Apparently, another one of Ren Zhengfei’s daughters is a computer science undergraduate at Harvard

The detaining of Meng Wanzhou, CFO of Huawei and daughter of its founder Ren Zhengfei, at Canadian airport on behest of the US authorities has made headlines the past several days. On that, I have little to say other than that she is quite attractive and also that the half-closeted nepotism via mother’s surname is both amusing and somewhat gross.

Reading her Wikipedia page, I pulled the following

Ren’s first wife was Meng Jun, the daughter of Meng Dongbo, a former deputy governor of Sichuan Province. They had two children: daughter Meng Wanzhou and son Meng Ping, both of whom took up their mother’s surname.[14] After their divorce, he married Yao Ling, with whom he had another daughter, Annabel Yao, who is 25 years younger than Meng Wanzhou. Annabel is a computer science student at Harvard University who made a high-profile debut at Le Bal des Débutantes in Paris in 2018.[14] Ren married for the third time to Su Wei, who was reportedly his former secretary.[14]

Ren’s eldest daughter, Meng Wanzhou, is deputy chairperson and chief financial officer (CFO) of Huawei.[15]

I’m somewhat surprised and also rather disappointed that Ren Zhengfei sent his other daughter, who also took her mother’s surname (and this is a different mother), to Harvard for undergrad. It reminds of how Xi Jinping’s daughter transferred to Harvard from Zhejiang University after her freshman year. I hate to say it but it’ll be hard to take the PRC elites too seriously when they send their children to elite US schools for undergrad. It is a sign that they still lack independence with a need to associate with global homo Anglo elite institutions. On the Chinese internet, people are questioning what other passports besides the Chinese one is possessed by Meng Wanzhou, and it was revealed somewhere that she had a green card from Canada.

Meng for undergraduate merely attended a good but not great university in China. That was when her father, who started Huawei in his 40s, was still a relative nobody. Remember that in China, aside from regional discrimination, wherein Beijing residents are allocated like 10 times more slots at Beida/Qinghua adjusted for population than residents in the provinces, admissions is based purely on gaokao and automatic admission via top performance on the national math and science olympiads. It’s basically impossible for filthy rich and/or well-connected parents to buy one’s kid’s way into Beida/Qinghua in China. So I guess when the kid of someone super high up in China turns out to be intellectually mediocre, attending an elite US school for undergrad becomes the default, sadly.

Something to note, relatedly, is that 富二代 (second generation rich) is a relatively new phenomenon in China. After all, after the communists took power, the rich people had their assets confiscated (like my amusingly pro-communist ABC friend stuck in the US at least for the near future, his family ended up sharing their four story house/building with a bunch of a poor people), and afterwards, people in high up positions did not make much more than the ordinary worker, with the salary difference, based on what I’ve heard, at most a factor of 10, barring very exceptional cases. It was really only after 1980 when it became possible for one to become extremely rich doing business. Between 1980 and 2010. there was a lot of low hanging fruit in China, economically. Plenty of mediocre people with the right entrepreneurial energy/personality became rich. And by that, the class of second generation rich was born, and it seems now they are forming their own culture, with studying abroad paying ridiculous tuition at US universities common, as well as their parents investing in US real estate. It’s very possible and perhaps likely that the class structure in China will solidify over the next generation, as is already the case in the US. How much so, that remains to be seen.

I’ve had some interactions with 富二代 kids and their parents in the US. Many of them are, predictably, not very smart academically. They are basically on permanent vacation in America. Their career/employment prospects in America are not surely not good for obvious reasons. It doesn’t seem like they’re well treated in school by their white peers in America. Yes, their families are economically well off, but that doesn’t mean they can integrate into American society. Their families are able to invest in real estate in America for now; even so, their existence in the US will be a marginal one much confined to the Chinese community. Many of them do not even know English well and are mostly here to invest in real estate, enjoy some material comfort and fresh air, and also send their kids to school here. In most cases, the father based in China only visits every once in a while. They’re obviously not really wanted in America other than for their money, spent on international tuition and luxury brands and so on.

As the reader can tell, I don’t have a high opinion of those 富二代 who send their kids to America and invest in real estate there. Being in China now, I’m sure I’ll learn more about them, including the process they go through to transfer all that money to the US, and maybe I can even influence them a bit. As one can see from the detention of Meng Wanzhou in Canada, there is little guarantee for them on US soil. Finance/economics tells us there is no such thing as a free lunch. They may feel they are winning now and that their assets in America are safe, but this could very much be a process of 温水煮青蛙 (a frog boiled in warm water), where initially one feels warmth, until one gets burned. They may own real estate in America on paper for now, but that real estate is physically in America, where the Chinese do not really control anything. They have no media power, no legal power, no military power there. It’s totally possible for them to suddenly lose everything. Unlikely, but it can happen and only needs to happen once.

