Upcoming TV series about Emperor Wu to be the most expensive TV series made in Chinese history

An upcoming series chronicling the life of China’s only female emperor, Emperor Wu Zetian of the Zhou Dynasty (she was the only emperor of that dynasty, the emperor to come before and after her were all emperors of the more famous Tang Dynasty), is set to become the most expensive TV series ever made in China (http://en.wikipedia.org/wiki/The_Empress_of_China). The TV series will star Chinese megastar Fan Bingbing, and is also produced by her company.

The story of Emperor Wu is one of the most famous in the history of China. She seemed to have done the impossible: she assailed the entrenched sexism and misogyny embedded in feudal China and ascended to the title of Emperor, making her both the nominal and actual leader of China. It is noted that only three women in Chinese history has wielded as much power, Wu Zetian being the second. The other two were Empress Lü Zhi of the Han Dynasty, wife of Emperor Gaozu, the founder of the Han Dynasty and Empress Dowager Cixi, the wife of Emperor Xianfeng and mother of Emperor Tongzhi of the Qing Dynasty. However, the other two never managed to claim the title of Emperor for themselves, instead having to rule from ‘behind the curtain’. 

Much of China’s history, understandably, is written by men and for men. It was noted earlier that in fact Chinese historians have a habit of writing towards a very specific audience: future rulers. Thus Chinese history is written as a warning to future emperors of the challenges they must conquer in order to hold onto power. The most deleterious of these challenges is undoubtedly guarding against the women you trust most. It is no surprise then that all of the above strong women who managed to climb to the peak of power were later demonized in the annals of Chinese history. Both Lü Zhi and Wu Zetian were ‘renowned’ for their immense cruelty. For instance, it was said that Lü Zhi turned Gaozu’s favored concubine, Mistress Zhao, into a human ‘stump’ by cutting off her limbs and mutilating her face and then blinding her, but kept her alive. Wu Zetian was said to employ ‘cruel ministers’ to administer extraordinary acts of cruelty and torture against political opponents and ruled by a campaign of terror. Cixi is widely regarded by modern Chinese history as the principal architect of the Qing Empire’s downfall, and the subsequent humiliation that the Chinese civilization suffered in the 20th century.

However, in more recent times, some familiar narratives of history have been challenged and re-imagined. One such case is that of the Qin Dynasty. Historically noted for their immense cruelty and extreme legal system and anti-Confucianism, the recent work 大秦帝国 has sought to re-tell the story of the rise of the Qin Empire, previously the Qin Kingdom in the Warring States Period, in a positive light. It sought boldly to re-imagine the Chinese civilization as having been born from a warrior nation (Qin), as opposed to a relatively meek and passive nation (context: after uniting China and establishing the Han Dynasty, Emperor Gaozu and his successors Wendi and Jingdi sought a policy of appeasement and compromise against the barbarians to the north, which was finally reversed by Emperor Wudi, who ordered several major military campaigns against China’s northern neighbors). The same has happened for the story of Wu Zetian, where more recent TV series have depicted her in a much more positive light, as a strong and wise leader far outweighing the weak Tang Emperors she replaced.

Underlying reasons for the unrest in Hong Kong

As some of you know, since earlier this year, there is civil unrest in Hong Kong centered around the so-called “Occupy Central” (referring to the Central Business district or 中环 in Chinese, not the Central Government of China, as several people I discussed this with mistakenly assumed) movement. The movement’s aims are to fight for true universal suffrage in 2017. While the current proposal drafted and now approved by the National People’s Congress offers universal suffrage to select the city’s leader, the Chief Executive, it also stipulates that all candidates must be approved by a majority the 1200 member Election Committee which contains many Beijing supporters, effectively ensuring that only those loyal to Beijing will be allowed to run for the top office. This is deemed unacceptable by the so-called pan-democratic persons in Hong Kong. The main goal of Occupy Central is to remove this stipulation in the 2017 election and allow anyone who chooses to run to be deemed a candidate.

