zyymat – Page 33 – Zyymat: Mathematics

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

Jun 202013

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

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.

Quanyu Huang’s book “Quality-oriented Education in United States”

book reviews, education No Responses »

Jun 082013

这几天在几大购物网站淘书, 突然想起了黄全愈的”素质教育在美国”. 这本书是少有的能让我心朝澎湃的书. 为了了解下这书的近况, 特意搜索了一下. 原来, 三年前,由中国人民大学出版社出了一个这本书的新版:

Quality-oriented Education in United States

于是, 订购了一本.

本书最早是在 1999 年由广东教育出版社推出. 2000年4月, 中国青年报连续使用三期的冰点, 选登了部分内容. 应该说, 中国青年报的影响力对这书的推广起了很大的作用, 至少我本人是从这里知道这书的.

The Shaw Prize in Mathematical Sciences 2013 is awarded to David L Donoho

Mathematician, Prize No Responses »

May 292013

The Shaw Prize in Mathematical Sciences 2013 is awarded to David L Donoho for his profound contributions to modern mathematical statistics and in particular the development of optimal algorithms for statistical estimation in the presence of noise and of efficient techniques for sparse representation and recovery in large data-sets.

The Shaw Prize in Mathematical Sciences for 2013 is awarded to David L Donoho, Anne T and Robert M Bass Professor of the Humanities and Sciences, and Professor of Statistics at Stanford University, USA for his profound contributions to modern mathematical statistics and in particular the development of optimal algorithms for statistical estimation in the presence of noise and of efficient techniques for sparse representation and recovery in large data-sets.

The dramatic developments in technology in the last half century present fundamental new challenges in theoretical and applied mathematical statistics. David Donoho has played a major role in developing new mathematical and statistical tools to deal with such problems ranging from large data-sets in high dimensions to contamination with noise. His work provides fast, efficient and often optimal algorithms which are founded on rigorous mathematical analysis.

Key themes introduced in his works, and which today are standard features of the theory, include the exploitation of sparseness of representation of complex objects, related adaptive nonlinear thresholding techniques and the deep relation between sparseness and certain penalty functions that are being minimized (specifically $L^1$ norms).

Many of these emerge from his development of algorithms for statistical estimators in the presence of noise. These are remarkable in that they overcome the difficulties associated with noise, with very little loss of efficiency or reliability. Along the way, he demonstrated the power of the mathematical theory of wavelets in dealing with such problems in statistics. The Donoho–Johnstone soft-thresholding algorithm has been widely used in statistical and signal processing applications.

During the last 15 years Donoho has developed a theory of sparse and multi-scale representations of signals and data-sets using nonlinear $L^1$ optimization methods. These combine very well with techniques of unstructured and redundant dictionaries of functions and provide a fundamental approach to lower the dimensionality of complex problems. Along with Candes and Tao, he made fundamental contributions to the development of “compressed sensing”. In terms of sparseness and recovery, this method which “compresses while sensing the data”, using dramatically fewer data points while retaining the ability to recover the correct signal, yields strikingly efficient and even optimal algorithms for compressing and decompressing complex signals (e.g. images). This area remains a very active area of research especially in view of its wide applications.

David L Donoho was born in 1957 in Los Angeles, USA and is currently Anne T and Robert M Bass Professor of the Humanities and Sciences, and Professor of Statistics at Stanford University, USA. He graduated from Princeton University in 1978 and received his PhD from Harvard University in 1983. From 1984 to 1990, he was on the faculty of the University of California, Berkeley before moving to Stanford. He is a fellow of the American Academy of Arts and Sciences, a SIAM Fellow, a foreign associate of the French Academy of Sciences, and a member of the US National Academy of Sciences.

Harald Helfgott has proved Goldbach’s weak conjecture

Advance, math.NT No Responses »

May 262013

Busy day in analytic number theory

On May 13, 2013, Harald Andres Helfgott uploaded to the arXiv his paper “Major arcs for Goldbach’s theorem” claimed that he has proved the ternary Goldbach conjecture, or odd Goldbach conjecture, asserts that every odd integer $n>5$ is the sum of three primes.