The Meng Wanzhou incident has been in Chinese media tied to the death of Stanford physics professor Shoucheng Zhang at age 55 which happened on the same day. The family statement said that it was due to depression, which we all know is just a euphemism. There has been speculation that it was related to the alleged tanking of his Chinese venture capital firm in Silicon Valley. It’s quite tragic that he took his ambition/careerism too far, assuming that his death was related to his VC activity. Had he just remained purely academic, he would have survived and likely eventually won his Nobel. Needless to say, using the prestige of a rare Stanford tenured professorship in physics to raise a Chinese VC is not really what he should be doing while officially in that position. Stanford probably should be somewhat embarrassed about his case as well, in addition to the recent Elizabeth Holmes case that was magnitudes worse. On this, I also have in mind the cases of Lucas Duplan and Evan Spiegel, both quite shameful.

It is my hope that talented, elite Chinese can focus more on developing their institutions at home instead of wasting so much money and energy associating with and working for elite US institutions for signaling. Doing so only inflates the value of a system hostile to Chinese while at the same time de-valuing Chinese institutions. If Chinese are really serious about this trade war, they should be going out of their way to devalue Harvard and Stanford, to devalue Google and Facebook, not eagerly trying to attend those schools for undergrad for a rather mediocre academic educational experience and making such a big deal about working for those companies, whose programmers are generally no better than ones in a good internet company in China as far as I can tell. If Chinese want other people to respect and value them, then they ought to first have confidence in their own institutions. I say this having been in prestigious US institutions myself, interacted substantially with numerous people from there, and frankly, many of them are not all that great, with some depressingly mediocre or even problematic. Of course, there are also plenty of genuinely brilliant, accomplished people in America in top American institutions. Overall, my point is that it’s clear that America will not judge the Chinese well, so why expend so much trying to associate and fit in with America? Why not instead expend more energy into creating a system where what America thinks simply becomes more or less irrelevant?

 

张首晟的去世

周四,我认识的一位同样搞理论凝聚态物理的人微信发给我了张首晟去世的消息,让我感到吃惊。这么牛逼的人,不光有资格得诺贝尔奖的理论物理学家,而且还很有钱,好像他90年代末凭他斯坦福大学教授的关系做了搞虚拟机器(virtual machine)的VMware(其创始人是斯坦福大学计算机系教授)的天使投资,赚了一大笔钱,也得到了为后来做风险投资的资历和关系。一流理论物理学家很少是多么有钱的,美国名校物理教授每年也就15万,最多30万的工资(当然,现在极少数最牛的理论物理学家可以通过什么突破奖发点大财了),而张首晟成为了极其罕见的即大物理学家又大发财的人,一种几乎不存在的令人惊叹不已的人生赢家。可惜他近年做得过度了,最终出现了悲剧。可以说他当初运气太好了,拿到的教职确是斯坦福的,才得到了这种天使投资的机会。反而他要是是什么麻省理工或哈弗的,很难想象到他这么去做。可是,反过来,也可以说最终好事变成了坏事,他的这种运气所迎来的成功引发了他后来自以为是的做一件物理教授不应该做的事情而运气不好彻底倒闭了,英年夭折失去了得诺贝尔奖的机会。

记得2015年底,我在湾区餐馆跟朋友吃饭看到了张首晟在另一个桌子和他老婆和有一对夫妇一起吃饭谈话,当时,我已经知道他是非常有名气的斯坦福理论物理学家了。可能那时候我也知道他搞丹华资本这件事情了,感觉是他在利用斯坦福大学物理教授的威望为中国人在硅谷争取一些地位,不光要当大物理学家,还要当什么民族领袖,民族英雄。当然,我也意识到他拿着美国名校的终身教职来为中国人搞VC真的不是他应该做的事情,会引起别人对他的不满,不过他在那种位置,别人也不能把他怎么样。不过,一些主流的白人VC若对他有仇,我想还真的能让他的风险投资商难以持续。我认识的那位与张首晟稍有过接触那位物理学家是说在风险投资界他根本不算啥,老美根本不会买他的帐,只有中国人会因为他的位置听他忽悠。比如,我在美国认识一位斯坦福大学的本科生,女生,也是学物理的,她跟我说她妈妈还会参加一些湾区张首晟讲VC的活动。那个人也说他的VC的钱也没有那么多,也就四亿,输一次就基本输光了,下面的图来自方舟子。

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如那个人所描述,输光了给他融资的人就要找他打官司了,而且会揭露他的一些非法的操作,就是不自杀也得坐牢了。