On the surface, this seems like the same story that has happened again and again throughout history. Even the organizers behind Occupy Central have compared their proposed campaign of civil disobedience to the likes of Martin Luther King and Nelson Mandela, with the goal of opposing a tyrannical and oppressive regime and unfair treatment of people. This may very well be true, but there are other factors that run deep beneath the surface.

One has to understand that Hong Kong, aside from being a British colony for a hundred years and a place where the residents are used to a British style of government and way of life, it is also a prime destination for mainland refugees fleeing from various situations in China after the establishment of Communist China in 1949. Many of the refugees were members of the bourgeois or land owner classes, which were mercilessly persecuted by Communist China. Others escaped the famine caused by the Great Leap Forward and the fearful period of the Cultural Revolution. The point is, many of the people in Hong Kong have a deep grudge against the Communist Party of China, as their own families have suffered greatly at the hands of China’s current rulers. They may not necessarily be pro-democracy, but they are definitely anti-Communist.

This is combined with a more delicate social history. Many of the people who fled to Hong Kong have found their way to a better way of life and prosperity after leaving their homes. There is a feeling of vindication, and definitely superiority, towards China. While the Mainland languished in the Cultural Revolution, Hong Kong boomed as a financial and economic hub. By the time China opened up its borders in 1978 under Deng Xiaoping’s economic reforms, Hong Kong was already a modern and cosmopolitan city. It is not even a secret that Hong Kongers widely feel they are superior to mainlanders, deeming mainlanders uncouth and uncivilized. For a long time, economic reality reflected much the same. This changed, however, in the past decade, when China’s unprecedented economic rise has placed it higher on the economic spectrum than Hong Kong. While Hong Kong enjoyed consistent economic growth since rejoining China in 1997, the growth rate is much slower than in the mainland. While in the late 90’s and 2000’s China’s economy boomed at double digit growth rates, Hong Kong’s economy grew at a relatively modest 4-5% range. This gap in growth rate has shifted economic realities. Now many mainlanders north of the border enjoy much more wealth and a higher standard of living than many middle class members of Hong Kong. However, the attitude of superiority of Hong Kongers towards mainlanders has not changed; and this has caused much resentment.

Further, the policy differences between Hong Kong and the mainland have exerted a lot of pressure on Hong Kong’s social fabric. Since Hong Kong’s laws are different than in the mainland, and usually much more transparent with a more or less independent judiciary, it is considered essentially a ‘foreign’ land in terms of investments. Thus many wealthy mainlanders see fit to invest in Hong Kong real estate, just as they would buy real estate in hot foreign markets like Vancouver or Australia, as a secure way to park their money. This has driven up prices in the most densely populated place in the world, dashing the dreams of home ownership for many middle class Hong Kongers. Undoubtedly this reality has caused significant resentment among Hong Kong’s populace. Secondly, because Hong Kong citizens enjoy a special status in the context of the People’s Republic of China, Hong Kong citizenship is deemed highly desirable. Many mainland pregnant women have essentially snuck past the border to give birth in Hong Kong, making their offspring Hong Kong citizens. This has caused undue stress on Hong Kong’s hospitals to handle women going through labour, and have created resentment as well. Further, Hong Kong is a place where foreign products are widely available; and things like baby formula are highly desirable for mainland Chinese (context: about 6 years ago there was a so-called ‘tainted milk’ scandal in China where bad baby formula caused death and serious injury or mental impairment for over 40,000 babies, thus scaring people off domestic brands). This has created shortage in Hong Kong.

My point is, the current turmoil is not as simple as a bunch of freedom-minded people fighting for rights and freedoms. It is a continuation of a long history of mistrust and hatred towards China’s ruling party, and continued economic tensions between the mainland and Hong Kong.

My thoughts on what makes a good thesis problem

The single most important decision of a PHD student in (pure) mathematics is undoubtedly the choice of research in their thesis work. Usually to get a PHD, one must prove at least one new result. We will refer to the achievement of this situation as ‘solving a problem’. Thus, we may adapt the ‘problem solver’ (see here) view that a PHD is centered around solving a thesis problem.

