計算共形幾何與應用

1. Introduction • Overview of Theories • Algorithms for Computing Conformal Mappings • Applications 2. Homotopy Group • Algebraic Topological Methodology • Surface Topological Classification • Homotopy of Continuous Mappings • Homotoy Group • Homotopy Invariant • Covering Spaces • Group Representation • Seifert-van Kampen Theorem 3. Homology and Cohomology • Simplicial Homology • Cohomology 4. Exterior Differential Calculus • Smooth Manifold • Differential Forms • Integration • Exterior Derivative and Stokes Theorem • De Rham Cohomology Group • Harmonic Forms • Hodge Theorem 5. Differential Geometry of Surfaces • Curve Theory • Local Theory of Surfaces • Orthonormal Movable Frame • Covariant Differentiation • Gauss-Bonnet Theorem • Index Theorem of Tangent Vector Field • Minimal Surface 6. Riemann Surface • Riemann Surface • Riemann Mapping Theorem • Holomorphic One-Forms • Period Matrix • Riemann-Roch Theorem • Abel Theorem • Uniformization • Hyperbolic Riemann Surface 10. Algorithms for Harmonic Maps • Piecewise Linear Functional Space, Inner Product and Laplacian • Newton’s Method for Open Surface • Non-Linear Heat Diffusion for Closed Surfaces • Riemann Mapping • Least Square Method for Solving Beltrami Equation • General Surface Mapping 11. Harmonic Forms and Holomorphic Forms • Computing Cohomology Basis • Harmonic 1-Form • Hodge Star Operator • Holomorphic 1-Form • Inner Product Among 1-Forms • Holomorphic Forms on Surfaces with Boundaries • Flat Metric Induced by Holomorphic 1-Forms • Conformal Invariants • Conformal Mappings for Multi-Holed Annuli

微積分

專題:60% 書報討論:40%

Computational Conformal Geometry, Xianfeng David Gu & Shing-Tung Yau, Higher Education Press, 2008.