我还记得有一次网上看到了他的一个视频,是在斯坦福大学举行的那种启发美国华裔中学生那种活动,网上可以搜到,那里边他讲了他的一些人生故事和价值观。记得一开始他说起他长大的文革时期中国的气候是不鼓励学习的,然后又说起他父母是工程师,然后家人有好多学过文科的人,自己小时候多么多么善于自己读书学习。然后还开始扯美国建国的人如Benjamin Franklin, John Adams的一些理念,说什么我自己要做一些实用的挣钱的事情,为了我的孩子或孙子能够从事更创造性的如艺术之类的工作。他说同样,我们这一代移民来到美国好多为了留下,为了挣钱,却选择了工程,他属于少数坚持他对理论物理热爱的人,但第二代就比他们有远远更好的条件,应该去追求自己的梦想,最自己喜欢的东西。他说的这些也有一定道理了,当然,他扯美国建国人的时候我有点感到不舒服,我想他是不是觉得自己在美国名校斯坦福举行这种明目张胆的华人利益活动有点不要脸了,他真的觉得中国人应该学习美国建国人那套吗,以白人至上主义和暴力逐出印第安土著人建设一个华人没有什么真正地位的国家吗?他跟那些孩子讲完了之后,在旧金山的中国领事馆的一个代表又开始采访他了。在那一段谈话里,我记得他说中国人在美国都混得不错,大多是在做工程。是,那些科技移民在硅谷都有比较好的工程工作,整一个相当高的工资,不过相当少能成为经理,很少能够创办自己的公司,也没有那么好,在美国依然算相对边缘。还记得他又提到自己的儿子在哈弗上学第一学期在上一门中国近代历史的课,不过都是用英文以美国的眼光讲,同时儿子中文也不错,也读了陈独秀的一些中文文集。我想这活动肯定是那儿的中国领事馆组织的,由于张首晟当时是拿了政府的钱本科时去德国留学的,中国政府的人依然觉得有资格“利用”他一下。

我还想说张首晟和他孩子都是虔诚的基督教徒,我也看到他的孩子对他们的基督教信仰还都比较公开,甚至到有点痴迷的程度。这种在美国华人里,据我所知,还是少见的,这种科学人及基督信仰混合在白人里我看到不少例子,华人这样却让我感到有点不对劲儿,至少感觉有点奇怪。我想这样做在中国会得到一定的排斥,毕竟太逆行于主流,尤其解放后,看到为此,方舟子也对张首晟表示了不满。同样,我认识的那位物理学家觉得基督徒的张首晟搞这种投机收利息的行为是比较讽刺的。

最后,为这位伟大的华人物理学家的不幸而逝默哀。他无法得诺贝尔奖真的是可惜了,我想好多中国人会为此感到遗憾。历史给予他什么样的地位,只有时间会告诉我们。

不知为何,突然想起测度论里的不可测度的维塔利集合

复制以我写的知乎文章

我在知乎上写的目前竟是一些有关于美国华人和ABC和犹太人的政治话题,自己快成了民族活动家了,其实对于学理工科的人而言,民族活动家比较贬义。民族活动家似的言论与活动,尤其在美国,其实是自然被有能力的人所藐视的,这原则很简单,它根本就是不“专业”的表现,甚至可以说是一种流氓耍诬赖的作为。在美国,中国人政治上都是特别老实的,从来不闹事儿,不抗议,就服从性的低调的埋头苦干。相反,我看到过一位根本不黑但有黑人血统的数学研究生,他的数学水平其实很差的,与其他人相比,可是他却公开的支持Black Lives Matter,然后学校媒体却非常支持他,以他宣传自己的diversity,公布的视频里还有他说I didn’t have to think about race。有意思的是他根本不黑,要他不说,其实都看不出来他是黑人。所预料,这些在学校没人敢说的,说了都怕给自己惹麻烦,其实好多人都为此感到不满,但不得不不了了之,最终政治赢者是谁就毫无疑问了。

我为什么开始写这些东西具体愿意也很难说,一个根本是我天性特别讨厌装傻扯淡的表现,而美国的政治在我眼中就是个大装傻戏(当然,未避免是好多人真的那么傻,或者自己太聪明了哈哈)。反正美国人关于中国和中国人的看法好多实在太傻,在那儿的中国人大多也懒得去“纠正“,大多也是不了了之,我也是,则好多负面的又不太符合客观事实的刻板印象逐渐形成。好多这些我也在我博客用英文写了,美国人爱看他们可以看,知识让一部人知道,并留个记录,也给中国人一点启发。

好,说起数学,我想稍微写写关于我对某非常具有纯数学性质又非常基础重要及美妙的一个观念和例子,那就是不可测度集合的体会。我记得曾经把他的英文维基百科页发给某北大物理毕业的人看了,他的反应就是这种鬼东西只有脱离现实的数学家才会在乎。其实测度这个观念大家是有的,即使不喜欢数学的人,长度宽度这些都是对人很直觉的。形式化一些,我们以实数集合模拟,一个实数集合的子集的度量,勒贝格度量(Lebesgue measure)定义为

\mu^*(E) = \inf\{\displaystyle\sum_{k=1}^{\infty} I_k : (I_k)_{n \in \mathbb{N}} \text{as open intervals}, \displaystyle\bigcup_{k=1}^{\infty} I_k \supset E\}

