Oct 292013
 

真心佩服 Springer! 出版的好书无数: 无数的系列, 每个系列都是几十, 几百. 很多的资讯都很独家, 极具价值!

代数几何最”浅”的书, 大概是 Vladimir I. Arnold 的 “Real Algebraic Geometry“!  Springer 刚刚出来英译本. 六章加一个附录, 刚好 100 页! 本书是面向高中生的讲座. 不过, 不懂一点拓扑学, 微积分, 射影几何, 是不可能完全看懂的!

虽然代数几何有不同的切入路径, 但是想入门代数几何, 最起码要在掌握基本的抽象代数之后, 最好能有较强的射影几何(Projective Geometry)基础.

不建议 David Cox, John Little, Donal O’Shea, Ideals, Varieties, and Algorithms, 以及 Harris, Algebraic Geometry: A First Course. 从这样的书, 不会学到多少东西, 尽管这些书都很容易读, 要求的预备知识也很少.

根据很多人的看法, Bertrametti, Carletti, Gallarati, Bragadin, Lectures on Curves, Surfaces and Projective Varieties 很精彩.

1. Daniel Bump, Algebraic Geometry.

The prerequisites for the textbook are fairly minimal. Although it does discuss commutative algebra, there is a flavor of geometry pervasive throughout the entire text.

2. Holme, A Royal Road to Algebraic Geometry.

3. Shafarevich, Basic Algebraic Geometry, vol. 1, 2

英文译本第三版, 出来没多久.

4. Perrin Algebraic Geometry an Introduction.

5. Miles Reid, undergraduate algebraic geometry.

Oct 212013
 

一个月前, Springer 推出了 Ronald L. Graham, Jaroslav Nešetřil, Steve Butler 的两卷本大作 “The Mathematics of Paul Erdős” 的第二版.

早在 2002 年, Springer 曾出版过 Gabor Halasz , Laszlo Lovasz, Miklos Simonovits, Vera T. Sós 的 “Paul Erdős and His Mathematics“, 也是两卷.不过后来合并为一本 1400 页的书.

“N Is a Number: A Portrait of Paul Erdős” 是一部 57 分钟的电影. The Story of  Paul Erdős–a Wandering Mathematician Obsessed with Unsolved Problems.

影片的导演是 George Paul Csicsery, 1988-1991 制作,  capturing Erdős in various countries along with some of his numerous collaborators. It covers his unusual career, his personal life, and many of his recurring jokes and anecdotes, including that of Erdős numbers.

这电影获得两个奖项:  the Gold Apple Award (National Educational Film & Video Festival), and the Gold Plaque Award.

这里有一个 youku 的链接:

Oct 132013
 

走向 IMO 数学奥林匹克试题集锦(2013)已经由华东师范大学出版社推出.

本书收集了 2012 年至 2013 年度国内数学奥林匹克的试题, 并对试题作详细解答. 试题包括: 全国高中数学联赛, 全国中学生数学冬令营, 国家队集训资料, 国家队选拔考, 女子奥林匹克, 西部奥林匹克, 东南地区数学奥林匹克, 俄罗斯数学奥林匹克, 美国数学奥林匹克以及国际数学奥林匹克.

这书对 2013 IMO 的题 6, 给出了两种不一般的解法. 建议认真的看一看.

书名: 走向 IMO 数学奥林匹克试题集锦(2013)
ISBN: 9787567511842
出版社: 华东师范大学出版社
作者: 2013 年IMO中国国家集训队教练组
装帧:平装
开本:32开
页码: 184 页
出版日期: 2013.9
定价: 22 人民币元

 Posted by at 8:41 am
Sep 202013
 

Lie algebra(李代数) 是 Sophus Lie 为了研究后来以他的名字命名的 Lie Groups 的代数工具而引进的. Lie algebra 这个术语, 是 Hermann Weyl 在 1930 年代引入的.

The reason why you want to study Lie algebras can have a great impact on what books one would recommend.

下面的书单, 都是以李代数为主. 所以, 谈及太多李群, 表示论的书, 就不列在这里了.

首先是中文书.

