A revisit of the drama behind the Poincaré

I recall back in 2008, when I first cared enough to learn about mathematicians, I read a fair bit of the media articles on the proof of the Poincaré conjecture. At that time, I was clueless about math, and these mathematicians seemed to me like these otherworldly geniuses. I do remember thinking once to myself that maybe it would be kind of cool to part of that world. Except at that time, I was way too dumb, and maybe I still am. However, now I actually have some idea of what math research is about, unlike back then, when my conception of math and mathematicians was more of a naive popular one.

Naturally, from that I learned about Shing-Tung Yau. I probably read that Manifold Destiny article by Sylvia Nasar and David Gruber that Yau was furious with, in response to which he hired a lawyer and had a PR site created for him to counter the libel (as perceived by him). That was pretty entertaining to read about.

The more I learned about math, about mathematicians, about how the world works, about the Chinese math establishment, and about Chinese language (which I’m pretty fluent with by now), the more accurately and deeply I could understand and thus appreciate all this. In particular, now that I know a little about Riemann surfaces, I feel closer to that rarefied world. I also read a fair bit in Chinese about that feud between Yau and Tian, which was also quite entertaining. If some of that stuff is actually true, then academia, even in its supposedly purest, hardest, and more meritocratic subject, is kind of fucked up.

Yesterday, I had the pleasure of talking with a Harvard math undergrad who is also an IMO gold medalist. And we both mentioned Yau. 😉

What can I say about all the politics and fight for credit over whole Poincaré conjecture? Surely, it was kind of nasty. It’s fair to say that Yau was pissed (or at least disappointed) that his school (of Chinese mathematicians) lost to this lone Russian Jew. Maybe in some years time, I’ll be able to judge for myself, but for now, it seems like Perelman’s proof was correct from the start and that what Cao and Zhu, along with the other two teams of two did were merely verification and exposition of Perelman’s result. Of course, attributing a proof entirely to an individual is somewhat misleading, because anyone who knows how math works knows that any proof of a big theorem employs sophisticated machinery and theory developed by predecessors. I’ve studied enough math now to recognize to some degree the actual substance, that is, what is genuinely original, versus what is merely derivative. In the case of Perelman, they say he was using the Ricci flow developed by Hamilton. I’ve encountered many times that in learning, it is much harder to learn about a topic I have little exposure to vastly different from anything I’ve seen before than to learn what is structurally similar (albeit different in its presentation and perhaps also level of generality) to something I had thought about deeply myself already, or at least seen.

Aside from the Poincaré, the focus of that New Yorker article, the authors of it also made it seem as if Lian, Liu, and Yau stole Givental’s proof of mirror symmetry as well about a decade earlier. After all, Givental published first. I suspected that might have been the case. The narrative even made it seem somewhat like Givental was this super genius whose arguments were somewhat beyond the comprehension of Lian, Liu, and Yau, who struggled to replicate his work. Maybe because I still see, or at least saw, Jews as deeper and more original than Chinese are. Again, I still know too little, but it does seem like Jews have contributed much more to math at the high end even in recent years, say, the past three decades.

Well, I found a writing on that doctoryau website by Bong Lian and Kefeng Liu documenting the flaws and deficiencies in Givental’s paper. It looked pretty thorough and detailed, with many objections. The most memorable one was

p18: Proposition 7.1. There was just one sentence in the proof. “It can be obtained by a straightforward calculation quite analogous to that in ‘[2]’.” Here ‘[2]’ was a 228-page long paper of Dubrovin.

And I checked that that was indeed true in Givental’s paper. This certainly discredited Givental much in my eyes. It’s like: how the fuck do you prove a proposition by saying it’s a straightforward calculation analogous to one in… a 228-page paper!!!!!!!!

Not just that. There is also

p27: Proposition 9.6. In the middle of its proof, a sentence read “It is a half of the geometrical argument mentioned above.” It’s not clear what this was referring to (above where? which half?)


p30: Proposition 9.9. This was about certain uniqueness property of the recursion relations. The proof was half a sentence “Now it is easy to check” But, again since we couldn’t check, it’s hard to tell if it was easy or not

So basically at least three times Givental proves with “it’s trivial,” once based on analogy with a 228-page paper.

There are far from all. There are many more instances of Givental’s arguing what Lian-Liu-Yau could not follow, according to that document, the list in which is also, according to its authors, who advise strongly the reader to “examine Givental’s paper make an informed judgment for himself”, “not meant to be exhaustive.” So they’ve listed 11 gaps in that paper, one of which is glaringly obvious of a rather ridiculous nature even to one who knows not the slightest about mathematics! And they suggest there is more that, to my guess, may be much more minor that they omitted in that document so as to avoid dilution.