这个其实是勒贝格外测度,是否可勒贝格测度有条件,那就是所有实数子集A 符合

\displaystyle \mu ^{*}(A)=\mu ^{*}(A\cap E)+\mu ^{*}(A\cap E^{c})

若可勒贝格测度,度值为上的外测度值。

好,我们去找一个非可勒贝格测度的集合。为此,我们将引用三个观察。

  • 测度平移守恒,那就是 \mu(S) = \mu(x+S), \forall x \in \mathbb{R}
  • \{A_k\}_{k \in \mathbb{N}} 互相不交则 \mu(\bigcup_{k=1}^{\infty} A_k) = \sum_{k=1}^{\infty} \mu(A_k)
  • S \subset T\mu(S) \leq \mu(T)

这些直觉上显而易见,形式化证也不难。

以前两点,我们发觉任何可以表示为可数无限多个平移同等的不交集的测度比为 0\infty ,因为所有不交集测度一样所以总测度必然是无限个零或无限个有限数。
那,若我们找到那样的一个集合,并且通过第三点,把他的测度加以有限上限及下限,则得以矛盾,则无可测度。

我们先取商加法群 \mathbb{R} / \mathbb{Q} ,并以选择公理在其所有同价类选在 [0,1] 范围内的一个元素构建一个集合,称之 V 。我们在以每一个 \mathbb{Q} \cap [-1,1] 的元素将 V 平移,这些平移互相不交叉,其并集又包含 [0,1] (也就是我们选择不大不小 [-1,1] ),但 [-1,2] 之内,所以以第三点,他的测度又在 13 之间。所以他若可测度就矛盾了。

我第一次看到这个好像是大三看到的,是自己在英文维基百科页看到的,当时,想这个脑子都有点晕了,还是太不数学成熟,使劲想了但还为此感到迷惑,这个构建的要素脑子里还未看透。可是,后来数学有了大的进步,今年初,我又在没有查任何资料的情况下十分钟左右就从新想了起来,接着把它板上解释给了一位芝加哥大学经济系毕业的人,可是这次,那个人却难以理解。

前天晚上,又想起这个了,感到它非常奇妙,尽然在数学的脱离于物质世界的抽象化及形式化存在这样度不可测的集合。当然,有一点是他依赖有一定哲学争议的选择公理,这我现在也没资格谈。这次根本没有想就能够清晰在脑子里看到这个构建,自然就回顾到了与其恰恰相反的无法理解之的曾经,觉得那时候自己脑子还处于一种半沉睡的状态。这也是数学一种奇妙之处吧。有的定理无论如何证出来不得不有点复杂繁琐,但也有一些定理或观念虽然简单但是从数学思想上却是天才般的,革命性的,之所以那么久才能被人发现到,之所以当初理解困难而经过正确深思后却一目了然永不忘。

纯数学我觉得还是最需要智力的学科,主要是他那种抽象度啊,是很少一部分人脑子先天条件足以接受的,像这种东西他跟计算机科学那些算法就很不一样了,算法还是相对具体的,离我们日常生活不太远,我当初接触有一定难度的算法题没问题,但是某些抽象的数学观念总是吃不透,让我感到自己就是不够聪明,天分有限,可以后来,突然就数学觉醒了。这些客观可严谨证明的抽象数学真理终于在我脑海里实现了,而之前虽然一直存在,对我当时还有问题的头脑却是不存在的。主要还是这些定理的构建与证明都有一些简单的抽象数学观念为基础,这些却很难抓住,在没有掌握的时候,你再费脑也无用,但一旦看透了就觉得其实很简单。所以我很佩服数学天才,他们的结果更多是靠一种天才般的智力和想象力,而非仅仅刻苦,他们能够看到一个远远更高的境界,而这不是什么毛泽东思想或耶稣这种人为的信仰世界,而是一种绝对的科学真理。

What’s wrong with the Ivy League

Very recently, a Chinese-American Yale undergrad cold emailed me expressing approval of this blog, and we not long after began to talk regularly.

A few days ago, in an email to Steve Hsu, me, and some others, he wrote:

I think the Ivy League is best understood as a giant money-making organism. Attached to it like a leech is a seminary for training priests and spreading the gospel of the American progressive religion. Attached to that leech is an even smaller leech which actually contains smart people doing good technical work. And gmachine1729 is right that the average Ivy student is not that great (but the top end does represent the best of the best).

Relatedly, I just saw an essay on Zhihu in Chinese by Yale finance professor Zhiwu Chen on the matter of 中国人那么优秀,为什么美国人还是更喜欢印度人?(Chinese are so exceptional, why do Americans still like Indians more). The essay was one of Chinese are smart and hardworking but modest, passive, conformist, filially pious per the Confucian tradition and emphasize hard skills at the expense of soft skills, while Indians, like Americans, are assertive and confident with strong leadership qualities. The comments, of which there were 27, were mostly dismissive. The first one, and the most memorable one was

你对比下毛主席和甘地还不能明白点?

