GTM(Graduate Texts in Mathematics) 系列, 目前是 \(269\) 册. 详细的书目可以在 wiki 找到, 亦可在 Springer 看到. 不仅如此, Springer 有每一本书的介绍, 也有部分电子书出售. 这里试着写出每本书最新的版本, 按学科对这些书进行分类, 并争取对每本书给出一个大致的评价.

## Number Theory 数论

7 A course in arithmetic, Serre

58 P-adic Numbers, p-adic Analysis, and Zeta-Functions, Neal Koblitz

**评论**:本书与 97, 114的作者都是 Neal Koblitz.

74 Multiplicative number theory, Harold Davenport&Hugh L.Montgpmery, 3rd edition (2000)

**评论**: 较为现代的解析数论经典书籍, 展示了这学科的困难所在.

84 A Classical Introduction to Modern Number Theory, Kenneth Ireland&Michael Rosen

97 Introduction to Elliptic Curves and Modular Forms, Neal Koblitz

114 A Course in Number Theory and Cryptography, Neal Koblitz

164 Additive Number Theory: The Classical Bases, Melvyn B. Nathanson

**评论**:本书和 165 讨论的主题都是堆垒数论中的经典问题. 164,165 与 195 的作者是同一个人.

165 Additive Number Theory: Inverse problems and the geometry of sumsets, Melvyn B. Nathanson

177 Analytic Number Theory, Donald J. Newman

195 Elementary Methods in Number Theory, Melvyn B. Nathanson

210 Number Theory in Function Fields, Michael Rosen

239 Number Theory Volume I: Tools and Diophantine Equations, Henri Cohen

240 Number TheoryVolume II: Analytic and Modern Tools, Henri Cohen

## Algebraic Geometry 代数几何

44 Elementary Algebraic Geometry, Keith Kendig, 1977

52 Algebraic Geometry, Robin Hartshorne, 1977

76 Algebraic Geometry: An Introduction to Birational Geometry of Algebraic Varieties, Shigeru Iitaka, 1981

133 Algebraic Geometry: A First Course, Joe Harris

168 Combinatorial Convexity and Algebraic Geometry, Gunter Ewald

185 Using Algebraic Geometry, David A.Cox

187 Moduli of Curves, Joe Harris&Ian Morrison

## Algebra Lie Groups, Lie Algebras, and Representations

9 Introduction to Lie Algebras and Representation Theory, James E.Humphreys

98 Representations of Compact Lie Groups, Theodor Brocker&Tammo tom Dieck

102 Lie Groups, Lie Algebras, and Their Representations, V. S. Varadarajan

129 Representation Theory: A First Course, William Fulton&Joe Harris

222 Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Brian C. Hall

## Topology, Manifold 拓扑 流形

47 Geometric Topology in Dimensions 2 and 3, Edwin E. Moise

**评论**:少有的几何拓扑.

176 Riemannian Manifolds: an introduction to curvature, John M. Lee

202 Introduction to Topological Manifolds, John M. Lee

218 Introduction to Smooth Manifolds, John M. Lee

## Differential Geometry, Riemann Geometry

51 A Course in Differential Geometry, Wilhelm Klingenberg, 1983

**评论**: 非常适合用作本科教材, 是微分几何的入门书. 本书写的很紧凑, 采用现代观点讲古典几何, 清晰的展示出了初等微分几何题材与(高级的)Riemann Geometry 的联系.

**先修课程**: 数学分析, 高等代数, 解析几何

115 Differential Geometry: Manifolds, Curves, and Surfaces, M.Berger&B.Gostiaux, 1988

166 Differential Geometry: Cartan’s Generalizations of Klein’s Erlangen Program, R. W. Sharpe

191 Fundamentals of Differential Geometry, Serge Lang

224 Metric Structures in Differential Geometry, Gerard Walschap, 2004

## Ergodic Theory 遍历论

79 An Introduction to Ergodic Theory, Peter Walters, 2000

259 Ergodic Theory: with a view towards Number Theory, Manfred Einsiedler$Thomas Ward, 2010

## Graph Theory 图论

54 Combinatorics with Emphasis on the Theory of Graphs, Jack E. Graver&Mark E. Watkins, 1977

63 Graph Theory: An Introductory Course, Béla Bollobás,1979

注意, 不要把这书与 184 混淆. Béla Bollobás 是 173 的作者Reinhard Diestel 的博士导师.

173 Graph Theory, Reinhard Diestel, 2010

2012年把第四版稍作修订后仍然当第四版推出. 中文版刚刚由高等教育出版社出版.

184 Modern Graph Theory, Béla Bollobás,

207 Algebraic Graph Theory, Godsil&Royle, 2001