I’ve noticed it’s often the Chinese scientists who have a bad reputation for plagiarism, made more believable by the dearth of first-rate science out of Chinese scientists in China, though that seems to be changing lately. On the latter, many Chinese are quite embarrassed about their not having won a homegrown Nobel Prize (until Tu Youyou in 2015 for what seemed to be more of a trial-and-error, as opposed to creative, discovery) or Fields Medal. On the other hand, I’ve also heard some suspicions that it’s the Jews who are nepotistic with regard to tenure decisions and prize lobbying in science, and what Givental did in that paper surely does not reflect well. I used to think that math and theoretical physics, unlike the easier and more collaborative fields in STEM (with many working in a lab or on an engineering project), revere almost exclusively individual genius and brilliance, but it turns out that to succeed nowadays typically involves recommendations from some super famous person, at Connes attests to here (on page 32), not surprising once one considers the sheer scarcity of positions. Now I can better understand why Grothendieck was so turned off by the mathematical community, where according to him, the ethics have “declined to the point that outright theft among colleagues (especially at the expenses of those who are in no position to defend themselves) has nearly become a general rule.” More reason why I still hesitate to go all out on a career in mathematics. It can get pretty nasty for a career with low pay and probability of job security, and I could with my talents make much more impact elsewhere. One could even say that unequivocally, one who can drastically increase the number of quality math research positions (not ridden with too many hours of consuming duties not related to the research) would do more to progress mathematics than any individual genius.

I’ll conclude with some thoughts of mine on this Olympiad math that I’ve lost interest in that many mathematicians express low opinion of, though it clearly has value as a method of talent encouragement and selection at the early stage, with many Fields Medalists having been IMO medalists, usually gold. I recall Yau had criticized the system of Olympiad math in China, where making its version of MOSP gives one a free ticket to Beida and Qinghua, as a consequence of which many parents force or at least pressure their kids into Olympiad math prep courses as early as elementary school. Even there, several of the IMO gold medalists have become distinguished mathematicians. I have in mind Zhiwei Yun, Xinyi Yuan, and Xuhua He, all speakers at this year’s ICM. So the predictive power of IMO holds for the Chinese just as well as for the non-Chinese. I personally believe that Olympiad math is beneficial for technical training, though surely, the actual mathematical content in it is not that inspiring or even ugly to one who knows some real math, though for many gifted high schoolers, it’s probably the most exciting stuff they’ve seen. I do think though that one seriously interested in mathematics would have nothing to lose from ignoring that stuff if one goes about the actual math the right way.

It’s kind of funny. A few days ago when I brought up on a chat group full of MOSP/IMO alumni that now, almost half of the top 100 on the Putnam (HM and higher) are Chinese, one math PhD quite critical of math contests was like: “ST Yau would weep.” Well, I don’t think ST Yau actually regards Olympiad math as a bad thing (half tongue-in-cheek, I even remarked on that chat that doing math contests (as a high schooler) is much better than doing drugs). Many of the Olympiad/Putnam high scorers do quite well, and in some cases spectacularly so, in math research. One point I shall make about them is that they are, unlike research, a 100% fair contest. Moreover, the Putnam, which I placed a modest top 500 on, solving three problems, has problems which do not require specialized technical training as do the inequalities and synthetic geometry problems in Olympiad math that have elegant solutions. On that, I have wondered based on their current dominance of those contests: could it be that at the far tail, the Chinese (who did not actually create the scientific tradition themselves) are actually smarter than the others, including the Jews? Could it be that the Chinese are actually somewhat disadvantaged job placement and recognition wise in math academia out of a relative lack of connections and also cultural bias? What I saw in that sound and unobjectionable rebuttal of Givental’s paper, in contrast to what was presented in the media, only makes this hypothesis more plausible. I am not denying that Givental did not make a critical contribution to the proof of mirror symmetry. That he did, along with some other predecessors, seems to be well acknowledged in the series of papers by Lian-Liu-Yau later that actually gave the first rigorous, complete proof of mirror symmetry. Idea wise, I read that Lian-Liu-Yau did something significant with so called Euler data, and though not qualified to judge myself, I have every reason to believe that to be the case for now.

Luboš Motl, and some thoughts on monopolies

I had the pleasure of reading some blog posts of Luboš Motl on present day academia. I first learned of him when I was a clueless undergrad. He seemed like this insanely smart theoretical physicist. Of course back then I was dumb and in awe of everything, so what else could I think? I know that he pissed off so many people that he was forced from resign from his tenure track position at Harvard physics in string theory. His academic work I am of course nowhere close to qualified to comment on, but people have said it’s first rate, and I’ll take their word. I even thought the guy was crazy. My very smart friend, in some online interaction with him, was scoffed off with: “You don’t understand vectors!” That guy later characterized the hypothetical combination of Luboš and this other guy I know, a PhD student in string theory, who is quite academically elitist and also so in terms of expecting good values and a fair degree of cultural/historical knowledge, as “a match made in heaven.” I also recall a commenter on Steve Hsu’s blog remark that Luboš has Aspergers syndrome or something like that. Anyhow, this time when reading the blog of Luboš, I no longer felt a sense of awe but rather a strong sense of clarity and reasonability in his thinking. He can be quite abrasive in some other contexts maybe, such as in his campaign against the climate change advocates (oh, on that I recently learned Freeman Dyson is also on the same side as Luboš on this one), but I believe it arises purely out of positive intentions on his part for the future of humanity, which many view as on a course of decline.