这论文仅仅证明了每个 $>10^{30}$ 的奇数可以表示为三个质数之和. 至于 $<10^{30}$ 的奇数, 已经通过计算机进行验证. 计算机实际上, 已经计算过, 对于 $<8.875\cdot10^{30}$ 的奇数, Goldbach’s conjecture 都是对的. 这样, 奇数 Goldbach’s conjecture 彻底终结.

这文章采用是基于圆法 (Hardy–Littlewood circle method), 大筛法(the large sieve) and exponential sums 的一种途径.

Goldbach’s conjecture 已经有 $271$ 年的历史了.

Yitang Zhang has proved a weak version of the twin prime conjecture

Advance, math.NT, Mathematician No Responses »

May 142013

On 14 May 2013, Mathematician Yitang Zhang claimed that he has proved there are infinitely many prime gaps shorter than 70 million, which was a weak version of the twin prime conjecture.

数学界对张的证明, 表示乐观, 应该没有错误.

[Update, May 21, 2013: 张的论文, 全文 $56$ 页已经可以在 Annals of Mathematics 的网站看到: Bounded gaps between primes(subscription required). 这文章的主要结果是证明了

\[\varliminf_{n\rightarrow\infty}(p_{n+1}-p_n)\lt7\times10^7,\]

这里 $p_n$ 表示第 $n$ 个质数.]

综合起来, 这故事有几个看点:

1. 成就太过突出
孪生质数猜想是数论中最古老的难题, 一直没有啥进展.

2. 用经典方法逆袭, 用弹弓打死了狗熊.
无数数学家企图使用弹弓打狗熊, 从没成功. 都已经放弃希望了, 突然有人宣布搞定. 不是崭新的思路, 这是很多数学家引以为憾的地方, 因此引来无数的酸葡萄, 大家都希望使用核武器来进攻, 甚至发明更猛的新式武器.

3.张益唐一直坎坷, 一举成名天下知.
很精彩的励志故事. 很可能将来的数论教科书在讲述他惊世骇俗的定理时, 也会用他送外卖糊口来思考数学的情事来鼓励后进.

石破天惊

4 月 17 日, 数学界最富盛名的数学杂志 Annals of Mathematics 的收件箱出现一篇论文. 这论文居然宣称在一个最古老的数学难题孪生质数猜想上取得重大突破. 专家们对作者张益唐感到陌生. 最要命的是, 张其实只是一所普通大学的讲师, 已经 50 好几.

著名数学杂志经常收到一些出自无名作者的号称解决了大问题的论文, 但这篇署名张益唐的数论论文不同. 这是一部严肃的作品: 论述清澈, 完全使用这学科当前的术语进行表述. 于是, 编辑们决定尽快审稿.

过了三个星期, 是的, 仅仅三周, 张就收到了对论文的评价: 一流!

一个没啥名气的研究人员取得重大进展的新闻, 迅速在数学界传开. 丘成桐邀请张益唐去 Harvard 做一个报告. 报告会于 5 月 13 日进行. 坐在教室的观众有 50 人, 没人之前听说过张的大名, 包括丘.(其实, 在1980年代, 张还在求学的时候, 与丘是打过交道的.) 于是, 张益唐的工作的一些细节为外界所知晓: 张没有使用崭新的办法, 而是通过改进已有的途径. 最顶尖的数论专家已经尝试过这种途径, 但张益唐在别人失败的地方取得了成功.

张益唐的定理令人惊讶, 是一个巨大的突破.

筛法

张益唐的成果可以追溯到八年前的一篇数论专家引用称为GPY的论文-以其三位作者 Goldston, János Pintz, Cem Yıldırım 的名字命名. 该论文已经非常非常接近, 但很遗憾的没能证明存在无限多对质数, 其差有限. 具体说来, GPY发展了一种称为筛法的方法. 研究人员把这种筛法与一个函数结合起来. 这个函数的效能是基于一个衡量质数多快才能呈现某种规律的称为 level of distribution 参数. level of distribution 至少是$\frac12$, 这就是得到 GPY 的结果的那个值. GPY的筛法要想得出存在无限多对质数, 其间隙有限, 必须提升 level of distribution, 使其 $>\frac12$, 哪怕只比 $\frac12$ 大那么一点点也足够了.