1. 万哲先, 李代数, 第二版, 高等教育出版社, 2013

2. 孟道骥, 复半单李代数引论, 北京大学出版社, 1998

3. 苏育才, 卢才辉, 崔一敏, 有限维半单李代数简明教程, 科学出版社, 2008

这三本书都是从代数角度, 来讲李代数. 确切的说, 这几本书都突出了线性代数的方法, 都主要论述李代数理论中最基本, 最完善的部分–复半单李代数的经典理论.

这三本书的门槛都很低, 要求的先修知识不多. 这是优点, 容易上手; 也是缺点, 看不到李代数与别的科目的联系. 李群, 方程, 流形, 在这三本书统统没有踪迹.

万哲先的书, 从头到尾, 只谈复李代数, 甚至第一页给出的李代数的定义, 也是复数域上的李代数. 所以, 阅读万哲先, 留心复数域上李代数与一般域上李代数的区别为好.

实际上, 在中文书中找复李代数, 前两本就够了. 苏育才的书最详尽. 孟道骥的书, 也很详细, 写出了所有的证明. 万哲先的书, 出现最早, 一些简单的证明留给了读者. 如果读者想寻找被万哲先省略的细节, 翻一翻孟道骥. 如果还是没发现, 也许可以在苏育才查到.

万哲先的书, 没有习题; 孟道骥, 在每一节留有几个题目, 大多数都很简单, 少数题的结论值得记住.

4. 严志达, 实半单李代数, 南开大学出版社

5. 严志达, 半单纯李群李代数表示论, 上海科技出版社

6. 孟道骥,朱林生, 姜翠波, 完备李代数, 科学出版社

7. 万哲先, Kac-Moody代数导引

Kac–Moody algebra 通常是无限维的.

本书有英文版 Introduction to Kac-Moody Algebra.

然后, 是外文参考书, 包括有中文译本的书籍.

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

9. J.P. Serre, Complex semisimple Lie algebras

Serre 的作品, 笔法都很清晰. 这本书写的很紧凑. 本书是一本非常有价值的进阶著作, 不适合初学者. 读者须有一定的李代数, 结合代数基础, 才可能看懂.

10. N. Bourbaki, Lie groups and Lie algebras

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.

Sep 162013
 

Abstract algebra(抽象代数)是本科生的基础课. 这里列出一些不错的参考书, 也写出评价. 这里, 暂时不涉及更深入的书.

非常值得一读的一本历史著作是 Israel Kleiner,  A History of Abstract Algebra, 2007, Birkhauser

首先是中文书籍.

1. 熊全淹, 近世代数

这是朕读过的第一本这科目的书, 是武汉大学出版社, 1991年第三版. 这是这里谈到这本书的第一个原因. 这本书现在还可以买到, 武大出版社 2004 年重印, 369 页, 与朕手中的那本是一样的.

熊老师是那种真心热爱数学, 用生命来做教学的人. 熊全淹是傅种孙教授的弟子. 熊全淹把数学当终身职业, 与傅先生之关怀与诱导有莫大关系. 熊全淹在武汉大学读书的时候, 从肖君绛教授那里学习了代数. 肖君绛教授是在中国介绍 Van der Waerden 的经典著作 Moderne Algebra的第一人.

这本书不是中国大陆出版的第一本关于近世代数的书, 但应该是属于较早出现的书之一. 该书第一版是 1963 年由上海科技出版社推出的. 据张寿武的经历, 他 1981 年在中山大学读二年级, 给数学系的老师讲抽象代数. 可见, 当时还没有几个数学系开设这个课程.

此外, 该书的体系, 大致类似 Van der Waerden 的书. 熊全淹在前言交待的很清楚了.

这书在每章的最后, 列出长长的参考文献. 这对于喜爱钻研的读者, 是非常重要的.

本书的内容, 大体就是本科生应该掌握的. 遗憾的是, 有些非常重要的概念, 在本书完全没有踪迹. 比如说, 群在集合的作用. 重要的 Sylow 定理, 没有写出证明, 也没有介绍完整.

可能没有哪本书是完美的, 这书当然不能例外. 本书语言有点晦涩, 描述性的话语相当多. 这对于数学书, 不是好的现象.

2. 聂灵沼, 丁石孙, 代数学引论

这是一本影响较大的书, 被很多学校拿来做教科书. 北大数学系多年来抽象代数的教学都遵循了这书. 虽然, 近几年北大的老师又写出了另外的两本书, 并且使用了新书, 但聂和丁的书, 依然是最重要的参考资料之一.