So the blog posts of Luboš read by me most memorable were on Scott Aaronson and feminism, a proposal for more political brainwashing requirements at Harvard, and Terence Tao’s silly mathematization of why Trump is not fit to be president respectively. On the first, I never knew Scott had followed the current feminist bandwagon. On the second, I’ve become more repulsed by and concerned with what I would characterize as absurd political notions (not matching with objective facts) held by many of elite school credentials, a sign that our elite selection is failing. On the third, I can’t believe Tao, a mathematician, would try to artificially mathematize a political matter. I would think that a mathematician would know better that substance trumps presentation in science.

Another friend of mine with a math PhD told me to my surprise a few years ago that now, we see many great scientists marginalized. I used to have the naive view that hard science fields like math and theoretical physics were almost entirely meritocratic and of a culture tolerant and supportive of independent, rational thinking and dismissive of the disingenuous marketing the norm in the business world, but now I increasing am doubting that, not that I deny at all that those fields are far better than the softer, less g-loaded areas of STEM, let alone non-technical subjects. It’s kind of sad that even mathematicians in high places like Aaronson and Tao are promoting such behavior with their prominent positions. So that friend of mine might be right on his somewhat of a verdict that the scientific community is in a catastrophic state right now.

I would say this is much owing to the scarcity of positions. Tenure is such a rare commodity nowadays that one who obtains it so often uses it to advance their political agenda, and sadly on that, it seems the bad guys are winning. Direct, honest, objective guys like Steve Hsu are few and fewer. Of course, different groups fighting for their own interests, for advancement of their own, be it their ethnic group, their political party, or their field of study, is deeply embedded in human nature and a necessity for survival. We now see in academia what in hyperbole are religious wars between different fields, different schools of thought, often in a manner that defies the so-called freedom of expression and thought that the university is in its ideal supposed to be for.

What I have just written holds within the theme of civilizational decline. On the matter of preservation of Western (white) civilization, my white American friend raised Christian remarked:

IDK the new divide is not “white vs nonwhite” it’s more like “people who have civilization worth preserving vs everyone else”

On that I asked with a chuckle: “what about Jews?” And he was like:

They have a country they should go there where they can’t parasitize everyone else

On that I recalled that my friend, another math PhD student, regards Jews more as a social class than as an ethnicity. He does have a point since as far as I know, the distinctiveness of Jews as an ethnic group is blurry in that they were this group in the Middle East with a religious culture of their own their seldom mixed with others despite often living amongst them. There, the leaks were more outwards with Jews converting to Christianity and thereby leaving permanently.

However, upper classes, especially ones in intellectual ability, within an ethnic group are still largely identified with and respected by the majority as emblematic of the group at large in some sense, which would contradict the aforementioned interpretation. I see that ordinary whites still view upper class whites as their own, as do ordinary Chinese with respect to intellectually elite Chinese, yet no other group really identifies with Jews the same way as far as I can tell.

Let me reiterate again that I, with many Jews I much respect and also some I talk to who have been major influences on me, am not anti-Semitic. Not that anti-X can be viewed as a binary variable. Lobos also said that in contrast, sex can be because there are X and Y chromosomes, so wise men think alike. 😉

I have commented before that

“Anti-Semitism” has become this political buzzword now. It basically is equivalent to anti-Jewish. So what? Many people in the world are also anti-Chinese, or anti-American, or anti-German, or anti-(any ethnic group or country), so what, they have the right to be, so long as they do not infringe too much. Also, keep in mind that anti-X is not binary; it’s very complex. Just like you almost never like or dislike everything about a person, you also can like certain things about a particular culture or people or country, and not like certain things.

I heartily believe that every group can be openly examined for their behavior as a collective. There is nothing wrong with that, and racist stereotypes are there for a reason after all. Pertaining to a specific one, Anti-Semitic conspiracy theorists (or most like cynical realists) might think that Jews want to absorb every competent group into their order so that they can have smart people working for them instead competing against them, and of course they will share power mostly amongst themselves.

Obviously, if you want to gain leverage over someone absorb him into your system make him dependent on you. We see this in international relations all the times. For example, in military technology, US and USSR created their own independent ecosystems, and many smaller countries had to more or less choose one or the other. There is a similar phenomenon in the software industry, with a very small number of widely used languages and frameworks. We’ve seen that many businesses are stuck with Microsoft once they use it for a while, and then there is a chain effect across the entire market.

