研究路径Geometry & Topology课程已上线

Symplectic geometry and topology

Optional research direction: the geometry of classical mechanics. Symplectic manifolds and Darboux's theorem, Hamiltonian flows, Moser's trick, moment maps, Gromov's non-squeezing theorem, the Arnold conjecture and Floer homology, and a glimpse of mirror symmetry.

math.geom.symplectic-geometry

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前置知识

Smooth manifolds Differential forms and Stokes' theorem Classical mechanics

45 项必需的前置知识（一直追溯到底）

你将学会

Define symplectic manifolds and Hamiltonian vector fields, and derive Hamilton's equations.
Prove Darboux's theorem using Moser's trick, showing symplectic manifolds have no local invariants.
State Gromov's non-squeezing theorem and the Arnold conjecture, and explain the idea of Floer homology, distinguishing proved cases from open ones.

定义

A symplectic form on a manifold is a closed, non-degenerate 2-form ω.
The Hamiltonian vector field X_H of a function H is defined by ι_{X_H} ω = dH.
A symplectomorphism is a diffeomorphism φ with φ*ω = ω.

定理

Darboux: every symplectic manifold is locally isomorphic to (ℝ²ⁿ, Σ dxᵢ ∧ dyᵢ).
Hamiltonian flows preserve ω and hence volume (Liouville's theorem).
Gromov non-squeezing (1985): a ball of radius r embeds symplectically in the cylinder B²(R) × ℝ²ⁿ⁻² only if r ≤ R.

常见误解

Thinking symplectic maps are just volume-preserving maps; in dimension at least 4, non-squeezing shows symplectic maps are far more rigid.
Believing symplectic geometry has curvature invariants like Riemannian geometry; by Darboux's theorem there are no local invariants at all.

可选发现

Complex and Kähler geometry; Hodge theory Every Kähler manifold is symplectic; mirror symmetry conjecturally exchanges the symplectic geometry of one Calabi–Yau manifold with the complex geometry of another.
Derived categories Kontsevich's homological mirror symmetry conjecture predicts an equivalence between a Fukaya category (symplectic) and a derived category of coherent sheaves (algebraic).
← Smooth manifolds

学习路径

Mathematical physics