有一种说法是, 本书的内容, 大体相当于 N. Jacobson 的三卷 Lectures in abstract algebra.

一般来说, 本科生只在课堂学到这里面内容的前四章, 加上第七, 八章的部分. 本书的一个特点是, 习题很多. 不少题目都是论文的结论, 因此很有难度. 如果你想搞定所有的习题, 要花一番功夫才行.

3. 丘维声, 抽象代数基础, 高等教育出版社

丘维声的书, 不论是他最擅长的线性代数, 还是解析几何教材, 或者表示论, 都是很一般的, 切不中要害, 观点一般. 不过, 拿来参考一下, 还是可以的. 这本抽象代数基础, 还行. 需要指出的是, 这书的自由群那一节的定理的证明是有错误的.

据说, 丘维声当年考大学的时候, 全国统一阅卷, 他是状元. 他在北大被多次评为十佳教师. 他在黑板的板书, 工工整整. 可是, 他教了十几次线性代数, 写了好几本线性代数的书, 处理行列式的定义, 依然乱七八糟.

4. 赵春来, 徐明曜, 抽象代数, I, II. 北京大学出版社

如果要在中文书里选出一本来入门抽象代数, 那么, 本书就是朕想推荐的.

本书分为 I,II 两册, II 是研究生教材, 而 I 适合本科生. II 的出版时间, 比 I 早一年半. 两本书的作者都是徐明曜和赵春来, 只是署名顺序不同.

5.  冯克勤, 李尚志, 章璞, 近世代数引论, 第三版, 中国科技大学出版社

中规中矩的一本教材. 不论是内容, 还是处理, 都没有特点. 作者还有一本配套的习题解答: 近世代数三百题, 高等教育出版社.

6. 姚慕生, 抽象代数学, 复旦大学出版社, 第二版

这本书反响不错.

7. 孟道骥 , 陈良云, 白瑞蒲, 抽象代数1:代数学基础, 科学出版社

8. 吴品三, 近世代数, 人民教育出版社

再来, 是 English book.

9. David S. Dummit and Richard M. Foote, Abstract Algebra, 3rd Edition

本书被广泛使用, 受到很高的评价. 这可能是最详尽的入门教科书了.

本书习题不算多, 难度适当. 读者完全可以自己独立作答. 实在遇到困难, 网上很容易找到全部的答案.

10. Joseph Gallian, Contemporary Abstract Algebra, 8th

这本书也很详细, 作者还写了一本习题解答.

本书最新是第八版. 不过, 其实即便第五版, 与第八版相比, 只在习题设置有些许差别.

11. Michael Artin, Algebra, second edition

 Posted by at 2:12 pm
Sep 122013
 

A new book A History in Sum: 150 Years of Mathematics at Harvard (1825-1975) has just been published by Harvard.

In the twentieth century, American mathematicians began to make critical advances in a field previously dominated by Europeans. Harvard’s mathematics department was at the center of these developments.A History in Sum is an inviting account of the pioneers who trailblazed a distinctly American tradition of mathematics–in algebraic geometry and topology, complex analysis, number theory, and a host of esoteric subdisciplines that have rarely been written about outside of journal articles or advanced textbooks. The heady mathematical concepts that emerged, and the men and women who shaped them, are described here in lively, accessible prose.

The story begins in 1825, when a precocious sixteen-year-old freshman, Benjamin Peirce, arrived at the College. He would become the first American to produce original mathematics–an ambition frowned upon in an era when professors largely limited themselves to teaching. Peirce’s successors–William Fogg Osgood and Maxime Bôcher–undertook the task of transforming the math department into a world-class research center, attracting to the faculty such luminaries as George David Birkhoff. Birkhoff produced a dazzling body of work, while training a generation of innovators–students like Marston Morse and Hassler Whitney, who forged novel pathways in topology and other areas. Influential figures from around the world soon flocked to Harvard, some overcoming great challenges to pursue their elected calling.

A History in Sum elucidates the contributions of these extraordinary minds and makes clear why the history of the Harvard mathematics department is an essential part of the history of mathematics in America and beyond.

