Sep 182013

Steve pointed out the thing that makes EGA difficult to read is not that it is dense, but rather that it is gigantic.

Robin Hartshorne’s book algebraic geometry is an edulcorated version of Grothendieck and Dieudonné’s EGA, which changed algebraic geometry forever.

EGA was so notoriously difficult that essentially nobody outside of Grothendieck’s first circle(roughly those who attended his seminars) could (or wanted to) understand it, not even luminaries like Weil or Néron .

Things began to change with the appearance of Mumford’s mimeographed notes in the 1960’s, the celebrated Red Book, which allowed the man in the street(well, at least the streets near Harvard) to be introduced to scheme theory.

Then, in 1977, Hartshorne’s revolutionary textbook  algebraic geometry was published. With it one could really study scheme theory systematically, in a splendid textbook, chock-full of pictures, motivation, exercises and technical tools like sheaves and their cohomology.

However the book remains quite difficult and is not suitable for a first contact with algebraic geometry: its Chapter I is a sort of reminder of the classical vision but you should first acquaint yourself with that material in another book.

GTM 52 的精华是第 2, 3章, 分别介绍 Scheme 和它上面的 Cohomollogy theory.

GTM 52 有习题 464 道. 这本书的习题, 非常重要! 当然, 习题也不一定必须一个一个全部做完.

May 082013

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

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