Annals, Volume 179, Issue 3 – May 2014, has just been published online. Yitang Zhang’s paper “Bounded gaps between primes” is the seventh paper, Pages 1121-1174.
虽然张的论文去年 5 月就已经可以下载, 但现在才是正式出版.
多数文章, 都是依靠 Annals 得到荣耀, 但张益唐的论文会给 Annals 带来光荣.
Yitang Zhang wins the 2014 Rolf Schock Prize in Mathematics, for his spectacular breakthrough concerning the possibility of an infinite number of twin primes. The Royal Swedish Academy of Sciences decided the laureate.
颁奖典礼将于 2014 年 10 月 22 日在瑞典斯德哥尔摩举行. 2014年肖克奖奖金合计 240 万瑞典克朗, 单个奖项的奖金为60万瑞典克朗(约合9万美元).
Acta Arithmetica(ISSN: 0065-1036(print) 1730-6264(online)) is a scientific journal of mathematics publishing papers on number theory. It was established in 1935 by Salomon Lubelski and Arnold Walfisz. The journal is published by the Institute of Mathematics of the Polish Academy of Sciences.
1935 年, Salomon Lubelski 和 Arnold Walfisz 创立了Acta Arithmetica.
Acta Arithmetica 是一个数学杂志, 发表数论方面的原创研究论文, 由 Polish(波兰)科学院的数学研究所出版. 从 1995 年开始, Acta Arithmetica 每年出版 5 卷(2012 年有 6 卷; 1996-2000 年间, 每年 4.5 卷), 刊登 80-100 篇论文.
目前, Acta Arithmetica 第 1-95 卷是 Open Access(开放存取), 而第 96 卷以及第 96 卷之后, 读者需要订阅才可以看到全文. 因为第 96 卷的后 2 期在 2001 年刊发, 因此, 任何人都可以及时, 免费, 不受任何限制地通过网络获取 2000 年以及之前的所有 Acta Arithmetica, 除了 2000 年最后的第 96 卷的前 2 期.
黎景辉, 赵春来合著的 “模曲线导引(Introduction to Modular Curves)” 出了新版. 北京大学出版社(Peking University press) 2014 年 1 月已出第二版.
Introduction to Modular Curves
本书的目的在于介绍模形式的几何理论的背景知识. 本书可供数学系的研究生作为教材, 也可以供从事数论, 代数几何等专业的数学工作者使用. 作者在2002年出版本书第一版之后, 近些年又做了大量的修订, 使得该书的内容更完善更前沿.
就内容而言, 首先是修正了一些错误. 其次, 第一章从范畴开始, 附带 Abel 范畴, 第四章谈到了 2-范畴理念, 补充了形变和叠, 第三章增加了层范畴和上同调群, 第七章加进了椭圆曲线, 第十章讲解了 Ramanujan 猜想的证明.
本书不是初级读物. 亲如果想修炼神功, 请先学一些代数几何, 模形式, 代数数论. 认真的搞懂本书后, 就可以登堂入室, 看懂最新的进展了.
黎景辉是澳大利亚悉尼大学数学系教授, 主要研究方向是代数数论. 他的博士是 1974 年在耶鲁大学拿到的.
赵春来是北京大学数学学院教授, 主要研究方向亦是代数数论.
第 1 章 范畴 1
第 2 章 模空间 43
第 3 章 层 51
第 4 章 叠 110
第 5 章 Hilbert 函子 139
第 6 章 Picard 函子 168
第 7 章 模曲线 187
第 8 章 微分形式 208
第 9 章 TATE 曲线 224
第 10 章 模形式 249
参考文献
索引
作者: 黎景辉, 赵春来
版次: 2
开本: 16开
装订: 平
字数: 267 千字
页数: 296
ISBN: 978-7-301-23438-9
条形码: 9787301234389
出版日期: 2014-01-09
定价: 35 人民币元
The 2014 Wolf Prize in Mathematics is awarded to Peter Sarnak, for his deep contributions in analysis, number theory, geometry, and combinatorics.
Peter Sarnak is on the permanent faculty at the School of Mathematics of the Institute for Advanced Study, Princeton, NJ, USA.
Peter Clive Sarnak (born December 18, 1953) graduated University of the Witwatersrand (B.Sc. 1975) and Stanford University (Ph.D. 1980), under the direction of Paul Cohen.
