经济领域的数学方法研究——曲面拓扑、微分几何和共形几何视角

这是一本数学家Norbert A’Campo先生在云南财经大学面向全国经济学者们一次公开培训班的讲稿实录，它记录了Norbert A’Campo先生带领大家迈向对纳什平衡（Nash equilibrium）的由基础向深层读解的全过程，希望能给中国广大的经济工作者和爱好者们带来更为透彻的数学理论视角解读。本书内容比较独特，中国国内没有竞争对手。对于促进我国经济数学的发展具有一定意义。

该书从掌握经济领域相关研究所需的共形几何理论基础开始，由基础微分几何逐步展开至黎曼几何，严谨、细致地对这一架构下的GeometryofManifolds（流形几何）、双曲几何（罗巴切夫斯基几何）、微分拓扑曲面、黎曼曲面等进行了系统阐述。同时，还引入在经济尤其是金融领域的研究问题求索中运用此系列数学方法的应用场景，从微分几何在经济尤其是金融领域的应用，推进至共形几何在经济尤其是金融领域的应用，对SABR模型、无套利几何、随机时间下的共形模型等进行了深入讨论。

朱南丽博士，在《Journal of Networks》等期刊上发表专业论文18篇，。主持或参与国家、省部及各级项目16项。Norbert A’Campo,瑞士数学家，于1972年获得法国巴黎大学博士学位。1974年，他成为国际数学家大会特邀讲者。 1988年当选瑞士数学学会主席，2012年入选美国数学学会。

1 Introduction
2 Basic Differential Geometry
2.1 Fields on open sets in real vector spaces
2.2 Closed forms are locally exact
2.3 Fixed Point Theorems
2.4 The abstract field C versus the R-algebra C of complex numbers
2.5 Coordinates and local smooth Rigidity Theorems
2.6 Differentiation in Banaeh spaces
2.7 Sard's Theorem
2.8 Morse Lemma, Morse functions
3 Geometry of Manifolds
3.1 Differential manifolds
3.2 Fields on manifolds
3.3 Integrability condition of Frobenius
3.4 Foliations on manifolds
3.5 Topology of connected, compact surfaces
4 Hyperbolic Geometry
4.1 Hyperbolic plane H=HI
4.2 Intermezzo: Higher Cross Ratios
4.3 Hyperbolic trigonometry
4.4 Hyperbolic Area
4.5 A compact hyperbolic surface of genus g ≥2
4.6 The Riemann Sphere □(数理化公式)
5 Examples of Geometry
5.1 The Space of Norms
5.2 Combinatorial Geometry
5.3 Spaces of Involutions
6 Differential Topology of Surfaces
6.1 o-and 1-deRham cohomology of surfaces
6.2 Hyperbolic plane again, now H=HJ
6.3 Reminder: Multi-Linear Algebra
6.4 Reminder: Holomorphie functions in one complex variable
6.5 J-Laplace operator and metric
6.6 J-surfaces
7 Riemann Surfaces
7.1 Riemann surfaces as z- and as J -surface
7.2 Natural structures on the space ~(TS)
7.3 J-fields and Integrability in higher dimension
7.4 Integrability of fibred J-fields
7.5 Analysis of Laplace operators on J -surfaces
7.6 Topology of the two point Green function
8 Differential Geometry in Economics and Finance
8.1 SABR model
8.2 Geometry of no arbitrage
9 Conformal Geometry in Economics and Finance
9.1 Models with Zero Instantaneous Spot Rate
9.2 Conformal Symmetries and Lorentz Transformations
9.3 Conformal Diffusion
9.4 Time Homogeneity and Hyperbolic (Lorentz) Evolution
9.5 The Conformal Model under Stochastic Clock
References
后记 ( Postscript )