Dec 182013
 

Springer 刚刚出版了 Vladimir Arnold 的 Collected Works II:

Vladimir I. Arnold-Collected Works II

Vladimir I. Arnold-Collected Works II

本书收集了Vladimir Arnold 的 45 篇文章: 33 篇是在1965-1972 年间写成, 8 篇是 80-89 年间发表, 73-79 有 2 篇, 91 与 92 年各有 1 篇.

Vladimir Arnold(June 12, 1937–June 3, 2010)

2. Vladimir I. Arnold – Collected Works: Representations of Functions, Celestial Mechanics, and KAM Theory 1957-1965, Springer, 2009

3. Real Algebraic Geometry, Springer, 2013 edition

4. Yesterday and Long Ago, Sringer, 2007

5. Huygens and Barrow, Newton and Hooke: Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals, Birkhäuser, 1990

6. Arnold’s Problems, Springer, 2nd edition, 2004

甚至有的书, 作者不是 Arnold, 但内容是他的, 学生把上课笔记出版. Abel’s Theorem in Problems and Solutions: Based on the lectures of Professor V.I. Arnold,  V.B. Alekseev, Springer, 2004

Dec 042013
 

Spriger 刚刚出版了三卷本的 “Geometric Trilogy: Axiomatic, Algebraic and Differential Approaches to Geometry“, 作者 Francis Borceux.

Geometric Trilogy

Geometric Trilogy

这三本书的合集厚达 \(1400\) 页. The Trilogy intends to introduce the reader to the multiple complementary aspects of geometry, paying attention to the historical birth and growth of the ideas and results, and concluding with a contemporary presentation of the various topics considered.

本书的特点是:

  • Focuses on historical aspects;
  • Supports contemporary approaches of the three aspects of axiomatic geometry: Euclidean, non-Euclidean and projective;
  • Includes full solutions to all famous historical problems of classical geometry and hundreds of figures.

An Axiomatic Approach to Geometry

An Algebraic Approach to Geometry

Unified treatment of the various algebraic approaches of geometric spaces; Provides a full treatment, perfectly accessible at a bachelor level, of all algebraic ingredients necessary to develop all the major aspects of  the theory of algebraic curves.

A Differential Approach to Geometry

Pays particular attention to historical development and preliminary approaches that support the contemporary geometrical notions; Links classical surface theory in the three dimensional real space to modern Riemannian geometry.

Nov 262013
 

次数 \(\geqslant5\) 的多项式方程根式不可解的证明, 不是只有 Galois theory 这一个办法. 1963 年, V. I. Arnold(June 12, 1937-June 3, 2010) 给出了一个拓扑证明. 这种新的思想, 可以用来解决另外一些经典问题. 随后, Topological Galois theory 就发展起来了. 该理论的建立者是 Askold G. Khovanskii.

Askold G. Khovanskii 于 2008 年在莫斯科出版了一本 Topological Galois theory. 该书的英译本会被 Springer 在 2015 年上半年出版.

This book provides a detailed and largely self-contained description of various classical and new results on solvability and unsolvability of equations in explicit form. In particular, a complete exposition of topological Galois theory is given. A unique feature of this book is that recent results are presented in the same elementary manner as classical Galois theory.

Contents

Introduction
Chapter 1: Liouville’s Theory
Chapter 2: Galois Theory
Chapter 3: Picard–Vessiot Theory
Chapter 4: Coverings and Galois Theory
Chapter 5: One-Dimensional Topological Galois Theory
Chapter 6: Solvability of Fuchsian Equations
Chapter 7: Multidimensional Topological Galois Theory
Appendix 1: Ruler and Compass Constructions
Appendix 2: Chebyshev Polynomials and Their Inverses
Appendix 3: Signatures of Branched Coverings
Appendix 4: On an Algebraic Version of Hilbert’s 13th Problem
Index

Olivia Caramello 在 arXiv 有一篇 82 页的 Topological Galois theory, 介绍了一种 topos-theoretic framework for building Galois-type theories in a variety of different mathematical contexts.

现在又兴起了所谓的 Semi-topological Galois theory.

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.

Jun 202013
 
Basic Algebraic Geometry 1

Basic Algebraic Geometry 1

The third Edition of  “Basic Algebraic Geometry” has just been published.

Shafarevich’s Basic Algebraic Geometry has been a classic and universally used introduction to the subject since its first appearance over 40 years ago. As the translator writes in a prefatory note, “For all [advanced undergraduate and beginning graduate] students, and for the many specialists in other branches of math who need a liberal education in algebraic geometry, Shafarevich’s book is a must.”

Shafarevich’s book is an attractive and accessible introduction to algebraic geometry, suitable for beginning students and nonspecialists, and the new edition is set to remain a popular introduction to the field.

The third edition, in addition to some minor corrections, now offers a new treatment of the Riemann–Roch theorem for curves, including a proof from first principles.

Basic Algebraic Geometry 2

Basic Algebraic Geometry 2

The second volume is in two parts: Book II is a gentle cultural introduction to scheme theory, with the first aim of putting abstract algebraic varieties on a firm foundation; a second aim is to introduce Hilbert schemes and moduli spaces, that serve as parameter spaces for other geometric constructions. Book III discusses complex manifolds and their relation with algebraic varieties, Kähler geometry and Hodge theory. The final section raises an important problem in uniformising higher dimensional varieties that has been widely studied as the “Shafarevich conjecture”.

The style of Basic Algebraic Geometry 2 and its minimal prerequisites make it to a large extent independent of Basic Algebraic Geometry 1, and accessible to beginning graduate students in mathematics and in theoretical physics.

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
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