Power, Simplicity and Beauty
Advances in Mathematics
7月7日，Thomas F. Bloom, Olof Sisask 在 arXiv 上传了一篇论文 Breaking the logarithmic barrier in Roth’s theorem on arithmetic progressions( arxiv.org/abs/2007.03528), 该文的主要结果是证明了: Theorem 1 如果 \(A\subset \{1, . . . , N\}\), …
The first non-trivial case of a conjecture of Erdős on arithmetic progressions Read More前几天传出了一个消息，英国杜伦大学的Andrew Lobb和波士顿学院的Joshua Greene这两位数学家解决了一个有109 年历史的著名难题：任何简单闭合曲线，都包含四个可以连接形成正方形的点。 然则，这则新闻有点耸人听闻。事实上，这两个数学家解决的只是一个附加了条件的 弱化版本，并没有彻底搞定 109 年前的那个原始的猜想。 我们先来看看这个猜想是一个什么问题。这个猜想(Toeplitz square peg problem)是猜测任意连续的简单闭曲线上存着四个点构成为一个正方形。 Andrew Lobb 和 Joshua Greene 证明的结果是： 对于任意光滑的 Jordan 曲线和长方形 R, 可以找到曲线上的四个点使得构成的长方形相似于 R. 换言之，Andrew Lobb 和 Joshua Greene 证明了 对于光滑的 Jordan 曲线上存着四个点构成为一个正方形，并且不仅仅如此，他们对于光滑的 Jordan 曲线得到的结果比猜想还要好很多。 于是 …
Toeplitz square peg problem hasn’t been solved yet Read MoreConjecture There exist elliptic curve groups \(E(\Bbb Q)\) of arbitrarily large rank. 用 \(r\) 表示 \(\Bbb Q\) 上的椭圆曲线 \(E\) 的秩—the rank of the Mordell–Weil group \(E(\Bbb Q)\). 一个悬而未决的著名难题是: \(r\) 是否可以任意大? Martin-McMillen 2000 …
Manjul Bhargava, the BSD conjecture and Fields medal Read More前些日子, 传出新闻, 北大数院的宗传明在堆球问题取得进展, 论文 On the translative packing densities of tetrahedra and cuboctahedra 发表在 Advances in Mathematics, Volume 260, 1 August 2014, Pages 130–190. 该文的主要结果是证明了 \[\delta^t(C)\leqslant\frac{90\sqrt{10}}{95\sqrt{10}-4}\quad\text{and}\quad\delta^t(T)\leqslant\frac{36\sqrt{10}}{95\sqrt{10}-4}\] 于是 \[0.9183673\dotsm\leqslant\delta^t(C)\leqslant0.9601527\dotsm\] 和 \[0.3673459\dotsm\leqslant\delta^t(T)\leqslant0.3840610\dotsm.\] 宗教授研究堆球已经很多年. 已经退休的项武义也在开普勒猜想(Kepler Conjecture)花了很多心思, 一度宣称解决了这个古老的著名问题, …
Chuanming Zong and Sphere Packings Read MoreEminent 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 has published a full proof of the Clay Navier-Stokes Millennium Problem Read MoreHere are some papers related to bounded gaps between primes: Erica Klarreich Together and Alone, Closing the Prime Gap Andrew Granville Bounded gaps between primes Primes in Intervals of bounded length James …
Progress on bounded gaps between primes Read MoreBusy 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, …
Harald Helfgott has proved Goldbach’s weak conjecture Read MoreOn 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. …
Yitang Zhang has proved a weak version of the twin prime conjecture Read More