We also see that Jews are also on top of arguably the premier credentialist hierarchy that is the Ivy League, with their accounting for arguably half or more of its presidents and senior administrators, and now people sort of need it to advance their career in America and even some other places, from which comes inevitably owing to our nature the political game of allotment of these scarce credentialist resources. Lately, Asians have realized by now that they can’t let Jews control too much of its distribution, favoring groups it fears not at the expense of those who pose more of a threat to themselves. On this, I have written that US higher education was and still is somewhat of a tool for cultivating (pseudo)-elite Chinese within an ecosystem wherein Jews have disproportionate influence. Chinese are a unique group in that they are intelligent, large, and a civilization and culture that emerged and evolved almost entirely independently of outsiders. (On the other hand, it is the modern science that Chinese are increasingly excelling at that is, in contrast, purely a product of Western civilization.) For this reason, Chinese have been very difficult if not impossible to absorb into any other system. Historically, even though the Mongols and Manchus had conquered China militarily, culturally they were much more absorbed into China than the other way round.

I believe cultural diversity (globally, not within every single country) is beneficial if not necessary for the overall health of human civilization. Referring back to the putative degraded state of US academia, Alain Connes, a French Fields Medalist, thinks the collapse of the Soviet science system, was catastrophic for science, since the USSR was a crucial counterweight to America. It was during the Cold War that was the golden period for STEM in America too, with Apollo 11 a climax. Now, with everybody absorbed into the American system sociologically, people are far less inclined to work on new things and instead play it safe in existent research programs, especially with grants and tenure-track, whereas in USSR in the research institutes, which he believes produced the best science, everyone basically had tenure from the start. That was quite an new and interesting perspective when I first saw it, and now, knowing more, I can see why he thinks that. Also, I think with China and Chinese, the mentality used to be, from the beginning of the reform and opening up, primarily one of how to gain approval from and integrate into what is globally prestigious along the (US-led) status quo, with say a sizable contingent obsessed with Ivy League, but that is taking a turn in the recent years now that China is far richer and more advanced than before. Still, one can say there was still back then a minority but one large enough to produce effect of talented people in China who thought all that prestige worship was silly and persisted in what they were doing to the extent that they gradually built more critical mass that while formerly much ignored by outsiders is now attracting ever more attention.

I’ve noted that different political factions and ethnic groups competing for resources for themselves will always be a thing, and one can think of scientific disciplines and schools of thought as political factions in some sense, which are in some cases even largely segregated by ethnic groups, with different countries having their own distinctive schools in various scientific disciplines. Sometimes, being too influenced by what others are doing and how others are thinking detracts from independent inquiry. Science in the long-term historical perspective values those who create new fields which turn out to be important. I have certainly seen the perspective that problem solvers in existent fields are a dime a dozen and it’s the theory builders who blaze new trails who are the real geniuses, one that resonates with me. For instance, the Greeks were the founders of the pure mathematics, and it was the step they took that was the more difficult and revolutionary, with Chinese civilization’s not having done so.

Politically in analogy, I admire the USSR for their having blazed a radically new trail that though ultimately unsuccessful, drastically altered the course of the 20th century and gave much to humanity in science and technology and the arts. Since China very successful today is in some sense an inheritor of the Soviet legacy, it surely hasn’t died out and is even rejuvenating. In contrast, I read on the Chinese QA site Zhihu an answer stating the proposition that after Qin Shihuang unified China in 212 BC, he forcibly made everything uniform across the whole country, burning books and burying scholars not in order with the official line of thought, enough that China as a civilization made little headway in intellectual thought for the next two millennia. Intellectuals only followed what was already there and could not escape it to create any tradition radically different, until superior forces without eventually forced change within.

The conclusion we can draw from all this is that monopoly of a form that discourages radically new ideas and development of alternative systems is detrimental to the advancement of human civilization.

Variants of the Schwarz lemma

Take some self map on the unit disk \mathbb{D}, f. If f(0) = 0, g(z) = f(z) / z has a removable singularity at 0. On |z| = r, |g(z)| \leq 1 / r, and with the maximum principle on r \to 1, we derive |f(z)| \leq |z| everywhere. In particular, if |f(z)| = |z| anywhere, constancy by the maximum principle tells us that f(z) = \lambda z, where |\lambda| = 1. g with the removable singularity removed has g(0) = f'(0), so again, by the maximum principle, |f'(0)| = 1 means g is a constant of modulus 1. Moreover, if f is not an automorphism, we cannot have |f(z)| = |z| anywhere, so in that case, |f'(0)| < 1.