In translation,

Can’t you better understand the situation by comparing Chairman Mao with Gandhi?

I’m very happy to see this. It means that Chinese are no longer dazed by Ivy League credentials. They are beginning to think more independently, to have more confidence in themselves. Maybe in another decade’s time, non-STEM professor at Ivy will become an explicit negative signal among Chinese in China.

On credentialism and selection systems

I’ve mentioned before that an Asian-American friend of mine, who is quite smart, disapproves of the whole campaign against Asian quotas spearheaded, or at least advocated, by Steve Hsu and others.

His words are the following:

  1. I don’t believe in legitimizing the credentialist culture of modern academia
  2. I don’t generically feel much kinship with Asian-Americans (who are the most affected by purported discrimination in admissions), even if I might feel more kinship with them on average than I would with any other large ethnic group in America (which is itself not necessarily true)
  3. I don’t find it implausible that there are legitimate reasons to discriminate against Asian-Americans in the admissions process, if by ‘discriminate’ we mean ‘weigh their formal accomplishments less than one would for a member of a different race’
  4. At the end of the line, I believe that persistent whining about this is a reflection of emotional immaturity on the part of Steve et al., in that they seem to have a ‘chip on their shoulder’ which they are incapable of overcoming, and if they were actually taking a principled approach, they would come together and try to create a superior alternative to the radically broken university system, which will likely not be saved by any infusion of Asian students

Here’s what I think.

On 1), I don’t like the credentialism culture of modern academia either. Much of it is a superficial and soulless arms race. Not that grades, test scores, publications, citations, impact factor aren’t strong signals but they are prone to manipulation and artificial inflation, and that there are qualities of work not well-captured by those metrics. People are more or less compelled to single-mindedly play this game, often at the expense of actually substantial scholarship, if they are to survive in academia nowadays.

On 2), I hate to say that this country has become more toxically consumed by identity politics over the years, not to mention that people are judged at least subconsciously by who one is associated with. So collective bargaining is crucial for a group’s position on the status hierarchy.

On 3), there is that due to Asian-Americans’ and Asians in general having traditionally been the underdog, as well as their lack of media presence, which is intimately tied to the alienness of their names in the Western linguistic context, some people are inclined to view Asians are grinds who aren’t actually as capable as they might appear on paper. Especially with the whole tiger mother phenomenon that Amy Chua popularized with her infamous book. Of course, China’s rise over the recent years has altered this perception somewhat, especially the one that Asians are smart but not creative, though surely, it does seem that controlling for grades and test scores, or IQ, Asians do seem less creative, though that may be due to environmental factors, such as de facto or implicit quotas imposed by diversity mandates and economic circumstances.

On 4), I mostly disagree. Asian-Americans don’t really have the power to create a sufficiently credible alternative in a world that runs so heavily on associating with prestigious, usually long-established, institutions like Harvard and Goldman-Sachs. In their ancestral countries, China and India, Asians can improve the university and research system and the economic and technological competitiveness of the country as a whole, so as to make their universities more credible as well. In America, all Asian-Americans can really do is make more noise around the issue to exert more pressure on the elite universities, and also donate more and enhance their media and political presence as their socioeconomic position improves, especially at the elite end, improves, so that the elite universities perceive themselves as having more to lose from discriminating against Asian-Americans based on race.

This is all I have to say as pertains exclusively to Asian-Americans. I shall now give my thoughts on credentialism and selection in general.

The job of admissions and hiring committees and HR is astronomically harder than in the pre-internet age. So many people apply for positions they are grossly under-qualified for, now that it’s so easy to shoot off a resume or application online. There are, of course, application fees for college and grad schools, but they are not enough to deter. This means in the selection process can be afford now significantly less time per candidate, and one can argue that as a consequence, the process becomes more bureaucratic and easier to game. Often, people will in the pre-screening stage eliminate all applicants who do not meet certain formal criteria, such as minimum GPA/test scores or a certain degree from a certain set of sufficiently credible universities. In the case of academia, to my limited second-hand knowledge, committees will look at publications lists with a focus on citation count and impact factor of the journals on which the papers were published and also verify the candidate against senior, tenured faculty in the same or at least similar area of research. In the case of industry jobs, what matters more is the interview, where for technical roles, technical questions will be asked to further test the technical aptitude and knowledge, as well as, the softer aspects of communication and personal chemistry. For non-technicals, I can only say it’s even more about credentials (school, companies, job titles, dates of employment) and how you present yourself. I can only conclude that way more energy is expended now in aggregate on application and selection than before, which is quite costly really. In the career world, people are mostly out for themselves and don’t really care about wasting other people’s time, so long as they can get away with it with impunity more or less.

