Symplectic geometry and topology

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عدد المتطلبات السابقة الإلزامية (حتى الأساس): 45

ستتعلّم

• 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.

تعريف

• 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.

مبرهنة

• 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.

مفاهيم خاطئة شائعة

• 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.

اكتشافات اختيارية

• 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

مسارات التعلّم

• Mathematical physics

قراءات إضافية

• Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology (1995)
• Ana Cannas da Silva, Lectures on Symplectic Geometry (2001)