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.”

28 日, Xiu-Xiong Chen(陈秀雄), Simon Donaldson, Song Sun(孙崧, 很年轻, 曾在科大少年班就读) 在 arxiv 上传了一篇文章 “Kähler-Einstein metrics and stability“, 给出了一个证明 K-稳定的 Fano 流形容许 Kähler-Einstein 度量(Yau-Tian-Donaldson conjecture)的轮廓, 工具是 Donaldson 新发展的连续性方法:
“We annnounce a proof of the fact that a K-stable Fano manifold admits a Kähler-Einstein metric and give a brief outline of the proof.”

2002年, 安徽省怀宁中学读高二的孙崧获得全国高中学生化学竞赛二等奖.  同年他参加高考, 成为怀宁县考进科大少年班的第一人.

Shinichi Mochizuki has released his long-rumored proof of the abc conjecture, in a paper called Inter-universal Teichmuller theory IV: log-volume computations and set-theoretic foundations.

If true, the proof would be one of the most astounding achievements of mathematics of the 21st century.

The homepage of  Professor Shinichi Mochizuki is here.

Excited, but caution

Terence Tao’s comment(from his blog): It’s still far too early to judge whether this proof is likely to be correct or not (the entire argument encompasses $$500$$ pages of argument, mostly in the area of anabelian geometry, which very few mathematicians are expert in, to the extent that we still do not even have a full outline of the proof strategy yet). For those that are interested, the Polymath wiki page on the ABC conjecture has collected most of the links to that discussion, and to various background materials.

Please  refer to  Low Dimensional Topology blog.

virtually Haken conjecture   states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken.

Virtual Haken 猜想  设$$M$$ 为紧致的可定向的不可约的基本群无限的$$3$$-流形, 则$$M$$有一个$$Haken$$ 流形的有限覆盖.

$$3$$-流形的有限覆盖一直是$$3$$-流形拓扑理论中重要但进展缓慢的课题.

Ian Agol (UC Berkeley) announced a proof  of  the  Wise’s  Conjecture  on March 12, 2012  when he was speaking  in a  seminar lecture  at  the Institut  Henri Poincaré.  In  particular, this implies  the Virtually Haken Conjecture.  His proof is based on joint work with Daniel Groves (UI Chicago) and Jason Manning (SUNY Buffalo).  It makes heavy use of the work of Dani Wise (McGill) on the Virtually Fibred Conjecture, as well as the proof of the Surface Subgroup Conjecture by Jeremy Kahn (Brown) and Vlad Markovic (Caltech). The Proof was subsequently outlined in three lectures March 26 and 28th at the Workshop on Immersed Surfaces in $$3$$-Manifolds at the Institut Henri Poincaré. A preprint of the claimed proof  has been posted on the ArXiv , pdf .

Terence Tao(陶哲轩)$$1$$月$$31$$日, 提交了一篇论文 “Every odd number greater than 1 is the sum of at most five primes“. 这篇文章的主要结果, 正如标题展示的, 每个奇数可以表示为不超过$$5$$个质数之和. 显然, 这个结果和 Goldbach’s conjecture(哥德巴赫猜想)有关, 把奇数情形的哥德巴赫猜想, 即弱哥德巴赫猜想(Goldbach’s weak conjecture)推进了一步, 也改进了 Ramare 的结论: 每个偶数可以表示为不超过$$6$$个质数的和.

Tao 的论文, 有 $$44$$ 页, 这里是pdf . 所采用的工具, 是哈代和立特伍德所创造的圆法（Hardy–Littlewood circle method), 结合了一些另外的技巧.