garywelz/programming_framework
0
1<!DOCTYPE html>2<html lang="en">3<head>4 <meta charset="UTF-8">5 <meta name="viewport" content="width=device-width, initial-scale=1.0">6 <title>Poisson Structures - Mathematics Process</title>7 <script src="https://cdn.jsdelivr.net/npm/mermaid@10.6.1/dist/mermaid.min.js"></script>8 <style>9 * { margin: 0; padding: 0; box-sizing: border-box; }10 body { font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; background: linear-gradient(135deg, #667eea 0%, #764ba2 100%); min-height: 100vh; padding: 20px; }11 .container { max-width: 1200px; margin: 0 auto; background: white; border-radius: 15px; box-shadow: 0 20px 40px rgba(0,0,0,0.1); overflow: hidden; }12 .header { background: #1e8449; color: white; padding: 30px; }13 .header h1 { margin: 0 0 10px 0; font-size: 2em; font-weight: 300; }14 .header-meta { display: flex; flex-wrap: wrap; gap: 15px; margin-top: 15px; font-size: 0.9em; opacity: 0.9; }15 .meta-item { background: rgba(255,255,255,0.2); padding: 5px 12px; border-radius: 20px; }16 .nav-links { padding: 15px 30px; background: #f8f9fa; border-bottom: 1px solid #ecf0f1; }17 .nav-links a { color: #1e8449; text-decoration: none; margin-right: 20px; font-weight: 500; }18 .nav-links a:hover { text-decoration: underline; }19 .content { padding: 30px; }20 .description { margin-bottom: 30px; }21 .flowchart-container { margin: 30px 0; }22 .flowchart-container h2 { color: #2c3e50; margin-bottom: 15px; }23 .mermaid { background: white; padding: 20px; border-radius: 10px; border: 1px solid #ecf0f1; overflow-x: auto; }24 .color-legend { background: #f8f9fa; padding: 20px; border-radius: 10px; margin: 30px 0; }25 .color-legend h3 { color: #2c3e50; margin-bottom: 15px; }26 .color-grid { display: grid; grid-template-columns: repeat(auto-fit, minmax(180px, 1fr)); gap: 15px; }27 .color-item { display: flex; align-items: center; gap: 10px; padding: 10px; background: white; border-radius: 5px; }28 .color-box { width: 30px; height: 30px; border-radius: 4px; border: 1px solid #ddd; }29 .info-card { background: #f8f9fa; padding: 20px; border-radius: 10px; margin-top: 20px; }30 .info-card h3 { color: #2c3e50; margin-bottom: 15px; }31 .info-card ul { list-style: none; padding: 0; }32 .info-card li { padding: 8px 0; border-bottom: 1px solid #ecf0f1; }33 .info-card li:last-child { border-bottom: none; }34 </style>35</head>36<body>37 <div class="container">38 <div class="header">39 <h1>Poisson Structures</h1>40 <div class="header-meta">41 <span class="meta-item">Symplectic Geometry</span>42 <span class="meta-item">Mathematics</span>43 </div>44 </div>45 <div class="nav-links">46 <a href="../../mathematics_index.html">← Back to Mathematics Index</a>47 <a href="symplectic_geometry-index.html">Symplectic Geometry Index</a>48 <a href="https://arxiv.org/list/math.SG/recent" target="_blank">arXiv: Symplectic Geometry (math.SG)</a>49 </div>50 <div class="content">51 <div class="description">52 <h2>Description</h2>53 <p>Poisson manifold: bracket {·,·} satisfying Leibniz and Jacobi. Symplectic leaves, Casimirs, and deformation quantization.</p>54 </div>55 <div class="flowchart-container">56 <h2>Dependency Flowchart</h2>57 <div class="mermaid">58graph TD59 D1["D1 Poisson bracket π\nSkew-symmetric, Leibniz, Jacobi identity"]60 D2["D2 Poisson manifold (M, π)\nBivector field with [π,π] = 0"]61 D3["D3 Symplectic leaf\nMaximal integral submanifold of π#(T*M)"]62 D4["D4 Casimir function C\n{C, f} = 0 for all f"]63 64 T1["T1 ω symplectic ⇒ {f,g} = ω(X_f,X_g)\nDefines Poisson"]65 T2["T2 Symplectic foliation\n(M,π) union of symplectic leaves"]66 T3["T3 Darboux-Weinstein\nLocal splitting: symplectic × cosymplectic"]67 T4["T4 Kostant-Souriau\nPrequantization of integral ω"]68 T5["T5 Kontsevich formality\nDeformation quantization exists"]69 70 D1 --> D271 D2 --> D372 D2 --> D473 D1 --> T174 D3 --> T275 D2 --> T376 D4 --> T277 T1 --> D278 T2 --> T379 T3 --> T580 81 classDef definition fill:#3498db,color:#fff,stroke:#2980b982 classDef theorem fill:#1abc9c,color:#fff,stroke:#16a08583 class D1,D2,D3,D4 definition84 class T1,T2,T3,T4,T5 theorem85 </div>86 </div>87 <div class="color-legend">88 <h3>Color Scheme</h3>89 <div class="color-grid">90 <div class="color-item"><div class="color-box" style="background:#3498db"></div><div><strong>Blue</strong><br><small>Definitions (D1–D4)</small></div></div>91 <div class="color-item"><div class="color-box" style="background:#1abc9c"></div><div><strong>Teal</strong><br><small>Theorems (T1–T5)</small></div></div>92 </div>93 </div>94 <div class="info-card">95 <h3>Info</h3>96 <ul>97 <li><strong>Subcategory:</strong> symplectic_geometry</li>98 <li><strong>Keywords:</strong> Poisson bracket, symplectic leaf, Casimir, Kontsevich, deformation quantization</li>99 <li><strong>Research frontier:</strong> <a href="https://arxiv.org/list/math.SG/recent" target="_blank">arXiv math.SG</a></li>100 </ul>101 </div>102 </div>103 </div>104 <script>105 mermaid.initialize({ startOnLoad: true, theme: 'default', flowchart: { useMaxWidth: true, htmlLabels: true, curve: 'step', nodeSpacing: 25, rankSpacing: 40 } });106 </script>107</body>108</html>109