I’ll say that there is a tradeoff between optimizing for one’s formal credentials and optimizing for one’s actual ability and knowledge. One loses out so much more now if one neglects the former too much due to more competition per position. Surely, there has been gross inflation of credentials. This is in its crudest form epitomized by college’s having become the new high school, thereby rendering prestige of institution a stronger signal. Furthermore, the largely consequent grade inflation and watering down of coursework has added more noise to school transcripts. Contest training, for math in particular, has become so much more popularized, that to not have credentials in those raises questions in some circles, and moreover, there is so much more of an obstacle course of summer programs and scholarships and grants and internships and jobs which one must pass through to some degree if one wants a reasonable chance of success at a specified level. In this sense, there is more pressure to conform to an existing, often complexity-ridden system. It may well be that people nowadays are not all that much better in terms of knowledge and proficiency than before, correcting for the positive effects of technology on learning, but they actually put in much more time and effort.

Now, if one expends much energy on actual substance, there is concern as to what would be lost if those translate not into formal credentials. Arguably more common is the other way round, where one turns into a soulless credential-chasing machine. I’ve been amazed at how many people manage to achieve much higher grades, test scores, and awards than what their knowledge and ability from interaction with them would reasonably indicate. Those people tend to be very boring and risk-averse, and they are often the types our current system selects for, like it or not.

I used to feel like to prove that one is actually smart, at least in STEM, one ought to do sufficiently well in one of those major math, physics, or computing olympiads or contests. I would say that for raw technical ability, that is probably still the strongest signal. Grades are somewhat noisy, because it’s not hard to copy or snipe homework solutions, and for tests, there is a large cramming and figuring out what’s gonna be on the test component. Perhaps they are more consequentially so as there are also some genuinely capable or even brilliant students who for related personality reasons have a hard time getting themselves to care too much about grades. I’ve personally seen some high GPA people, even in college, who signal in what they say or write complete idiocy that would make you wonder if they were pretending stupid, especially if said person were female. Some people learn much more deeply and also much more broadly, outside of what the system teaches them, to a high level of retention, much of which is not captured through any formal credential. From my personal experience, tests of a wide range of knowledge, sufficiently substantial but not too esoteric, are stronger signals since they cannot be crammed for, but they are, for the difficulty of organization, seldom administered.

In the real world and in academia though, what matters is the ability to deliver actual projects and conduct meaningful research, and those, while correlated with ability to learn, are not quite the same. Those are also way more context-dependent, which means more noise, both due to more variance and more ambiguity of judgment.

I will say that at times or even often, society is met with the problem of people finagling themselves into a position to judge what they are not really qualified to, per their ability and expertise, which means some resume-padding bozos rising up and actual competents being passed over. This problem I believe has been accentuated by the ever more credentialist culture that has emerged over the recent years. What’s kind of sad is how the more conformism and risk-aversion rises, the more these traits are pressured and selected for.

I’ve come to notice that there tends to be some difference between maverick genius and the conformist first-rate professional. If one looks at history, real genius, the ones who create paradigm shifts, tends to have more very lopsided profiles, though surely, it might go too far to say that *most* of the real geniuses were out of it in a Stallman or Galois like fashion, especially as it’s the deranged ones which garner more attention. But one can say with high level of confidence that there were many real geniuses who had a hard time fitting in even into the elite mainstream of his profession, who have even been marginalized. I’ve been told that the real genius mathematicians like Perelman, Langlands, and Shimura more or less cut contact with the mathematical community apparently out of disgust. There is also evidence that plenty would-be real geniuses did not actually make it, with their enormous potential having been thwarted by the system at some point and hardly realized. In an ideal world that optimizes for collective value, if somebody else can do the job much better than you and actually really wants to, you should let him do the job and get out of his way. Of course, reality is far from that. I have personally felt that way with regard to my mathematical ability, often feeling that I wasn’t good enough when I failed to derive something on my own, yet I see so many people worse than I am even so eager to play the whole credentialist game without recognizing how deficient they really are. This suggests that I am very partial towards a certain side of the spectrum. I even feel that in some sense, nothing is more embarrassing then formally being much higher than what one’s ability actually merits, since it demonstrates not only incompetence but poor character. However, I am, regrettably, or not, feeling that circumstances are pressuring me ever more towards the opposite direction.