Review

This book tells the tale of how mathematics developed at Harvard–and by extension in the United States–since early days. It is filled with fascinating stories about some of the legendary names of modern mathematics. Both fans of mathematics and readers curious about the history of Harvard will enjoy it. (Edward Witten, Professor Of Physics, Institute For Advanced Study)

A History in Sum is a beautiful tribute to a beautiful subject, one that illuminates mathematics through the lens of some of its most remarkable practitioners. The authors’ love of mathematics shines through every chapter, as they use accessible and spirited language to describe a wealth of heady insights and the all-too-human stories of the minds that discovered them. There is perhaps no better book for immersion into the curious and compelling history of mathematical thought. (Brian Greene, Professor Of Mathematics & Physics, Columbia University)

The book is written in a leisurely style, the scope is remarkably broad, and the topics covered are explained astonishingly well. Once I started the book, I simply couldn’t put it down and I was ecstatic to easily understand important mathematics far from my own research interests. (Joel Smoller, Professor Of Mathematics, University Of Michigan)

A History in Sum contains a wealth of good stories, stories that go to the heart of the development of mathematics in this country. The authors succeed in humanizing and enlivening what might otherwise be a dry treatment of the subject. (Ron Irving, Professor Of Mathematics, University Of Washington)

  • Author: Steve Nadis and Shing-Tung Yau
  • Hardcover: 280 pages
  • Publisher: Harvard University Press (October 7, 2013)
  • Language: English
  • ISBN-10: 067472500X
  • ISBN-13: 978-0674725003
  • Price: $39.95
  • Product Dimensions: 6 x 9 inches
Aug 032013
 

Lizhen Ji’s new book “Great Mathematics Books of the Twentieth Century: a Personal Journey” has just been published. 季理真的新书”二十世纪伟大的数学书:个人之旅” 刚刚由高等教育出版社推出.

Great Mathematics Books of the Twentieth Century: a Personal Journey

Great Mathematics Books of the Twentieth Century: a Personal Journey

海量的数学书,哪些值得我们认真读, 哪些读后让我们对数学有更好的认识, 这些对门外汉, 学生, 年轻的学者和专家都是非常需要解决的问题. 季理真教授的新作”二十世纪伟大的数学书――个人之旅”)great mathematics books of the twentieth century: a personal journey)在这个问题上为我们带来了极大的便利. 本书比较全面收列了二十世纪以来最有影响的数学书并恰当地加以简评和引述其他评论. 本书收列的书目范围之广, 数量之大令人吃惊, 这需要作者广阔的视野, 艰辛的工作, 并花大量的时间请教很多不同方向的专家. 本书的作者完成了一项很有意义的工作, 本书定会让喜爱数学的读者受益.

另外本书还包含了作者从University of Michigan图书馆特藏馆图书中, 亲自拍摄 \(150\) 多幅著名的数学家及其著作的珍贵的照片. 其中有Newton伟大书的第一版, Euclid几何原本 \(15\) 世纪版本, 第一彩色版等.作者对这些数学家的成就以及一些奇闻趣事给于简单的介绍, 这也使得本书内容更加丰富.

书是非常的厚实, \(667\) 页, 装帧的非常漂亮, 价格不高, 绝对值得向大家推荐. 开卷有益是这本书最好的注脚. 本书应当成为每一个数学工作者书架上的必备.

Preface

In this book, we list and introduce some interesting, important or useful mathematics books.  Most selected books were published during the twentieth century. For the convenience of the reader, we have arranged books according to topics. Besides some introductions and comments, we also quote from informative reviews of these books from sources including MathSciNet, Zentralblatt Math and the Bulletin of the American Mathematica Society. A common way for people to pick out books to read is to follow  recommendations of either book reviews or experts.  The list of books is probably the most interesting part of this book. Once the titles or authors’ names are known, it is  relatively easy to find valuable information and reviews about the book from many different sources  (but it might take some efforts to find good books on subjects outside one’s expertise.) In spite of this, we hope that additional information provided here about these books might be helpful and convenient. We  hope that such a list of books in contemporary mathematics might be helpful to other people and students who are interested in finding out what kind of mathematics books  exist and have been read or enjoyed by others, or who are simply interested in mathematics books.

