基础教育均衡发展是现代教育发展的新境界，是教育未来发展的方向。本书是首次对中国基础教育均衡发展作理论和实证研究的专著。以广阔的理论视野，首次从宏观、中观、微观三个维度、十五个内涵指标分析入手，对教育均衡发展进行深入的理论探索和科学阐释，提出了具有独到见解的教育均衡发展理论体系。构建了教育均衡发展的基本体系框架，首次提出了基础教育均衡发展四个阶段的发展理论。首次把指数引入教育均衡分析，构建了分析研究教育均衡发展的指数体系和教育均衡发展指数。

The launch of this Advanced Lectures in Mathematics series is aimed at keepingmathematicians informed of the latest developments in mathematics, as well asto aid in the learning of new mathematical topics by students all over the world.Each volume consists of either an expository monograph or a collection of signifi-cant introductions to important topics. This series emphasizes the history andsources of motivation for the topics under discussion, and also gives an overviewof the current status of research in each particular field. These volumes are thefirst source to which people will turn in order to learn new subjects and to dis-cover the latest results of many cutting-edge fields in mathematics.

Introduction

1.1 Overview of Theories

1.1.1 RiemannMapping

1.1.2 Riemann Uniformization

1.1.3 Shape Space

1.1.4 General Geometric Structure

1.2 Algorithms for Computing Conformal Mappings

1.3 Applications

1.3.1 Computer Graphics

1.3.2 Computer Vision

1.3.3 Geometric Modeling

1.3.4 Medical Imaging

Further Readings

Part I Theories

Homotopy Group

2.1 Algebraic Topological Methodology

2.2 Surface Topological Classification

2.3 Homotopy of Continuous Mappings

2.4 Homotopy Group

2.5 Homotopy Invariant

2.6 Covering Spaces

2.7 Group Representation

2.8 Seifert-van Kampen Theorem

Problems

Homology and Cohomology

3.1 Simplicial Homology

3.1.1 Simplicial Complex

3.1.2 Geometric Approximation Accuracy

3.1.3 Chain Complex

3.1.4 Chain Map and Induced Homomorphism

3.1.5 Simplicial Map

3.1.6 Chain Homotopy

3.1.7 Homotopy Equivalence

3.1.8 Relation Between Homology Group and Homotopy Grou

3.1.9 Lefschetz Fixed Point

3.1.10 Mayer-Vietoris Homology Sequence

3.1.11 Tunnel Loop and Handle Loop

3.2 Cohomology

3.2.1 Cohomology Group

3.2.2 Cochain Map

3.2.3 Cochain Homotopy

Problems

4 Exterior Differential Calculus

4.1 Smooth Manifold

4.2 Differential Forms

4.3 Integration

4.4 Exterior Derivative and Stokes Theorem

4.5 De Rham Cohomology Group

4.6 Harmonic Forms

4.7 Hodge Theorem

Problems

5 Differential Geometry of Surfaces

5.1 Curve Theory

5.2 Local Theory of Surfaces

5.2.1 Regular Surface

5.2.2 First Fundamental Form

5.2.3 Second Fundamental Form

5.2.4 Weingarten Transformation

5.3 Orthonormal Movable Frame

5.3.1 Structure Equation

5.4 Covariant Differentiation

5.4.1 Geodesic Curvature

5.5 Gauss-Bonnet Theorem

5.6 Index Theorem of Tangent Vector Field

5.7 Minimal Surface

5.7.1 Weierstrass Representation

5.7.2 Costa Minimal Surface

Problems

Riemann Surface

6.1 Riemann Surface

6.2 Riemann Mapping Theorem

6.2.1 Conformal Module

6.2.2 Quasi-Conformal Mapping

6.2.3 Holomorphic Mappings

6.3 Holomorphic One-Forms

6.4 Period Matrix

6.5 Riemann-Roch Theorem

6.6 Abel Theorem

6.7 Uniformization

6.8 Hyperbolic Riemann Surface

6.9 Teichmiiller Space

6.9.1 Quasi-Conformal Map

6.9.2 Extremal Quasi-Conformal Map

6.10 Teichm011er Space and Modular Space

6.10.1 Fricke Space Model

6.10.2 Geodesic Spectrum

Problems

Harmonic Maps and Surface Ricci Flow

7.1 Harmonic Maps of Surfaces

7.1.1 Harmonic Energy and Harmonic Maps

7.1.2 Harmonic Map Equation

7.1.3 Rad6's Theorem

7.1.4 Hopf Differential

7.1.5 Complex Form

7.1.6 Bochner Formula

7.1.7 Existence and Regularity

7.1.8 Uniqueness

7.2 Surface Ricci Flow

7.2.1 Conformal Deformation

7.2.2 Surface Ricci Flow

Problems

Geometric Structure

8.1 (X, G) Geometric Structure

8.2 Development and Holonomy

8.3 Affine Structures on Surfaces

8.4 Spherical Structure

8.5 Euclidean Structure

8.6 Hyperbolic Structure

8.7 Real Projective Structure

Problems

Part II Algorithms

Topological Algorithms

9.1 Triangular Meshes

9.1.1 Half-Edge Data Structure

9.1.2 Code Samples

9.2 Cut Graph

9.3 Fundamental Domain

9.4 Basis of Homotopy Group

9.5 Gluing Two Meshes

9.6 Universal Covering Space

9.7 Curve Lifting

9.8 Homotopy Detection

9.9 The Shortest Loop

9.10 Canonical Homotopy Group Generator

Further Readings

Problems

10 Algorithms for Harmonic Maps

10.1 Piecewise Linear Functional Space, Inner Product and Laplacian

10.2 Newton's Method for Open Surface

10.3 Non-Linear Heat Diffusion for Closed Surfaces

10.4 Riemann Mapping

10.5 Least Square Method for Solving Beltrami Equation

10.6 General Surface Mapping

Further Readings

Problems

11 Harmonic Forms and Holomorphic Forms

11.1 Characteristic Forms

11.2 Wedge Product

11.3 Characteristic 1-Form

11.4 Computing Cohomology Basis

l 1.5 Harmonic 1-Form

11.6 Hodge Star Operator

11.7 Holomorphic 1-Form

11.8 Inner Product Among 1-Forms

11.9 Holomorphic Forms on Surfaces with Boundaries

11.10 Zero Points and Critical Trajectories

11.11 Flat Metric Induced by Holomorphic 1-Forms

11.12 Conformal Invariants

11.13 Conformal Mappings for Multi-Holed Annuli

Further Readings

Problems

12 Discrete Ricci Flow

12.1 Circle Packing Metric

12.2 Discrete Gaussian Curvature

12.3 Discrete Surface Ricci Flow

12.4 Newton's Method

12.5 Isometric Planar Embedding

12.6 Surfaces with Boundaries

12.7 Optimal Parameterization Using Ricci Flow

12.8 Hyperbolic Ricci Flow

12.9 Hyperbolic Embedding

12.9.1 Poincare Disk Model

12.9.2 Embedding the Fundamental Domain

12.9.3 Hyperbolic Embedding of the Universal Covering Space

12.10 Hyperbolic Ricci Flow for Surfaces with Boundaries

Further Readings

Problems

A Major Algorithms

B Acknowledgement

Reference

Index