Cauchy’s integral formula in complex analysis

I took a graduate course in complex analysis a while ago as an undergraduate. However, I did not actually understand it well at all, to which is a testament that much of the knowledge vanished very quickly. It pleases me though now following some intellectual maturation, after relearning certain theorems, they seem to stick more permanently, with the main ideas behind the proof more easily understandably clear than mind-disorienting, the latter of which was experienced by me too much in my early days. Shall I say it that before I must have been on drugs of something, because frankly the way about which I approached certain things was frankly quite weird, and in retrospect, I was in many ways an animal-like creature trapped within the confines of an addled consciousness oblivious and uninhibited. Almost certainly never again will I experience anything like that. Now, I can only mentally rationalize the conscious experience of a mentally inferior creature but such cannot be experienced for real. It is almost like how an evangelical cannot imagine what it is like not to believe in God, and even goes as far as to contempt the pagan. Exaltation, exhilaration was concomitant with the leap of consciousness till it not long after established its normalcy.

Now, the last of non-mathematical writing in this post will be on the following excerpt from Grothendieck’s Récoltes et Semailles:

In those critical years I learned how to be alone. [But even] this formulation doesn’t really capture my meaning. I didn’t, in any literal sense learn to be alone, for the simple reason that this knowledge had never been unlearned during my childhood. It is a basic capacity in all of us from the day of our birth. However these three years of work in isolation [1945–1948], when I was thrown onto my own resources, following guidelines which I myself had spontaneously invented, instilled in me a strong degree of confidence, unassuming yet enduring, in my ability to do mathematics, which owes nothing to any consensus or to the fashions which pass as law….By this I mean to say: to reach out in my own way to the things I wished to learn, rather than relying on the notions of the consensus, overt or tacit, coming from a more or less extended clan of which I found myself a member, or which for any other reason laid claim to be taken as an authority. This silent consensus had informed me, both at the lycée and at the university, that one shouldn’t bother worrying about what was really meant when using a term like “volume,” which was “obviously self-evident,” “generally known,” “unproblematic,” etc….It is in this gesture of “going beyond,” to be something in oneself rather than the pawn of a consensus, the refusal to stay within a rigid circle that others have drawn around one—it is in this solitary act that one finds true creativity. All others things follow as a matter of course.

Since then I’ve had the chance, in the world of mathematics that bid me welcome, to meet quite a number of people, both among my “elders” and among young people in my general age group, who were much more brilliant, much more “gifted” than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle—while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things that I had to learn (so I was assured), things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates, almost by sleight of hand, the most forbidding subjects.

In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still, from the perspective of thirty or thirty-five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve all done things, often beautiful things, in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have had to rediscover in themselves that capability which was their birthright, as it was mine: the capacity to be alone.

Grothendieck was first known to me the dimwit in a later stage of high school. At that time, I was still culturally under the idiotic and shallow social constraints of an American high school, though already visibly different, unable to detach too much from it both intellectually and psychologically. There is quite an element of what I now in recollection with benefit of hindsight can characterize as a harbinger of unusual aesthetic discernment, one exercised and already vaguely sensed back then though lacking in reinforcement in social support and confidence, and most of all, in ability. For at that time, I was still much of a species in mental bondage, more often than not driven by awe as opposed to reason. In particular, I awed and despaired at many a contemporary of very fine range of myself who on the surface appeared to me so much more endowed and quick to grasp and compute, in an environment where judgment of an individual’s capability is dominated so much more so by scores and metrics, as opposed to substance, not that I had any of the latter either.

Vaguely, I recall seeing the above passage once in high school articulated with so much of verbal richness of a height that would have overwhelmed and intimidated me at the time. It could not be understood by me how Grothendieck, this guy considered by many as greatest mathematician of the 20th century, could have actually felt dumb. Though I felt very dumb myself, I never fully lost confidence, sensing a spirit in me that saw quite differently from others, that was far less inclined to lose himself in “those invisible and despotic circles” than most around me. Now, for the first time, I can at least subjectively feel identification with Grothendieck, and perhaps I am still misinterpreting his message to some extent, though I surely feel far less at sea with respect to that now than before.

Later I had the fortune to know personally one who gave a name to this implicit phenomenon, aesthetic discernment. It has been met with ridicule as a self-congratulatory achievement one of lesser formal achievement, a concoction of a failure in self-denial. Yet on the other hand, I have witnessed that most people are too carried away in today’s excessively artificially institutionally credentialist society that they lose sight of what is fundamentally meaningful, and sadly, those unperturbed by this ill are few and fewer. Finally, I have reflected on the question of what good is knowledge if too few can rightly perceive it. Science is always there and much of it of value remains unknown to any who has inhabited this planet, and I will conclude at that.