I have made the following comments in person to many people, but I thought I would record them here as well.

When selecting a thesis problem, there are many factors to be considered, some of them quite deleterious. Firstly, mathematics is, contrary to popular belief, a social endeavor. Mathematicians do not work in isolation. As the recent controversy involving Shinichi Mochizuki and his purported proof of the abc-conjecture has shown, a problem is not solved and a result not proven until there is consensus among the mathematical community that it is. Therefore, progress or results that are not engaged with the mathematical community basically do not exist; just like the old paradigm involving trees falling in the woods. Thus one must ensure that their work is in sync with the current mathematical culture. One should therefore not choose a problem that is so far off in the fringes that few, if anybody, is aware of its existence and fewer still have any vested interest in whether it is resolved or not.

However, one should not pick something that is too hot or attractive, relative to its difficulty. Perhaps the best example of this would be the bounded gaps between primes problem. Prior to May 2013, the problem was considered intractable, and few people dedicated any amount of time to solving it. However when Yitang Zhang announced that he had proved that there exist infinitely many pairs of consecutive primes that differ by no more than 70 million, he sent shock waves around the mathematical world. Soon there was a flurry of results improving upon Zhang’s work in various ways. It would have been an extraordinarily bad idea to choose any results directly related to Zhang’s work as a thesis problem. There were literally dozens of top level experts around the world working on this topic, and the probability that your result will be scooped is very high. Thus, one must pick a problem that is either not a very obvious one to ask but is nonetheless interesting, or one that is interesting but not so interesting that it will attract the attention of many experts.

The final, and perhaps most subtle point, is that one should not pick a problem that is conducive to an ad-hoc solution that does not generalize. The best example of this is perhaps the so-called ‘elementary’ proof of the prime number theorem. In this case, while it was extraordinary that the prime number theorem, originally proved using arguments from complex analysis, could be proved in an elementary way, the proof fell drastically short of expectations. In particular, it was widely expected that an elementary proof of the prime number theorem would shed light on the distribution of primes and perhaps even pave the way to proving the Riemann hypothesis. The elementary proof of the prime number theorem did no such thing and to this day the technique remains restricted to only being able to prove the prime number theorem. One does not want a thesis problem like this. Instead, ideally during the course of solving one’s thesis problem, one discovers that the same techniques and machinery can be applied to a plethora of other problems and thus pump out a consistent stream of papers after and establish a solid research program. Whether or not this would be the case is extremely difficulty to predict before one actually attempts to solve the problem seriously.

Usually as a new PHD student who has limited understanding of your subject area and have limited perspective on the landscape of your research domain, it is extremely important to find an advisor who does have the right expertise and perspective. Even then it is a very delicate process to select a thesis problem which you would be expected to work on for several years.

Some statistics for hearthstone

This took a bit longer than I would care to admit, but I have finally computed the expected gain of stars per game. The answer depends a little bit on information available and interpretation.

Let p denote the probability of you winning a game, which is considered to be constant. If you are on a win streak, meaning you’ve won the previous two games, then the expected winning of your next game is 2p - 1(1-p) = 3p-1. The probability that you’ve won your previous two games is p^2, so this case contributes p^2(3p-1) to the total expected value. Otherwise, the expected winning is p - (1-p) = 2p-1. The probability of this is 1 - p^2, so the contribution for this case is (1-p^2)(2p-1). Summing, we obtain that the total expected value of the gain of stars is

3p^3 - p^2 + 2p - 1 - 2p^3 + p^2 = p^3 + 2p - 1.

Thus, if you win half the time (so that p = 1/2), you should expect to gain roughly one star every 8 games or so.

Another Hearthstone related problem, which I worked out with my officemate, is to ask what is the probability of you getting the card you need on turn 1 assuming you are going first and that you are willing to discard all other cards in your initial draw in order to get the card that you want. First, the probability that the card you need will appear in the initial draw of three cards is

\displaystyle \frac{\binom{2}{1} \binom{28}{2} + \binom{2}{2} \binom{28}{1}}{\binom{30}{3}}.

