卷绕:拓扑、几何和分析中的卷绕数

   卷绕数是拓扑学中基本的不变量之一。它测量一个动点P绕一个不动点Q运动的次数，前提是P的运动路径不经过Q并且P的最终位置和它的起始位置相同。这个简单的想法有着深远的应用。通过《卷绕：拓扑、几何和分析中的卷绕数》的学习，读者将了解以下内容：卷绕数如何帮助我们证明每个多项式方程都有一个根（代数基本定理），保证通过单个平面切割对空间中三个对象进行公平划分（火腿三明治定理），解释为什么每个简单的闭曲线都有内部和外部（Jordan 曲线定理），将微积分与曲率和向量场的奇点联系起来（Hopf指数定理），允许从无穷中减去无穷并得到一个有限的答案（Toeplitz算子），推广给出关于矩阵群拓扑的一个基本且美丽的洞见（Bott周期性定理）。

《卷绕：拓扑、几何和分析中的卷绕数》适合对卷绕数的概念及其在分析、微分几何和拓扑等数学领域中的应用感兴趣的本科生和研究生阅读。

   Foreword：MASS and REU at Penn State University

Preface

Chapter 1.Prelude： Love，Hate，and Exponentials

1.1.Two sets of travelers

1.2.Winding around

1.3.The most important function in mathematics

1.4.Exercises




Chapter 2.Paths and Homotopies

2.1.Path connectedness

2.2.Homotopy

2.3.Homotopies and simple-connectivity

2.4.Exercises




Chapter 3.The Winding Number

3.1.Maps to the punctured plane

3.2.The winding number

3.3.Computing winding numbers

3.4.Smooth paths and loops

3.5.Counting roots via winding numbers

3.6.Exercises




Chapter 4.Topology of the Plane

4.1.Some classic theorems

4.2.The Jordan curve theorem Ⅰ

4.3.The Jordan curve theorem Ⅱ

4.4.Inside the Jordan curve

4.5.Exercises




Chapter 5.Integrals and the Winding Number

5.1.Differential forms and integration

5.2.Closed and exact forms

5.3.The winding number via integration

5.4.Homology

5.5.Cauchy's theorem

5.6.A glimpse at higher dimensions

5.7.Exercises




Chapter 6.Vector Fields and the Rotation Number

6.1.The rotation number

6.2.Curvature and the rotation number

6.3.Vector fields and singularities

6.4.Vector fields and surfaces

6.5.Exercises




Chapter 7.The Winding Number in Functional Analysis

7.1.The Fredholm index

7.2.Atkinson's theorem

7.3.Toeplitz operators

7.4.The Toeplitz index theorem

7.5.Exercises

……

Chapter 8.Coverings and the Fundamental Group

Chapter 9.Coda： The Bott Periodicity Theorem

Appendix A.Linear Algebra

Appendix B.Metric Spaces

Appendix C.Extension and Approximation Theorems

Appendix D.Measure Zero

Appendix E.Calculus on Normed Spaces

Appendix F.Hilbert Space

Appendix G.Groups and Graphs

Bibliography

Index

Foreword： MASS and REU at Penn State University

Preface

Chapter 1.Prelude： Love，Hate，and Exponentials

1.1.Two sets of travelers

1.2.Winding around

1.3.The most important function in mathematics

1.4.Exercises

Chapter 2.Paths and Homotopies

2.1.Path connectedness

2.2.Homotopy

2.3.Homotopies and simple-connectivity

2.4.Exercises

Chapter 3.The Winding Number

3.1.Maps to the punctured plane

3.2.The winding number

3.3.Computing winding numbers

3.4.Smooth paths and loops

3.5.Counting roots via winding numbers

3.6.Exercises

Chapter 4.Topology of the Plane

4.1.Some classic theorems

4.2.The Jordan curve theorem I

4.3.The Jordan curve theorem II

4.4.Inside the Jordan curve

4.5.Exercises

Chapter 5.Integrals and the Winding Number

5.1.Differential forms and integration

5.2.Closed and exact forms

5.3.The winding number via integration

5.4.Homology

5.5.Cauchy's theorem

5.6.A glimpse at higher dimensions

5.7.Exercises

Chapter 6.Vector Fields and the Rotation Number

6.1.The rotation number

6.2.Curvature and the rotation number

6.3.Vector fields and singularities

6.4.Vector fields and surfaces

6.5.Exercises

Chapter 7.The Winding Number in Functional Analysis

7.1.The Fredholm index

7.2.Atkinson's theorem

7.3.Toeplitz operators

7.4.The Toeplitz index theorem

7.5.Exercises

Chapter 8.Coverings and the Fundamental Group

Chapter 9.Coda： The Bott Periodicity Theorem

Appendix A.Linear Algebra

Appendix B.Metric Spaces

Appendix C.Extension and Approximation Theorems

Appendix D.Measure Zero

Appendix E.Calculus on Normed Spaces

Appendix F.Hilbert Space

Appendix G.Groups and Graphs

Bibliography

Index