1980 年代后期, IAS 的 Fields Medal 得主 Enrico Bombieri, Toronto大学的 John Friedlander, 和 Rutgers大学的 Henryk Iwaniec 设法修改level of distribution 的定义, 使得这个修订后的参数达到 $\frac47$. GPY 的文章在 2005年出笼以后, 研究人员一窝蜂想把这个修改后的 level of distribution 与 GPY的筛法组合起来, 但没有什么成效.

张益唐的工作

与此同时, 张益唐独自游走在 GPY 与质数的有界间隙之间, 想要完成 GPY 未尽的事业.

张读过 GPY 这论文. 论文里有一句话是如此振奋人心. 这句话指出, 质数间隙有界已经近在咫尺! 经过三年孤独的奋斗, 张没有任何进展.

想进行一点休息, 张益唐去年夏天访问了一个在 Colorado 的朋友. 就在这期间, 7 月 3 日, 在离开朋友家去一个音乐会之前, 在后院休息的半小时里, 张益唐突然想出了答案. 张的想法, 不是直接使用Selberg 的筛法, 而是做一些修正: 不是使用所有的数来过滤, 只考虑那些没有大的质数因子的数.

Goldston认为, 张的筛法, 没有那么强大, 效果也差一点, 但在 GPY 会有一点奇效. 这样一来, 张把 level of distribution 提高到了 $\frac12+\frac1{584}$, 这足以使用 Bombieri, Friedlander, 和 Iwaniec 的方法. “新筛法得出了张的惊天动地的结果, 但不太可能证明孪生质数猜想. 即便假定 level of distribution 最好的结果成立, 从 GPY 的方法只能得出有无穷多对质数, 其差不超过 $16$.” Goldston 说.

张也使用了 Enrico Bombieri, John Friedlander, 和 Henryk Iwaniec 所发展的技巧, 比如有限域上的特征和, 自守形式的理论, 然后独创性的把所有东西结合在一起. 他也优雅的借用其他领域的工具, 比如间接用到有限域上代数簇的 Riemann hypothesis.

张益唐花费了几个月才完成所有的细节. 最后的论文表述清晰. 这是解析数论的巅峰之作.

[Update, June 8, 2013: 去年7月3日, 张益唐前往在科罗拉多州立大学音乐系任教的好友, 音乐指挥家齐雅格家中作客. 当时他与齐雅格正准备离家去看排练, 临走前20分钟, 张益唐想到齐家院子后看不请自来的梅花鹿, 顺便抽根烟.

Yitang Zhang

齐雅格回忆, 张益唐破解孪生素数的关键就是在那20分钟里,”有如神明启示一般地”想出来. 他那次到他家作客, 纯粹为了放松, “身上没带一本书,没有任何资料,也不上电脑.这似乎是个奇迹”.

张益唐则表示, 这是长期研究的积累, 一旦有机遇, 就成功地突破难题, 找到別人没有想到的特別突破口, “这也是运气”.[9] ]

张益唐其人

张益唐, 北大 78 级, 本科学习的是计算数学. 1982年毕业之后, 拜潘承彪教授为师继续在北大学习三年, 获得硕士学位. 然后, 他赴美, 到 Purdue University攻读博士学位, 导师是莫宗坚教授. 莫宗坚的大名很多人应该是知道的, 这要归功于他的两本”代数学”.

学术界也讲究”血统”! 张益唐可算得上是华罗庚的”徒孙”. 张的硕士导师潘承彪尽管不是华罗庚正式的学生, 但显然受到华巨大的影响. 潘承彪曾追随闵嗣鹤教授学习广义解析函数, 但他的数论知识应该主要来自他的哥哥潘承洞. 潘承洞是闵嗣鹤的研究生, 但也被认为是华罗庚的学生, 尤其在对Goldbach猜想的研究上. 更不待言闵嗣鹤教授本人也被华深深影响. [1]

张益唐的博士题目是 Jacobian conjecture. 其实这 Jacobian conjecture 现在仍在考验人类的智慧. 张在博士毕业之前, 认为自己解决了 Jacobian conjecture. 但是, 他的证明使用的其导师莫宗坚的一个引理后来被发现是错的, 于是张几年的心血付之东流. 张益唐这博士论文没发表, 而且和导师莫宗坚的关系不好, 于是博士毕业即为失业, 连博士后都没找到.[2]

接下来的事情, 很多新闻都有报道: 张一边做零工糊口, 一边思考数学! 最后, 在他的两个师弟, 北大80级的同学唐朴祁, 尤其是葛力明的帮助下, 在一个偏僻的地方, University of New Hampshire, 找到一个讲师职位. 这样张益唐才算有了稳定的工作, 能在更好的条件下思考质数分布的规律.