If you did not get either copy of the card that you need, then you may discard all three cards and try again, knowing that the three cards you tossed cannot come back. In this case, the probability of you getting the card you need is the product of the probability of you not getting the card you want in the initial draw and the probability of getting at least one copy of the card that you need in the second draw.

\displaystyle \frac{\binom{28}{3}}{\binom{30}{3}} \cdot \frac{\binom{2}{1} \binom{25}{2} + \binom{2}{2} \binom{25}{1}}{\binom{27}{3}} .

Adding these, we get some pretty neat cancelations and end up with the simple expression of p = 1/5.

However, we have not yet accounted for the random card that you top deck after your initial draw on turn 1. The only remaining case that needs to be considered is when you failed to get the card you need after tossing all of your cards on the initial draw. In this case, the probability of getting the card you need is

\displaystyle \frac{\binom{27}{3}}{\binom{30}{3}} \cdot \frac{\binom{25}{3}}{\binom{27}{3}} \cdot \frac{2}{27}

which is about 0.041963. Adding, the total probability is about 0.241963, or slightly less than one in four.

习大是汉武帝还是雍正爷(清世宗)?

第一次用中文在此博客打文章,如果语法不通请见谅。一下意见纯属给人。

放眼望天下,都可感受到现在是个动荡时期。中国久违的崛起近在眼前。新任领导人习近平明显的要成为毛泽东和邓小平之后的下一位伟大领袖。本人曾经多次说中国共产党其实就是中国第十个“王朝“。那么说,个代领导人则是君王。当然,现代毕竟不是家天下,所以不可能有像康熙乾隆之类的能从小当到老的那样统治六十年,所以现代的”皇帝“只有十来年的统治生涯。尽管如此,仍然能与前朝君主做对比。

毛泽东毛主席,毫无疑问的是本朝开国之君,所以封他为”太祖“当仁不让。历代来说,当太祖的未必能把国家管理好。汉高祖刘邦推行的无为之治被后人推翻,因为过于消极。唐太祖李渊威德不够,被次子李世民取而代之。明朝也是第二任皇帝明成祖朱棣(按理来说是第三任)把它推向顶峰地。

那么如果把本朝与汉朝比的话,那邓小平应该就是汉文帝,江泽民/胡锦涛是汉景帝之类。。。在任时通过”文景之治“把中国国力搞起来了。但是,跟文帝景帝一样,他们并没有彻底让中国真正站起来,以雪前一百年所受的种种不平等待遇和国耻。那么习近平则可被看为汉武帝,因为他勇于铲除前朝的腐败和政治堵塞并强积极于增强军队力量,以强硬的态度对待一向不把中国看在眼里而任意欺辱的西方列强和日本。

如果拿清朝来做对比的话,习近平也有可能是雍正。如果把邓小平/江泽民/胡锦涛都算上是”康熙“的话,那么他们所达到的是一个基本上国泰民安的状况,但是宽仁的政治同时滋生了无数的贪腐和败坏的风气,需要一个刚正不阿的君主来清理。习近平有效并且大规模的反贪反腐则可湊美雍正年代。再者,习近平对文字和言论也管的较严,有点类似雍正年代的文字狱。

不论武帝也好,世宗也罢,中国后面十年的前景是很好地。

A beautiful answer to why pure mathematics matters

Credit goes to Khalid Bou-Rabee, who posted this on MathOverflow, and Benson Farb, to whom the quote belongs.

“Since I am a pure mathematician, Dean Hefley suggested as a possible topic for this talk: “Why the square root of negative 1 is necessary”. I could take up this challenge of justifying pure science on its vast applicability; indeed the square root of negative 1, the basic “imaginary number”, underlies a huge swath of modern technology, from the design of circuits, airplanes and skyscrapers, to the construction of economic and financial models, to robotics. I have decided, however, to take the opposite point of view. I want to defend the value of basic science for its own sake…

…the purpose of pure mathematics, of basic science, is not the quick harvest. It is nothing less than an attempt to bring human thought and understanding to a higher level. It is an attempt to change not just what we think about the world, but how we think about it. The importance of this for human evolution is incalculable. As British physicist JJ Thomson said: “Research in applied science leads to reforms, research in pure science leads to revolutions.”

