​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 $$A\subset \{1, . . . , N\}$$, 且 $$A$$ 不含非平凡的三项等差数列，即 $$x+y=2z$$ 的解, $$x\ne y$$. 则

$|A|\ll \frac{N}{(\log N)^{1+c}}$

$$c\gt 0$$ 是绝对常数.

Thomas F. Bloom, Olof Sisask 的这个结果改进了Roth 的一个关于整数不含三项等差数列的上界的定理。

$|A|\ll Ne^{-O((\log N)^c)}$

Erdos 的著名猜想

Erdos 有一个著名的猜测是：如果 $$A\subset\Bbb N$$, 且  $$\sum\limits_{n\in A}\frac1n=\infty$$，那么 $$A$$ 包含任意长的等差数列。

Corollary 2 如果 $$A\subset\Bbb N$$, 且  $$\sum\limits_{n\in A}\frac1n=\infty$$，那么 $$A$$ 含无穷多非平凡的三项等差数列。

Proof.  若不然，假定 $$A\subset\Bbb N$$, 且 $$A$$ 仅仅含有有限个非平凡的三项等差数列。于是，对于任意的 $$N$$

$F(N)\colon=|A\cap\{1, . . . , N\}|\ll\frac{N}{(\log N)^{1+c}}+1,$

$\sum_{n\in A\atop n\leq N}\frac1n=\frac{F(N)}{N}+\int_1^N\frac{F(t)}{t^2}\mathrm dt\ll \int_1^N\frac{1}{t(\log t)^{1+c}}\mathrm dt+1\ll1.$

$$A$$ 的阶的下界

$|A|\geq Ne^{-c\sqrt{\log N } }$

# 非欧几何是一个时代的结束？还是开始？

If you ask a mathematician “why?” the mathematician will give you a proof.
But when you ask a historian “why?” the historian will tell you a story.

## 设准 vs. 公理

Postulate = Special Notion vs. Common Notion = Axiom

We will raise this conjecture (the purport of which will hereafter be called the “Principle of Relativity”) to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity:c which is independent of the state of motion of the emitting body. These two postulates suffice for the attainment of a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell’s theory for stationary bodies. The introduction of a “luminiferous aether” will prove to be superfluous inasmuch as the view here to be developed will not require an “absolutely stationary space” provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place.

## 数学真理的相关议题

“Snow is white” is true if snow is white.

## 爱因斯坦的反思！

Morris Kline在《数学：确定性的失落》中写道：「倘若数学命题是对现实(reality)的描述，它们就不是确定的(certain)；倘若它们是确定的，那它们就不是描述现实。……然而另一方面，可以确定的是，不论就数学整体或是单就几何而言，它们的存在都是我们想要得知实际物体的性质。

## 非欧几何的历史

15世纪文艺复兴绘画开始发展的「透视」和「无穷远点」等射影几何概念，不被想成「非」欧氏几何，而是关联于欧氏几何发展的领域。

1. 绝对欧氏空间的扬弃，把实际空间还给物理学，把欧氏几何想成一种可能的几何（最简单的）。
2. 非欧几何必须相应处理原来欧氏几何的几何概念：长度、角度、面积、体积等。

## 测量高斯曲率

Gauss-Bonnett定理之原版：三角形内角和

$\theta_1 + \theta_2+ \theta_3=\iint_{\triangle} K\mathrm dA.$

## 康德的复仇

John Horton Conway 小传

1964年，他在剑桥获得博士学位，后成为普林斯顿大学数学系教授。

1966年8月，在莫斯科，一位博士生向他介绍了“Leech晶格”：在24维的欧几里得空间里，一堆球体紧密排布，每个球都挨着周围196560个球。他的想象力被激活了，开始寻找这个晶格的空间对称群。他成功了。

Siobhan Roberts 写过一本Conway 的传记 Genius At Play: The Curious Mind of John Horton Conway

John Horton Conway 的著述

John Conway 写过很多精彩的书籍。一个不完整的清单包括:

Conway, J. H. (1970): Regular machines and regular languages；

Conway, J. H. (1976): On numbers and games；

John B Conway: Functions of One Complex Variable[经评论提醒，这书作者不是本文主角John H. Conway ]

Conway, J. H.; Berlekamp E. R.; Guy, R. K. (1982): Winning ways for your mathematical plays；

Conway, J. H.; Sloane, N.J.A. (1988): Sphere packings, lattices and groups；

Conway, J. H.; Guy, R. K. (1982): The book of numbers；

John B Conway: A Course in Functional Analysis[经评论提醒，这书作者不是 John H. Conway]

John B Conway:  A Course in Operator Theor[经评论提醒，这书作者不是 John H. Conway]

Conway, J. H.; Smith D.A. (2003): On Quaternions and Octonions

John Conway | Nov 27, 2012: All Yesterdays: Unique and Speculative Views of Dinosaurs and Other Prehistoric Animals

G. Polya and John H. Conway | Oct 27, 2014: How to Solve It: A New Aspect of Mathematical Method

John H. Conway, Heidi Burgiel, 2008: The Symmetries of Things

Tao 的纪念

I was greatly saddened to learn that John Conway died yesterday from COVID-19, aged 82.

