Symplectic Manifold

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In Symplectic Geometry, a Symplectic Manifold is a smooth manifold equipped with a symplectic form, a special type of structure that plays a central role in Hamiltonian mechanics.

Formally, a symplectic manifold is a pair $(M, \omega)$, where $M$ is a smooth manifold, and $\omega$ is a closed, non-degenerate 2-form on $M$.

Properties:

• Closedness: The symplectic form $\omega$ is closed, meaning $d\omega = 0$.
• Non-degeneracy: The symplectic form $\omega$ is non-degenerate at each point in $M$.
• Even Dimension: A symplectic manifold must have even dimension.

Examples:

• Phase Space in Classical Mechanics: The cotangent bundle of a configuration space can be given a natural symplectic structure.
• Complex Projective Spaces: These can be equipped with a Fubini-Study symplectic form.

Applications:

• Hamiltonian Dynamics: Symplectic manifolds provide the geometric framework for Hamiltonian mechanics, where the flow of a Hamiltonian function corresponds to the motion of a mechanical system.
• Quantum Mechanics: The mathematical structure of quantum mechanics can be described using symplectic manifolds.

Symplectic manifolds serve as a unifying geometric structure in both classical and quantum physics and continue to be a rich area of research in mathematics and physics.

(edit) Topics in Statistical Mechanics and Thermodynamics x (📁) Fundamentals Ensemble Theory • Canonical Ensemble • Microcanonical Ensemble • Grand Canonical Ensemble • Phase Space • Partition Function • Equiparition Theorem • Boltzmann Distribution • Entropy • Free energy (Gibbs, Helmholtz • Enthalpy • Thermodynamic potentials • Thermodynamic equilibrium • Laws of Thermodynamics Phase Transitions First Order Phase Transition • Second Order Phase Transition • Critical Exponents • Renormalization Group • RG flow • Scaling Theory • Universality Classes • Phase transition • Critical point • Order Parameter • Correlation Function • Universality • Hysteresis • Non-Equilibrium Phase Transitions • Kosterlitz-Thouless Transition • Symmetry Breaking • Ergodicity Breaking • Replica Symmetry Breaking Dynamical Systems Stochastic processes • Irreversible Process • Adiabatic process • Isothermal process • Onsager Relations • Linear response theory • Fluctuation-dissipation theorem • Brownian motion • Fokker-Planck equation • Langevin equation • Master equation • Hopf decomposition Disordered Systems Spin glass • Random field Ising model • Griffiths phase • Percolation theory • Localization Models Ising Model • Landau Theory • Curie-Weiss Model • Bragg-Williams Approximation • Spin glass • Random field Ising model • Quenched disorder • Annealed disorder • Griffiths phase • Percolation theory • Localization • Hopfield Networks • Attractor Networks Computational Methods Monte Carlo Simulations • Molecular Dynamics Simulations • Mean-Field Simulations • Percolation Algorithms • Tensor Networks • Path Integral Methods • Quantum Annealing • Simulated Annealing Advanced Topics Neural Mean Field Theory • Energy-Based Models • Statistical Mechanics of Deep Learning • Topological Mean-Field Theory • Order Parameter-Based Theory • Generalized Rigidity • Emergence in Statistical Mechanics • Anderson Localization • Jammed Systems • Jammey Transitions • Gibbs Measures • Metastable States • Glassy Dynamics • Lyapunov Functions • Synchronization Phenomena • Self-Organized Criticality • Complex Systems in Statistical Mechanics • Information Theory in Statistical Mechanics • Quantum Statistical Mechanics • Geometric Deep Learning • Topological Data Analysis in Statistical Mechanics • Statistical Field Theory • Variational Methods in Statistical Mechanics • Effective Field Theory • Coarse-Graining Methods • Stochastic Optimization • Bayesian Inference in Statistical Mechanics • Maximum Entropy Methods• Statistical Mechanics of Coupled Networks • Statistical Mechanics of Social Networks • Statistical Mechanics of Financial Markets • Statistical Mechanics of Epidemics • Statistical Mechanics of Traffic Flow • Statistical Mechanics of Ecosystems

(edit) Topics in Topological Data Analysis (📁) Background Point-set topology • Algebraic topology Manifolds Manifold • Configuration space • Differentiability • Vector space (bundle) • Tangent space (bundle) • Cotangent space (bundle) • Vector field (Flow, Critical points) • Braid • Transversality • Whitney embedding theorem • Stratified space Complexes CW Complex • Δ-Complex • Regular Complex (Simplicial, Cubical, Vietoris-Rips, Čech, Witness, Flag, Strategy • Decision Task) • Nerve • Phylogenetic tree • Discretized graph configuration space • State complex Euler Characteristic Euler characteristic • Curvature • Nonvanishing vector field • Fixed point index • Tame topology • Euler calculus • Target enumeration • Fubini theorem • Euler integral transform • Intrinsic volume • Gaussian random field Exact Sequences Chain homotopy • Snake lemma • Good pair • Mayer-Vietoris sequence • 5-lemma • Cellular homology • Coverage • Degree • Sperner's Lemma • Borsuk-Ulam theorem • Short exact sequence • Lefschetz index • Nash equilibrium • Hex • Barcode • Space of natural images • Zigzag persistence Homology Simplicial homology • Künneth theorem • Coefficient • Singular homology • Reduced homology • Čech homology • Relative homology • Excision Theorem • Local homology • Relation homology • Functoriality • Inverse kinematics • Winding number • Brouwer fixed point theorem Cohomology Duality • Cochain complex • Cohomology • Universal coefficient theorem • Poincaré duality • Alexander duality • Helly's Theorem • Numerical Euler integration • Forms • De Rham cohomology • Cup product • Helicity Theorem • Laplacian • Hodge Theorem • Cohomological coordination Morse Theory Morse homology • Excursion set • Definable functional • Euler-Fourier transform • Stratified Morse theory • Conley index • Forcing theory • Attractor-repeller pairs • Floer homology • Symplectic form • Hamiltonian field • Arnold's Conjecture • Lefschetz-Hopf theorem • Stratified Lefschetz-Morse Index • Discrete Morse theory • LS category • Unimodal decomposition Homotopy Fundamental group • Van-Kampen theorem • Covering space • Knot • Braid • Seifert conjecture • Higher homotopy groups • Biaxial nematics • Hurewicz Theorem • Whitehead Theorem • Eilenberg-MacLane space • Topological social choice • Fiber bundle • Section • Path space • Farber lemma • Sectional category • Topological complexity • Fibration • Homotopy extension property • Model category • Homotopy type theory Sheaves Cellular sheaf • Local cohomology • Fiber homology • Sheaf cohomology • Global sections theorem • Linear sheaf • Flow sheaf • Network coding sheaf • Information flow theorem • Čech cohomology • Preseaf • Stalk • Gluing axiom • PDE sheaf • Operations on sheaves • Vietoris mapping theorem • Sheaf-theoretic Nyquist-Shannon • Fourier-Sato transform • Cosheaf • Bézier curve • Barcodes • Interval Indecomposables • Étale Space • Leray spectral sequence • Spectral sheaf theory • Sheaf neural network Categorification Category • Groupoid • Monoid • Morphism • Equalizer • Functor • Kan complex • Nerve • Clustering • Natural transformation • Cone • Interleaving Limit • Terminal object • Product • Pullback • Homology Colimit • Initial object • Coproduct • Pushout • Cohomology Topological sheaf • Sheaf operations • Derived functors