Prof. Sarnak is a mathematician of an extremely broad spectrum with a far-reaching vision. He has impacted the development of several mathematical fields, often by uncovering deep and unsuspected connections. In analysis, he investigated eigenfunctions of quantum mechanical Hamiltonians which correspond to chaotic classical dynamical systems in a series of fundamental papers. He formulated and supported the “Quantum Unique Ergodicity Conjecture” asserting that all eigenfunctions of the Laplacian on negatively curved manifolds are uniformly distributed in phase space. Sarnak’s introduction of tools from number theory into this domain allowed him to obtain results which had seemed out of reach and paved the way for much further progress, in particular the recent works of E. Lindenstrauss and N. Anantharaman. In his work on L-functions (jointly with Z. Rudnick) the relationship of contemporary research on automorphic forms to random matrix theory and the Riemann hypothesis is brought to a new level by the computation of higher correlation functions of the Riemann zeros. This is a major step forward in the exploration of the link between random matrix theory and the statistical properties of zeros of the Riemann zeta function going back to H. Montgomery and A. Odlyzko. In 1999 it culminates in the fundamental work, jointly with N. Katz, on the statistical properties of low-lying zeros of families of L-functions. Sarnak’s work (with A. Lubotzky and R. Philips) on Ramanujan graphs had a huge impact on combinatorics and computer science. Here again he used deep results in number theory to make surprising and important advances in another discipline.
By his insights and his readiness to share ideas he has inspired the work of students and fellow researchers in many areas of mathematics.
Professor Gerd Faltings, winner of the prize in science, is the Director at the Max-Planck Institute for Mathematics in Bonn. He has made groundbreaking contributions to algebraic geometry and number theory. His work combines ingenuity, vision and technical power. He has introduced stunning new tools and techniques which are now constantly used in modern mathematics.
Faltings’ deep insights into the p-adic cohomology of algebraic varieties have been crucial to modern developments in number theory. His work on moduli spaces of abelian varieties has had great influence on arithmetic algebraic geometry. He has introduced new geometric ideas and techniques in the theory of Diophantine approximation, leading to his proof of Lang’s conjecture on rational points of abelian varieties and to a far-reaching generalization of the subspace theorem. Professor Faltings has also made important contributions to the theory of vector bundles on algebraic curves with his proof of the Verlinde formula.
数学一入深似海, 从此红尘是路人
数论导引
堆垒素数论
指数和的估计及其在数论中的应用
数论的方法
哥德巴赫猜想
模形式导引
解析数论基础
代数数论
素数定理的初等证明
初等数论 第三版
二次数域的高斯猜想
模形式讲义
模曲线导引 第二版 黎景辉 赵春来
二阶矩阵群的表示与自守形式 黎景辉 蓝以中
模形式与迹公式
数论及其应用
模形式和三元二次型
算法数论
分圆函数域
非同余数和秩零椭圆曲线
代数数论
平方和
代数数论简史
谈谈不定方程
初等数论 100 例
趣味数论 单墫
谈谈不定方程 单墫, 余红兵
初等数论 冯志刚
数论(原名”数学竞赛中的数论问题”) 余红兵
初等数论难题集 刘培杰
很偶然的, 看到了几个韩京俊传出来的数论问题. 据说问题来自牟晓生.
这些问题显然非常的有难度. 第 3 个问题, 俺多年前就见过, 是 Paul Erdős 在美国数学月刊上提的问题(编号 A6431).
俺特意调查了其他几个问题的出处.
问题 5 也是 Paul Erdős 提出的, 但证明是 R.L. Graham (也可能是 Sierpsinski) 给的. R.L. Graham 证明了
\(\dfrac pq\) can expressed as the finite sum of reciprocals of distinct squares if and only if
\[\frac pq\in[0, \frac{\pi^2}6-1)\cup[1,\frac{\pi^2}6).\]
问题 6 的答案也是 R. L. Graham 提供的: Graham published a proof in 1963 as “A Theorem on Partitions”, Journal of the Australian Mathematical Society 3 (1963), pp. 435-441.
If \(n\) is an integer exceeding \(77\) then there exist positive integers \(k\), \(a_1\), \(a_2\), \(\dotsc\), \(a_k\) such that:
His proof is constructive and fairly short, but it does require a long table of decompositions for relatively small values of \(n\). It would be interesting to see a non-constructive proof that doesn’t require such a long list.
问题 7 也不简单.
Granville and Selfridge, Product of integers in an interval, modulo squares: “We prove a conjecture of Irving Kaplansky which asserts that between any pair of consecutive positive squares there is a set of distinct integers whose product is twice a square.”
The details are Electronic Journal of Combinatorics, Volume 8(1), 2001.
有比问题 8 更普遍的结果. More precisely, let \(\zeta_1\), \(\dotsc\), \(\zeta_k\) be \(n\)-th roots of unity. If
\[|\sum_{i=1}^k n_i\zeta_i|= 1,\]
where \(n_i\in\mathbb Z\), then \(\sum\limits_{i=1}^k n_i \zeta_i\) is also an \(n\)-th root of unit.