This book could not exist without the kind help from many experts in various fields. I would like to thank the following people for their opinions, suggestions, comments, criticism, corrections, interests and encouragement: Maxim Arap, Tim Austin, Jinho Baik, Oliver Baues, Vitaly Bergelson, Jean-Michel Bismut, Andreas Blass, Patrick Boland,  Martin Bridson, Dan Burns, Peter Buser, Richard Canary, John Coates, Joseph Conlon, S.G. Dani, Igor Dolgachev, Peter Duren, Weinan E, Alexandre Eremenko, Tom Farrell, Sergey Fomin, Jacques Franchi, Kenji Fukaya, Bill Fulton, Francis Fung, Sergei Gelfand, David Harbater, William Harvey, Elton Hsu, Mattias Jonsson, Manfred Karbe, Linda Keen,  Hans Koelsch, Kai Kohler, Igor Kriz, Jeff Lagarias, Robert Lazarsfeld, Enrico Leuzinger,  Tien-Yien Li, Wenbo Li, Eduard Looijenga,  Hugh Montgomery, Kumar Murty, Louis Nirenberg, Peter Olver,  Athanase Papadopoulos,  Hugo Parlier, Katrik Prasana, Stratos Prassidis,  Mikael Ragstedt, Andrei Rapinchuk, Frank Raymond, John Schotland, Leonard Scott,  Mei-Chi Shaw,  Reyer Sjamaar, Peter Smereka, Ralf Spatzier,  Christopher Stark, Alejandro Uribe, Roman Vershynin, Divakar Viswanath, Charles Weibel, Trevor Wooley, Scott Wolpert, Ping Zhang, Weiping Zhang, Michael Zieve, and Steve Zucker.

I would also  like to thank Dr. Graeme Fairweather,  the director of MathSciNet, for his permission to quote from reviews  in MathSciNet  and Zentrablatt Math for providing me  the full access to its data base during  the preparation of this book.

Especially I would  like to thank my wife, Lan Wang, for suggesting the key word “journey” in the subtitle and for her encouragement, my oldest daughter Lena for drawing the picture for this book which is based on many famous books related to mathematics, and my youngest daughter Karen for proof-reading the preface and introduction.

Finally I would like to thank Liping Wang of the Higher Education Press for her interests in this unusual project,  and her time and efforts in carefully editing this book. She made suggestions for many new features and improvements to this book. For example, the idea of inclusion of pictures of  libraries and other buildings of University of Michigan came from her, and the idea of inclusion of many pictures of old and rare mathematics books arose from discussion with her. I would also like to thank Shannian Lu of the Higher Education Press for preparing the index of books which have been translated into Chinese or reprinted in China.

One reason for including these pictures of mathematics books is that though ancient and classical mathematics books are not read by many people now, they had greatly influenced the development of mathematics and are still interesting to many mathematicians. According to one Chinese proverb, “One picture is worth ten thousand words”. We hope that these pictures and the accompanying comments might be both interesting and informative to the reader. For example, by putting them together, we can see from these pictures of old books how book printing has changed over the past centuries, and how the authors of these books have left their permanent marks on the history of mathematics. At the beginning of each chapter and section, we have tried to select pictures which fit the topics under discussion, but it is not clear whether we have succeeded due to many obvious constraints and the lack of knowledge of the author.

All pictures used in this book come from the Special Collections Library  and the other libraries of the University of Michigan. Except for the picture of a collection of books on the cover and for the picture of the Galileo manuscript at the beginning of Chapter 10, almost all other pictures were taken by the author. I would like to thank the staff members of the  Special Collections Library at the University of Michigan, in particular the curator Peggy Daub, for providing these two special pictures and for their help  which made it possible for me to view and take pictures of over one hundred rare mathematics books at the special collections library. It was the first time I came in close contact with these great mathematics books by old masters, and flipping through these books was both a humbling and inspiring experience for me.

书名: Great Mathematics Books of the Twentieth Century: a Personal Journey(二十世纪伟大的数学书:个人之旅)
作者: Lizhen Ji(季理真)
出版社: 高等教育出版社; 第1版 (2013年6月1日)
精装: 667页
语种: 英语
开本: 16
定价: 89 人民币元
ISBN: 9787040375428
条形码: 9787040375428

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本文内容, 大部来自季理真先生的博客. 我自己的话, 只有几句.

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