form, is closely related to the study of certain types of fiber bundles.
• Phase Space: Phase space is a key concept in symplectic geometry, representing the state of a dynamical system in terms of position and momentum coordinates.
• Symplectic Field Theory: Symplectic field theory is a variant of Gromov-Witten theory, studying pseudoholomorphic curves in symplectic manifolds and their algebraic invariants.
• Lagrangian Submanifolds: Lagrangian submanifolds are a special type of submanifold in symplectic geometry, playing an important role in various constructions and theorems.
• Poisson Brackets: Poisson brackets are a generalization of the usual commutator of vector fields, and are closely related to symplectic geometry through the Poisson bracket on functions.
• Smooth Space: Symplectic manifolds are a type of smooth manifold with additional structure.
• Symplectic Reduction: Symplectic reduction is a technique for constructing new symplectic manifolds from old ones, by taking quotients with respect to certain group actions.
• Geometric Quantization: Geometric quantization is a program for constructing quantum theories from classical symplectic manifolds, providing a link between symplectic geometry and quantum mechanics.
• Symplectic Capacities: Symplectic capacities are numerical invariants of symplectic manifolds, measuring the symplectic size of subsets and providing obstructions to symplectic embeddings.
• Mirror Symmetry: Mirror symmetry is a remarkable phenomenon relating symplectic geometry and complex geometry, providing deep connections between these two areas.
• Moment Maps: Moment maps are a tool for studying the symmetries of symplectic manifolds and their associated Hamiltonian actions.
• Moser’s Trick: Moser’s trick is a powerful technique in symplectic geometry for constructing symplectic diffeomorphisms between symplectic manifolds.
• Weinstein’s Neighborhood Theorem: Weinstein’s neighborhood theorem provides a local normal form for symplectic manifolds near a Lagrangian submanifold, and is an important tool in symplectic geometry.
• Hamiltonian Systems: Hamiltonian systems are a fundamental concept in symplectic geometry, describing the dynamics of systems with a symplectic structure.
• Darboux’s Theorem: Darboux’s theorem is a fundamental result in symplectic geometry, stating that locally, every symplectic manifold is isomorphic to a standard symplectic vector space.
• Floer Homology: Floer homology is a powerful tool in symplectic geometry, providing invariants of Lagrangian submanifolds and their intersections.
• Symplectic Manifolds: Symplectic manifolds are the central objects of study in symplectic geometry, being smooth manifolds equipped with a closed, non-degenerate 2-form.
• Riemannian Geometry: Symplectic Geometry is closely related to Riemannian Geometry, as it studies manifolds with a symplectic form instead of a Riemannian metric.
• Complex Geometry: Symplectic geometry is the study of geometric structures on manifolds equipped with a symplectic form, which has applications in complex geometry, particularly in the study of Calabi-Yau manifolds and mirror symmetry.
• Gromov-Witten Invariants: Gromov-Witten invariants are important algebraic invariants in symplectic geometry, counting pseudoholomorphic curves in symplectic manifolds.