So, one of the theorems in that class was of course Cauchy’s integral formula, one of the most central tools in complex analysis. Formally,

Let D be a bounded domain with piecewise smooth boundary. If f(z) is analytic on D, and f(z) extends smoothly to the boundary of D, then

f(z) = \frac{1}{2\pi i}\int_{\partial D} \frac{f(w)}{w-z}dw,\qquad z \in D. \ \ \ \ (1)

This theorem was actually somewhat elusive to me. I would learn it, find it deceptively obvious, and then forget it eventually, having to repeat this cycle. I now ask how one would conceive of this theorem. On that, we first observe that by continuity, we can show that the average on a circle will go to its value at the center as the radius goes to zero. With dw = i\epsilon e^{i\theta}d\theta, we can with the w - z in the denominator, vanish out any factor of f(z + \epsilon e^{i\theta}) in the integrand. From this, we have the result if D sufficiently small circle. Even with this, there is implicit Cauchy’s integral theorem, the one which states that integral of holomorphic function inside on closed curve is zero. Speaking of which, we can extend to any bounded domain with piecewise smooth boundary along the same principle.

Cauchy’s integral formula is powerful when the integrand is bounded. We have already seen this in Montel’s theorem. In another even simpler case, in Riemann’s theorem on removable singularities, we can with our upper bound on the integrand M, establish with M / r^n establish that for n < 0, the coefficient in the Laurent series about the point is a_n = 0.

This integral formula extends to all derivatives by differentiating. Inductively, with uniform convergence of the integrand, one can show that

f^{(m)}(z) = \frac{m!}{2\pi i}\int_{\partial D} \frac{f(w)}{(w-z)^{m+1}}dw, \qquad z \in D, m \geq 0.

An application of this for a bounded entire function would be to contour integrate along an arbitrarily large circle to derive an n!M / R^n upper bound (which goes to 0 as R \to \infty) on the derivatives. This gives us Liouville’s theorem, which states that bounded entire functions are constant, by Taylor series.


More politics from that China-hating Jew

The guy I keep referring to here, who does combinatorics, just pinged me on Facebook. The conversation goes as follows:

“He’s now president for life. President for life. No, he’s great. And look, he was able to do that. I think it’s great. Maybe we’ll have to give that a shot some day.” -Trump on Xi Jinping
china’s cultural values are reaching the US
the cultural values of a society that has statues of the greatest killer in history

Lol I had seen that
You’re referring to Mao?

and in america, we worry about robert lee statues
I refuse to give a fuck until they take down the mao statues in beijing
it’s comical

It’s funny how America regards Mao as the greatest killer in history
Once in China, somebody was like: I don’t even think he ever fired a gun once in his life.
He was mostly an intellectual

did hitler fire lots of guns
he was anti intellectual
he destroyed university education in china

You’re referring to during the cultural revolution
when the gaokao was cancelled
I’m curious about that decision
Who made it
Most likely, these leftists in the ministry of education eventually pressured the decision
Gang of Four types
On Baidu there are rumors that the poems he wrote weren’t actually written by him
And were written for him by Hu Qiaomu instead
Same with many of his writings
Reference Archive: Mao Zedong
Mao Zedong archive
Honestly I would bet that he, like Lenin, was simply prolific, writing wise.
What are the odds that Lenin didn’t actually write the books/essays attributed to him.
Honestly I think it’s most likely both of them were superhumanly smart, at least at verbal and politics.
With encyclopedic knowledge on relevant history and politics.
Lol with my blog, I’m becoming similar.


Weierstrass products

Long time ago when I was a clueless kid about the finish 10th grade of high school, I first learned about Euler’s determination of \zeta(2) = \frac{\pi^2}{6}. The technique he used was of course factorization of \sin z / z via its infinitely many roots to

\displaystyle\prod_{n=1}^{\infty} \left(1 - \frac{z}{n\pi}\right)\left(1 + \frac{z}{n\pi}\right) = \displaystyle\prod_{n=1}^{\infty} \left(1 - \frac{z^2}{n^2\pi^2}\right).

Equating the coefficient of z^2 in this product, -\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^2\pi^2}, with the coefficient of z^2 in the well-known Maclaurin series of \sin z / z, -1/6, gives that \zeta(2) = \frac{\pi^2}{6}.

This felt to me, who knew almost no math, so spectacular at that time. It was also one of great historical significance. The problem was first posed by Pietro Mengoli in 1644, and had baffled the most genius of mathematicians of that day until 1734, when Euler finally stunned the mathematical community with his simple yet ingenious solution. This was done when Euler was in St. Petersburg. On that, I shall note that from this, we can easily see how Russia had a rich mathematical and scientific tradition that began quite early on, which must have deeply influenced the preeminence in science of Tsarist Russia and later the Soviet Union despite their being in practical terms quite backward compared to the advanced countries of Western Europe, like UK and France, which of course was instrumental towards the rapid catching up in industry and technology of the Soviet Union later on.

I had learned of this result more or less concurrently with learning on my own (independent of the silly American public school system) what constituted a rigorous proof. I remember back then I was still not accustomed to the cold, precise, and austere rigor expected in mathematics and had much difficulty restraining myself in that regard, often content with intuitive solutions. From this, one can guess that I was not quite aware of how Euler’s solution was in fact not a rigorous one by modern standards, despite its having been noted from the book from which I read this. However, now I am aware that what Euler constructed was in fact a Weierstrass product, and in this article, I will explain how one can construct those in a way that guarantees uniform convergence on compact sets.