- Benson Farb, 2012

Some gems from analytic number theory

I am currently attending the Counting Arithmetic Objects summer school at Centre de Recherches Mathematiques in Montreal. During the second day of lectures I heard a remarkable talk from Professor Andrew Granville. He spoke with outstanding clarity on the basic aim of analytic number theory, and I wish to remark on some of the epiphanies I had during the talk.

Without a doubt, one of the most important identities in analytic number theory is the following:

\int_0^1 e^{2\pi i n t}dt = \begin{cases} 1, & \text{ if } n = 0 \\    0, & \text{ if } n \ne 0 \end{cases}

The key insight is that the above identity serves as an “indicator function” for whether a quantity is zero or not. Indeed, this basic observation is behind behind the solutions to Waring’s problem and the proof of Vinogradov’s theorem on the sum of three primes. This “smooth” indicator function allows the use of methods in analysis to deal with arithmetic problems.

A related identity is Perron’s formula. Here again we start with an “indicator” function, defined for c > 0

\frac{1}{2\pi i} \int_ {c-i\infty}^{c+i\infty} \frac{y^s}{s} ds = \begin{cases} 1, & \text{ if } y > 1 \\ 0, & \text{ if } y < 1 \end{cases}

This allows to handle sums of the shape

\displaystyle \sum_{n \geq 1} a_n.

To see this, we start with the observation that for each n in the range of summation we have (provided x is not an integer) that x/n > 1. Hence we have

\displaystyle \sum_{n \leq x} a_n = \sum_{n \leq x} a_n \frac{1}{2 \pi i } \int_{c - i\infty}^{c + i\infty} \left(\frac{x}{n} \right)^s \frac{1}{s} ds.

The outer sum is finite, but we would not change the sum at all if we included all positive integers n > x because of the indicator function nature. For well behaved a_n‘s and taking c sufficiently large to ensure absolute convergence, we have that

\displaystyle \sum_{n \geq 1} a_n \frac{1}{2 \pi i} \int_{c - i\infty}^{c + i\infty} \frac{1}{s} \left(\frac{x}{n}\right)^s ds = \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} \left(\sum_{n\geq1} \frac{a_n}{n^s}\right) \frac{x^s}{s} ds.

We note that g(s)= \displaystyle \sum_{n \geq 1} \frac{a_n}{n^s} is the Dirichlet series associated to the a_n‘s. Summarizing, we obtain the equation

\displaystyle \sum_{n \leq x} a_n = \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} g(s) \frac{x^s}{s}ds.

Now we can apply this to the familiar Von Mangoldt function \Lambda(n) defined by

\displaystyle \Lambda(n) = \begin{cases} \log p, & \text{ if } n = p^m \\ 0, & \text{ otherwise.} \end{cases}

where as usual p denotes a prime. Now using Perron’s formula, we have

\displaystyle \sum_{n \leq x} \Lambda(n) = \sum_{p^m \leq x} \log p = \frac{1}{2 \pi i} \int_{c - i\infty}^{c + i\infty} - \frac{\zeta'(s)}{\zeta(s)} \frac{x^s}{s}ds

Where \zeta(s) denotes the Riemann zeta function. This gives rise, after evaluating the integral on the right using residue theory and noting that the poles of \zeta'(s)/\zeta(s) are 1 and the zeroes of \zeta, we have

\displaystyle \sum_{p^m \leq x} \log p = x - \sum_{\rho: \zeta(\rho) =0} \frac{x^\rho}{\rho}+\frac{\zeta'(0)}{\zeta(0)}.

This is likely one of the most striking equations in mathematics as the left hand side is a discrete sum while the right hand side involved values of a meromorphic function. Somehow, the zeroes of the zeta function “knows” where the primes are.