On grad school, science, academia, and also a problem on Riemann surfaces

I like mathematics a ton and I am not bad at it. In fact, I am probably better than many math graduate students at math, though surely, they will have more knowledge than I do in some respects, or maybe even not that, because frankly, the American undergrad math major curriculum is often rather pathetic, well maybe largely because the students kind of suck. In some sense, you have to be pretty clueless to be majoring in just pure math if you’re not a real outlier at it, enough to have a chance at a serious academic career. Of course, math professors won’t say this. So we have now an excess of people who really shouldn’t be in science (because they much lack the technical power or an at least reasonable scientific taste/discernment, or more often both) adding noise to the job market. On this, Katz in his infamous Don’t Become a Scientist piece writes:

If you are in a position of leadership in science then you should try to persuade the funding agencies to train fewer Ph.D.s. The glut of scientists is entirely the consequence of funding policies (almost all graduate education is paid for by federal grants). The funding agencies are bemoaning the scarcity of young people interested in science when they themselves caused this scarcity by destroying science as a career. They could reverse this situation by matching the number trained to the demand, but they refuse to do so, or even to discuss the problem seriously (for many years the NSF propagated a dishonest prediction of a coming shortage of scientists, and most funding agencies still act as if this were true). The result is that the best young people, who should go into science, sensibly refuse to do so, and the graduate schools are filled with weak American students and with foreigners lured by the American student visa.

Even he believes that now the Americans who go into science are often the ones who are too dumb or clueless to realize that they basically have no future there. I can surely attest to how socially inept, or at least clueless, many math grad students are, as I interact with them much more now. The epidemic described by Katz is accentuated by the fact that professors in science are not encouraging of students who seek a plan B, which everyone should given the way the job market is right now, and even go as far as to create an atmosphere wherein even to express a desire to leave academia is a no-no. I am finding that this type of environment is even corroding my interest in mathematics itself, which is sad. In any case, I sort of disagree with Katz in that I feel like the very top scientific talent of my generation still mostly ends in top or at least good graduate schools, though surely there are many who feel alienated or don’t find the risk worth taking, and end up leaving science. I myself am thinking of forgetting about mathematics altogether. So that I can concentrate my motivation and time and energy on developing expertise in some area of software engineering that is in demand, for the money and (relative) job security, and hopefully also find it a sufficiently fulfilling experience. There are a lot of morons in tech of course, but certain corners of it do provide refuge. I had always thought of mathematics as being a field with a much higher threshold cognitively in its content, enough to filter out most of the uninteresting people, but that’s, to my disappointment, less so than I expected. I do have reason to be scared, because one of the smartest and most interesting people I know took like five years following his math PhD to make his way into full employment, in a programming/data science heavy role of course, despite being arguably much better at programming than most industry software engineers with a computer science degree, which he lacked, an indicator of the perverse extent to which our society now runs on risk-aversion and (artificial) credential signaling. I can only consider myself fortunate that I do have a computer science degree from a reputable place, and with that, I have already made a modest pot of gold, despite being frankly quite mediocre at real computer stuff, which I have had difficulty becoming as interested in as I have been in mathematics. Maybe I was even fortunate to have not been all that gifted in the first place, which in some sense compelled me to be more realistic, as there is arguably nothing worse than becoming an academic loser, which academia is full of nowadays, sadly. This type of thing can happen to real geniuses too. Look at Yitang Zhang for instance, the most prominent case to come to mind. Except he actually made it afterwards, spectacularly and miraculously, with his dogged belief in himself and perseverance under adversity. For every one of him, I would expect like 10 real geniuses (in ability) who were under-nurtured, under-recognized, or even screwed, left to fade into obscurity.

I’ll transition now to a problem that I’ve been asked to solve. Its statement is the following:

Let f be holomorphic on a simply-connected Riemann surface M, and assume that f never vanishes. Then there exists F holomorphic on M such that f = e^F. Show that harmonic functions on M have conjugate harmonic functions.

Every p_0 \in M corresponds to an open connected neighborhood U =  \{p : \lVert F(p) - F(p_0) \rVert < F(p_0)\}. Let \{U_{\alpha}\} be the system consisting of these neighborhoods, (\log F)_{\alpha} a continuous branch of the logarithm of F in U_{\alpha}. From this arises a family F_{\alpha} = \{(\log F)_{\alpha} + 2n\pi i, n \in \mathbb{Z}\}.

In Schlag, there is the following lemma.

Lemma 5.5. Suppose M is a simply-connected Riemann surface and

\{D_{\alpha} \subset M : \alpha \in A\}

is a collection of domains (connected, open). Assume further that these sets form an open cover M = \bigcup_{\alpha \in A} D_{\alpha} such that for each \alpha \in A there is a family F_{\alpha} of analytic functions f : D_{\alpha} \to N, where N is some other Riemann surface, with the following properties: if f \in F_{\alpha} and p \in D_{\alpha} \cap D_{\beta}, then there is some g \in F_{\beta} so that f = g near p. Then given \gamma \in A and some f \in F_{\gamma} there exists an analytic function \psi_{\gamma} : M \to N so that \psi_{\gamma} = f on D_{\gamma}.

Using the families of analytic function F_{\alpha} as given above, it is clear that near p \in D_{\alpha} \cap D_{\beta}, (\log F)_{\alpha} + 2n_{\alpha}\pi i = (\log F)_{\beta} + 2n_{\beta}\pi i when n_{\alpha} = n_{\beta}, which means the hypothesis of Lemma 5.5 is satisfied by the above families.