Given a finite number of points on the complex plane, one can easily construct an analytic function with zeros or poles there for any combination of (finite) multiplicities. For a countably infinite number of points, one can as well the same way but how can one know that it, being of a series nature, doesn’t blow up? There is quite some technical machinery to ensure this.

We begin with the restricted case of simple poles and arbitrary residues. This is a special case of what is now known as Mittag-Leffler’s theorem.

Theorem 1.1 (Mittag-Leffler) Let z_1,z_2,\ldots \to \infty be a sequence of distinct complex numbers satisfying 0 < |z_1| \leq |z_2| \leq \ldots. Let m_1, m_2,\ldots be any sequence of non-zero complex numbers. Then there exists a (not unique) sequence p_1, p_2, \ldots of non-negative integers, depending only on the sequences (z_n) and (m_n), such that the series f (z)

f(z) = \displaystyle\sum_{n=1}^{\infty} \left(\frac{z}{z_n}\right)^{p_n} \frac{m_n}{z - z_n} \ \ \ \ (1.1)

is totally convergent, and hence absolutely and uniformly convergent, in any compact set K \subset \mathbb{C} \ {z_1,z_2,\ldots}. Thus the function f(z) is meromorphic, with simple poles z_1, z_2, \ldots having respective residues m_1, m_2, \ldots.

Proof: Total convergence, in case forgotten, refers to the Weierstrass M-test. That said, it suffices to establish

\left|\left(\frac{z}{z_n}\right)^{p_n}\frac{m_n}{z-z_n}\right| < M_n,

where \sum_{n=1}^{\infty} M_n < \infty. For total convergence on any compact set, we again use the classic technique of monotonically increasing disks to \infty centered at the origin with radii r_n \leq |z_n|. This way for |z| \leq r_n, we have

\left|\left(\frac{z}{z_n}\right)^{p_n}\frac{m_n}{z-z_n}\right| < \left(\frac{r_n}{|z_n|}\right)^{p_n}\frac{m_n}{|z_n|-r_n} < M_n.

With r_n < |z_n| we can for any M_n choose large enough p_n to satisfy this. This makes clear that the \left(\frac{z}{z_n}\right)^{p_n} is our mechanism for constraining the magnitude of the values attained, which we can do to an arbitrary degree.

The rest of the proof is more or less trivial. For any K, pick some r_N the disk of which contains it. For n < N, we can bound with \displaystyle\max_{z \in K}\left|\left(\frac{z}{z_n}\right)^{p_n}\frac{m_n}{z-z_n}\right|, which must be bounded by continuity on compact set (now you can see why we must omit the poles from our domain).     ▢

Lemma 1.1 Let the functions u_n(z) (n = 1, 2,\ldots) be regular in a compact set K \subset C, and let the series \displaystyle\sum_{n=1}^{\infty} u_n(z) be totally convergent in K . Then the infinite product \displaystyle\sum_{n=1}^{\infty} \exp (u_n(z)) = \exp\left(\displaystyle\sum_{n=1}^{\infty} u_n(z)\right) is uniformly convergent in K.

Proof: Technical exercise left to the reader.     ▢

Now we present a lemma that allows us to take the result of Mittag-Leffler (Theorem 1.1) to meromorphic functions with zeros and poles at arbitrary points, each with its prescribed multiplicity.

Lemma 1.2 Let f (z) be a meromorphic function. Let z_1,z_2,\ldots = 0 be the poles of f (z), all simple with respective residues m_1, m_2,\ldots \in \mathbb{Z}. Then the function

\phi(z) = \exp \int_0^z f (t) dt \ \ \ \ (1.2)

is meromorphic. The zeros (resp. poles) of \phi(z) are the points z_n such that m_n > 0 (resp. m_n < 0), and the multiplicity of z_n as a zero (resp. pole) of \phi(z) is m_n (resp. -m_n).

Proof: Taking the exponential of that integral has the function of turning it into a one-valued function. Take two paths \gamma and \gamma' from 0 to z with intersects not any of the poles. By the residue theorem,

\int_{\gamma} f(z)dz = \int_{\gamma'} f(z)dz + 2\pi i R,

where R is the sum of residues of f(t) between \gamma and \gamma'. Because the m_is are integers, R must be an integer from which follows that our exponential is a one-valued function. It is also, with the exponential being analytic, also analytic. Moreover, out of boundedness, it is non-zero on \mathbb{C} \setminus \{z_1, z_2, \ldots\}. We can remove the pole at z_1 with f_1(z) = f(z) - \frac{m_1}{z - z_1}. This f_1 remains analytic and is without zeros at \mathbb{C} \setminus \{z_2, \ldots\}. From this, we derive

\begin{aligned} \phi(z) &= \int_{\gamma} f(z)dz \\ &= \int_{\gamma} f_1(z) + \frac{m_1}{z-z_1}dz \\ &= (z-z_1)^{m_1}\exp \int_0^z f_1(t) dt. \end{aligned}