张益唐在大学教书, 一周上课六个小时, 空余时间不少. 虽然学校並不重视研究, 但自己始终没有放弃思考和钻研自己热爱的数学数论问题. 在这三, 四年的过程中, 遇到许多令人沮丧的挫折.

张益唐不重视金钱和名利, 几乎任何时间都在思考数学, 甚至休息的时候. 这也部分回答了为什么他能成功. 用他自己的话来说,”The idea is based on an accumulation of my thinking for several years,I had tried various methods. To answer why others could not get it and I could, I may say that I had been working harder and never gave up.” [10]

闲暇之际, 张喜爱阅读莎士比亚(Shakespeare), 哈姆雷特, 罗密欧与朱丽叶(Hamlet and Romeo and Juliet) 是他的最爱.[8]

张益唐的学生对他评价很好, 字写的很漂亮.

他太太在 California 工作, 两人没有孩子.[5]

References

季理真, 素数不再孤单: 孪生素数和一个执着的数学家张益唐, May 20, 2013.
汤涛, 张益唐和北大数学 78 级, May 19, 2013.
Erica Klarreich, Unheralded Mathematician Bridges the Prime Gap, simons foundation, May 19, 2013.
Kenneth Chang, Solving a Riddle of Primes, The New Yorks Times, May 20, 2013.
Carolyn Y. Johnson, Globe Staff, Obscure University of New Hampshire math professor takes major step toward elusive proof, May 23,2013.
Dan Goldston, Zhang’s Theorem on Bounded Gaps Between Primes.
Henryk Iwaniec, a email to Shing-Tung Yau: Subject: Re: Yitang zhang, May 24,2013.
Liam O’brien, That figures: Professor who had to work at Subway dazzles world of maths after solving centuries-old prime number riddle, May 21, 2013
唐嘉丽, 张益唐破解千古数学难题, June 6, 2013.
Paul Feely, UNH professor solves ancient mathematics riddle, June 2, 2013.

A classification and review of GTM

book reviews No Responses »

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

Input “Paul Erdős” with $\rm\TeX$

Mathematician, software No Responses »

Apr 182013

想知道复旦学子对最近发生在自己学校的某件事怎么议论, 今天特意进复旦大学日月光华 BBS 上逛逛. 意外的, 在数学区, 看到了网友 spline 在上个月的 26 日, Paul Erdős 的诞辰100年, 发布的”纪念 Paul Erdős 诞辰 100 周年”的帖子. 这个帖子的最后, 专门指出, 用 $\rm\TeX$ 输出 Erdős, 其正确的语法结构是这么:

Erd\H{o}s

现在特意记下来.

大家都知道的, Paul Erdős (1913.03.26-1996.09.20)是著名数学家, Wolf 奖得主, 以高产和合作的数学家多著称. 我还不清楚自己的 Erdős number 是多少, 你呢?

查了下 $\rm\TeX$ 的书籍, 原来

\H{ }

这个命令产生重音号. 遗憾的是, 本博客暂时还不支持这个命令. 准确点说, 是 MathJax 暂不支持.

注: 至于 “某件事”, 显然指的就是那件震惊全国, 同时也拷问国人心灵, 肯定也上了很多很多国家的报纸和网站的大案: 复旦医学院在读 2010 级研究生黄洋, 疑被同寝室某同学在寝室饮水机投毒,于 2013 年 4 月 1 日因身体不适入院, 经抢救无效, 逝世在上海中山医院.

The Princeton Companion to Mathematics

book reviews No Responses »

Apr 082013

“The Princeton Companion to Mathematics” 由世界图书出版公司影印出版. 中文书名定为 “数学指南”, 甚为滑稽.