I’ll present the proof of the above lemma here, to consolidate my own understanding, and also out of its essentiality in the construction of a global holomorphic function matching some function in each family. It does so in generality of course, whereas in the problem we are trying to solve it is on a specific case.

Proof. Let

\mathcal{U} = \{(p, f) | p \in D_{\alpha}, f \in F_{\alpha}, \alpha \in A\} / \sim

where (p, f) \sim (q, g) iff p = q and f = g in a neighborhood of p. Let [p, f] denote the equivalence class of (p, f). As usual, \pi([p, f]) = p. For each f \in F_{\alpha}, let

D'_{\alpha, f} = \{[p, f] | p \in D_{\alpha}\}.

Clearly, \pi : D_{\alpha, f}' \to D_{\alpha} is bijective. We define a topology on \mathcal{U} as follows: \Omega \subset D_{\alpha, f}' is open iff \pi(\Omega) \subset D_{\alpha} is open for each \alpha, f \in F_{\alpha}. This does indeed define open sets in \mathcal{U}: since \pi(D'_{\alpha, f} \cap D'_{\beta, g}) is the union of connected components of D_{\alpha} \cap D_{\beta} by the uniqueness theorem (if it is not empty), it is open in M as needed. With this topology, \mathcal{U} is a Hausdorff space since M is Hausdorff (we use this if the base points differ) and because of the uniqueness theorem (which we use if the base points coincide). Note that by construction, we have made the fibers indexed by the functions in F_{\alpha} discrete in the topology of \mathcal{U}.

The main point is now to realize that if \widetilde{M} is a connected component of \mathcal{U}, then \pi : \widetilde{M} \to M is onto and in fact is a covering map. Let us check that it is onto. First, we claim that \pi(\widetilde{M}) \subset M is open. Thus, let [p, f] \in \widetilde{M} and pick D_{\alpha} with p \in D_{\alpha} and pick D_{\alpha} with p \in D_{\alpha} and f \in F_{\alpha}. Clearly, D'_{\alpha, f} \cap \widetilde{M} \neq \emptyset and since D_{\alpha}, and thus also D'_{\alpha, f}, is open and connected, the connected component \widetilde{M} has to contiain D'_{\alpha, f} entirely. Therefore, D_{\alpha} \subset \pi(\widetilde{M}) as claimed.

Next, we need to check that M \setminus \pi(\widetilde{M}) is open. Let p \in M \setminus \pi(\widetilde{M}) and pick D_{\beta} so that p \in D_{\beta}. If D_{\beta} \cap \pi(\widetilde{M}) = \emptyset, then we are done. Otherwise, let q \in D_{\beta} \cap \pi(\widetilde{M}) and pick D_{\alpha} containing q and some f \in F_{\alpha} with D'_{\alpha, f} \subset \widetilde{M} (using the same “nonempty intersection implies containment” argument as above). But now we can find g \in F_{\beta} with the property that f = g on a component of D_{\alpha} \cap D_{\beta}. As before, this implies that \widetilde{M} would have to contain D'_{\beta, g} which is a contradiction.

To see that \pi : \widetilde{M} \to M is a covering map, one verifies that

\pi^{-1}(D_{\alpha}) = \bigcup_{f \in F_{\alpha}} D'_{\alpha, f}.

The sets on the right-hand side are disjoint and in fact they are connected components of \pi^{-1}(D_{\alpha}).

Since M is simply-connected, \widetilde{M} is homeomorphic to M (proof given in the appendix). We thus infer the existence of a globally defined analytic function which agrees with some f \in F_{\alpha} on each D_{\alpha}. By picking the connected component that contains any given D_{\alpha, f}' one can fix the “sheet” locally on a given D_{\alpha}.     ▢

By this, we can construct an analytic F such that for all \alpha,

f_{|U_{\alpha}} = (\log F)_{\alpha} + n_{\alpha} \cdot 2\pi i, \qquad n_{\alpha} \in \mathbb{Z}.

from which follows e^F = f.

For the existence of harmonic conjugates, we do similarly. Take a connected open cover of M, \{U_{\alpha}\} where each U_{\alpha} is conformally equivalent to the unit disc, and v_{\alpha} is a harmonic conjugate of u in U_{\alpha} (which exists uniquely up to constant on the unit disc. Let F_{\alpha} = \{v_{\alpha} + c, \quad c \in \mathbb{R}\}. Then by the same lemma, there exists v such that for all \alpha,

v_{|U_{\alpha}} = v_{\alpha} + c_{\alpha}, \quad \text{some } c_{\alpha} \in \mathbb{R}

that is harmonic and conjugate to u since it is the harmonic conjugate to u on every element of the cover, again with choise of c_{\alpha}s to ensure that on intersection of cover elements there is a match.