My own mathematical areas of expertise are somewhat far from Conway’s; I have played for instance with finite simple groups on occasion, but have not studied his work on moonshine and the monster group.  But I have certainly encountered his results every so often in surprising contexts; most recently, when working on the Collatz conjecture, I looked into Conway’s wonderfully preposterous FRACTRAN language, which can encode any Turing machine as an iteration of a Collatz-type map, showing in particular that there are generalisations of the Collatz conjecture that are undecidable in axiomatic frameworks such as ZFC.  [EDIT: also, my belief that the Navier-Stokes equations admit solutions that blow up in finite time is also highly influenced by the ability of Conway’s game of life to generate self-replicating “von Neumann machines“.]

I first met John as an incoming graduate student in Princeton in 1992; indeed, a talk he gave, on “Extreme proofs” (proofs that are in some sense “extreme points” in the “convex hull” of all proofs of a given result), may well have been the first research-level talk I ever attended, and one that set a high standard for all the subsequent talks I went to, with Conway’s ability to tease out deep and interesting mathematics from seemingly frivolous questions making a particular impact on me.  (Some version of this talk eventually became this paper of Conway and Shipman many years later.)

Conway was fond of hanging out in the Princeton graduate lounge at the time of my studies there, often tinkering with some game or device, and often enlisting any nearby graduate students to assist him with some experiment or other.  I have a vague memory of being drafted into holding various lengths of cloth with several other students in order to compute some element of a braid group; on another occasion he challenged me to a board game he recently invented (now known as “Phutball“) with Elwyn Berlekamp and Richard Guy (who, by sad coincidence, both also passed away in the last 12 months).  I still remember being repeatedly obliterated in that game, which was a healthy and needed lesson in humility for me (and several of my fellow graduate students) at the time.  I also recall Conway spending several weeks trying to construct a strange periscope-type device to try to help him visualize four-dimensional objects by giving his eyes vertical parallax in addition to the usual horizontal parallax, although he later told me that the only thing the device made him experience was a headache.

About ten years ago we ran into each other at some large mathematics conference, and lacking any other plans, we had a pleasant dinner together at the conference hotel.  We talked a little bit of math, but mostly the conversation was philosophical.  I regrettably do not remember precisely what we discussed, but it was very refreshing and stimulating to have an extremely frank and heartfelt interaction with someone with Conway’s level of insight and intellectual clarity.

Conway was arguably an extreme point in the convex hull of all mathematicians.  He will very much be missed.

（本文参考了几篇文章，部分段落从这里来：

1. 中国青年报，4月17日的《被新冠病毒带走的数学大玩家

2. 掌桥科研，4月12日在知乎的文章《因新冠肺炎逝世的天才数学家约翰·康威 John Horton Conway》 ）

7月4日，加州大学圣地亚哥分校细胞与分子医学系付向东教授实名举报中科院上海神经所80后明星教授 Yang H 学术抄袭、造假。

“2018 年 6 月 14 号，受蒲慕明所长特邀学术报告，付向东在中科院神经所报告了我 们这项未发表的、治疗帕金森综合征的研究成果，详细介绍了此项研究工作的科学 思路、全部实验设计和研究结果;同时，我还分享了将抗 PTBP1 因子成功应用到视 网膜疾病治疗的一项合作研究工作。杨辉和神经所百余名科研人员参加了我的学术 报告。报告之后，Yang H 和几位研究员与我共进晚餐，在晚餐期间Y H向我咨询了许 多关于实验细节问题。”

​6月17日，人教版数学八年级下册自读课本写到爱因斯坦用相对论中的质能方程论证勾股定理，但是摆了乌龙的消息刷屏。这里不去讨论这个错误的证明，虽然在官方教科书出现这种低级错误实在不该。

1. ​ 相似三角形的面积之比等于相似比的平方；
2.  相似三角形的面积与某条对应边边长平方之比为一个常数。

$mc^2=ma^2+mb^2.$

IMO 2019 官方网站的解答

IMO 2019 solutions