We can continue this process for the remainder of the z_is.      ▢

Theorem 1.2 (Weierstrass) Let F(z) be meromorphic, and regular and \neq 0 at z = 0. Let z_1,z_2, \ldots be the zeros and poles of F(z) with respective multiplicities |m_1|, |m_2|, \ldots, where m_n > 0 if z_n is a zero and m_n < 0 if z_n is a pole of F(z). Then there exist integers p_1, p_2,\ldots \geq 0 and an entire function G(z) such that

F(z) = e^{G(z)}\displaystyle\prod_{n=1}^{\infty}\left(1 - \frac{z}{z_n}\right)^{m_n}\exp\left(m_n\displaystyle\sum_{k=1}^{p_n}\frac{1}{k}\left(\frac{z}{z_k}^k\right)\right), \ \ \ \ (1.3)

where the product converges uniformly in any compact set K \subset \mathbb{C} \ \{z_1,z_2,\ldots\}.

Proof: Let f(z) be the function in (1.1) with p_is such that the series is totally convergent, and let \phi(z) be the function in (1.2). By Theorem 1.1 and Lemma 1.2, \phi(z) is meromorphic, with zeros z_n of multiplicities m_n if m_n > 0, and with poles z_n of multiplicities |m_n| if m_n < 0. Thus F(z) and \phi(z) have the same zeros and poles with the same multiplicities, whence F(z)/\phi(z) is entire and \neq 0. Therefore \log (F(z)/\phi(z)) = G(z) is an entire function, and

F(z) = e^{G(z)} \phi(z). \ \ \ \ (1.4)

Uniform convergence along path of integration from 0 to z (not containing the poles) enables term-by-term integration. Thus, from (1.2), we have

\begin{aligned} \phi(z) &= \exp \displaystyle\sum_{n=1}^{\infty} \left(\frac{z}{z_n}\right)^{p_n} \frac{m_n}{t - z_n}dt \\ &= \displaystyle\prod_{n=1}^{\infty}\exp \int_0^z \left(\frac{m_n}{t - z_n} + \frac{m_n}{z_n}\frac{(t/z_n)^{p_n} -1}{t/z_n - 1}\right)dt \\ &= \displaystyle\prod_{n=1}^{\infty}\exp \int_0^z \left(\frac{m_n}{t - z_n} + \frac{m_n}{z_n}\displaystyle\sum_{k=1}^{p_n}\left(\frac{t}{z_n}\right)^{k-1}\right)dt \\ &= \displaystyle\prod_{n=1}^{\infty}\exp \left(\log\left(1 - \frac{z}{z_n}\right)^{m_n} + m_n\displaystyle\sum_{k=1}^{p_n}\frac{1}{k}\left(\frac{t}{z_n}\right)^k\right) \\ &= \displaystyle\prod_{n=1}^{\infty}\left(1 - \frac{z}{z_n}\right)^{m_n} \exp \left(m_n\displaystyle\sum_{k=1}^{p_n}\frac{1}{k}\left(\frac{t}{z_n}\right)^k\right).\end{aligned}

With this, (1.3) follows from (1.4). Moreover, in a compact set K, we can always bound the length of the path of integration, whence, by Theorem 1.1, the series

\displaystyle\sum_{n=1}^{\infty}\int_0^z \left(\frac{t}{z_n}\right)^{p_n}\frac{m_n}{t - z_n}dt

is totally convergent in K. Finally, invoke Lemma 1.1 to conclude that the exponential of that is total convergent in K as well, from which follows that (1.3) is too, as desired.     ▢

If at 0, our function has a zero or pole, we can easily multiply by z^{-m} with m the multiplicity there to regularize it. This yields

F(z) = z^me^{G(z)}\displaystyle\prod_{n=1}^{\infty}\left(1 - \frac{z}{z_n}\right)^{m_n}\exp\left(m_n\displaystyle\sum_{k=1}^{p_n}\frac{1}{k}\left(\frac{z}{z_n}^k\right)\right)

for Weierstrass factorization formula in this case.

Overall, we see that we transform Mittag-Leffler (Theorem 1.1) into Weierstrass factorization (Theorem 1.2) through integration and exponentiation. In complex, comes up quite often integration of an inverse or -1 order term to derive a logarithm, which once exponentiated gives us a linear polynomial to the power of the residue, useful for generating zeros and poles. Once this is observed, that one can go from the former to the latter with some technical manipulations is strongly hinted at, and one can observe without much difficulty that the statements of Lemma 1.1 and Lemma 1.2 are needed for this.


  • Carlo Viola, An Introduction to Special Functions, Springer International Publishing, Switzerland, 2016, pp. 15-24.