Conjecture

There exist elliptic curve groups $$E(\Bbb Q)$$ of arbitrarily large rank.

Martin-McMillen 2000 年有一个 $$r\geq24$$ 的例子:

\begin{equation*}\begin{split}y^2+xy+y&=x^3-120039822036992245303534619191166796374x\\&+ 504224992484910670010801799168082726759443756222911415116\end{split}\end{equation*}

Hasse-Weil $$L$$-function $$L(s, E)$$ 在 $$s=1$$ 处的零点的阶数 $$r_a$$ 称为 $$E$$ 的 analytic rank(解析秩).

Manjul Bhargava, Christopher Skinner, Wei Zhang(张伟) 7 月 7 日在 arXiv 上传的论文 “A majority of elliptic curves over $$Q$$ satisfy the Birch and Swinnerton-Dyer conjecture“, 宣布了取得的进展:

1. $$\Bbb Q$$ 上的椭圆曲线, when ordered by height(同构类以高排序), 至少有 $$66.48\%$$ 满足 BSD conjecture;
2. $$\Bbb Q$$ 上的椭圆曲线, when ordered by height, 至少有 $$66.48\%$$ 有有限 Tate–Shafarevich group;
3. $$\Bbb Q$$ 上的椭圆曲线, when ordered by height, 至少有 $$16.50\%$$ 满足 $$r=r_a=0$$, 至少有 $$20.68\%$$ 满足 $$r=r_a=1$$.

$\delta^t(C)\leqslant\frac{90\sqrt{10}}{95\sqrt{10}-4}\quad\text{and}\quad\delta^t(T)\leqslant\frac{36\sqrt{10}}{95\sqrt{10}-4}$

$0.9183673\dotsm\leqslant\delta^t(C)\leqslant0.9601527\dotsm$

$0.3673459\dotsm\leqslant\delta^t(T)\leqslant0.3840610\dotsm.$

1. Sphere Packing, Springer-Verlag, New York, 1999;
2. Strange Phenomena in Convex and Discrete Geometry, Springer-Verlag, New York, 1996.

2009 年初, 科学出版社推出他的”离散几何欣赏”, 是姜伯驹主编的科普丛书”七彩数学”中的一本. 这个的厚度是前一本的两倍, 175 页, 有一些细节的证明.

Eminent Kazakh mathematician Mukhtarbay Otelbaev, Prof. Dr. has published a full proof of the Clay Navier-Stokes Millennium Problem  in “Mathematical Journal” (2013, v.13 , № 4 (50))

The area of Muhtarbay Otelbaev’s scientific interests included spectral theory of operators, theory of operators’ contraction and expansion, investment theory of functional spaces, approximation theory, computational mathematics, inverse problems.

Mukhtarbay Otelbaev 已经发表超过 $$200$$ 篇论文, 指导了超过 $$70$$ 个博士.

[Update, Feb 7, 2014: Terence Tao 已经向 J. Amer. Math. Soc. 投了一篇论文 “Finite time blowup for an averaged three-dimensional Navier-Stokes equation”. 同时, 他也把文章传到了 arXiv: Finite time blowup for an averaged three-dimensional Navier-Stokes equation. 参看他 4 日的博客.]

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.

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

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.

[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

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

1. 成就太过突出

2. 用经典方法逆袭, 用弹弓打死了狗熊.

3.张益唐一直坎坷, 一举成名天下知.

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.

筛法

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

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.

[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

张益唐其人

References

1. 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.
2. 汤涛, 张益唐和北大数学 78 级, May 19, 2013.
3. Erica Klarreich, Unheralded Mathematician Bridges the Prime Gap, simons foundation, May 19, 2013.
4. Kenneth Chang, Solving a Riddle of Primes, The New Yorks Times, May 20, 2013.
5. Carolyn Y. Johnson, Globe Staff, Obscure University of New Hampshire math professor takes major step toward elusive proof, May 23,2013.
6. Dan Goldston, Zhang’s Theorem on Bounded Gaps Between Primes.
7. Henryk  Iwaniec, a email to Shing-Tung Yau: Subject: Re: Yitang zhang, May 24,2013.
8. 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
9. 唐嘉丽, 张益唐破解千古数学难题, June 6, 2013.
10. Paul Feely, UNH professor solves ancient mathematics riddle, June 2, 2013.

On Mar 10, Ciprian Manolescu posted a preprint on ArXiv proving that the Triangulation Conjecture is false:

Pin(2)-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture

Ciprian Manolescu(born December 24, 1978) is currently a Professor of Mathematics at the University of California, Los Angeles. 他曾在 IMO 取得三个满分: 1995,1996,1997. 他的学士(2001) 和博士(2004)两个学位都在 Harvard 大学完成.

Gang Tian(田刚) has just uploaded to the arXiv his paper “K-stability and Kähler-Einstein metrics“(Nov 20, 2012). The motivation of this paper is:
“In this paper, we prove that if a Fano manifold $$M$$ is K-stable, then it admits a Kähler-Einstein metrics. It affirms a folklore conjecture. Our result and its outlined proof were lectured on Oct. 25 of 2012 during the Blainefest at Stony Brook University.”

“This is the first of a series of three papers which provide proofs of results announced recently in arXiv:1210.7494.”