Israel, China, and more

I figured that as interested in Jews and Jewish achievement (and shenanigans) as I am, I should at least learn something real about Israel, which I know little about at the detailed factual level. That part of the world has, predictably, always felt rather remote in my life, though it is in some sense the cradle of civilization. While on the bus with nothing to do, I was just last week, trolling some of my friends on Facebook with some Hebrew I copy pasted. Like, ברוך השם (Baruch HaShem), which literally means “blessed his name.” On that I’m pleased to say that I’m now sort of paying attention to the letters of the Hebrew alphabet when I visit English Wiki pages on Jewish matters with the English transliteration of Hebrew words alongside the Hebrew original. It’s kind of cute that it, like Arabic, reads right to left, a fact I had not known.

Last night, I had the pleasure of going through the history of Israel (particularly its formation) in some degree of detail. So now I know what Haganah, Irgun, Lehi, Palmach are. Interestingly, there was tension between the IDF (headed by Ben-Gurion) and the Irgun (headed by Menachim Begin), which resulted in the Altalena Affair in which a ship containing armaments of the Irgun was ordered to be sunk by Ben-Gurion on high seas by the air force. I was rather surprised there was actually this much discord among the Zionist leaders, as stereotype is of course that Jews are super cohesive. I was also somewhat surprised that the Zionists had the nerve to assassinate Western politicians like Lord Moyne and Folke Bernadotte they did not like who were mediating truces between the Israelis and the Arabs during the 1948 war. Overall, my impression of the war was that neither side had substantial military experience or ability and the war was on a relatively small scale, being in a very small region.

I was not fully aware that American support for Israel really only became substantial following the Six Day War. Around that time, France, which had provided Israel with high end military technology before, had announced an embargo there. Israel’s nuclear weapons was provided to it largely by France as well, and some say that at the time of Six Day War, Israel already had a functional nuke to use as a last resort. Almost certainly, it did in the Yom Kippur War seven years later. Details regarding Israel’s secret nuclear weapons program were revealed to the public via Mordechai Vanunu, who had worked as a technician on classified projects at the Negev Nuclear Research Center, who was eventually caught and shamed for life by Mossad agents.

From this reading, I can better appreciate Israel’s vulnerability due to its small size, in land and in population, the latter especially, that makes it impossible to sustain itself without external aid. This will hold regardless of how advanced it becomes, so even with nuclear ICBMs, they still have much to fear. I’ve seen pro-Jewish sites characterize Israel’s military and survival as a miracle. I’ve also seen that Israel is scared shit of North Korea, which could potentially transfer its nuclear and missile technology to Iran, Syria, etc. Israel’s attitude is of course that the Arab nations cannot obtain nukes at all costs, and Israel will send Mossad agents to assassinate anyone suspected to be assisting them on that, which it has already done many times.

Readers of my blog might know that I write here about that Jew in math I talk with quite a bit, who has some interesting views for sure. As an update there, I didn’t quite expect him to say to me that “Israel surviving is not that impressive.” I would somewhat agree actually given how much support Israel has gotten from the West, which does not apply at all to North Korea, whose survival I would say is much more of a miracle. Only time will tell who will last longer, and I would think that both will remain intact for quite a while.

That guy also tells me that China is very pro-Israel, which I’m not so sure about. China only developed at first secret relations with Israel in the late 70s/early 80s. China at the time was very interested in procuring some high-end Western military tech from Israel, which it did to a significant degree in the 80s, and surely, America is not terribly happy about this. This guy responds with “China” when I ask him how Israel will fare on a weakened America, and I’m not sure how serious he is on that, since I could hardly imagine China actually going to the lengths to rescue Israel under the hypothetical scenario that it is about to be run over.

The aggressive and often overtly biased political attitudes of Jews and Israelis are understandable given how precarious their situation is. They faced life or death and though their situation is much more secure now, they still do, being too small. On this, recalled to me was this physics professor at Washington University of St. Louis whose infamous essay Don’t Become a Scientist I had read and reread, who also has on his web page a collection of political pieces against Iraq and North Korea, with provocative titles such as Anyone Who Bombs Baghdad [when Saddam was in power] Gets My Vote. I haven’t seen yet any mention of Israel and its nukes in his pieces and I sure wonder why. In another one of those, Limiting the Nuclear Club—Iraq, North Korea et al., he characterized Stalin (when it was mostly his USSR that defeated Hitler) very one-sidedly as “a man and system which murdered tens of millions of people with bullets, famine, and prison camps.” On the nuclear club, he also wrote that “fortunately, most of these countries are stable democracies and therefore not aggressors; the two chief exceptions (the Former Soviet Union and China) were successfully contained for many years, and the more powerful of these is making a transition, one hopes successful and irreversible, to democracy,” which again goes to show his blatant bias and lack of rigorous thinking (that he would exhibit in physics) when it comes to politics.

On this, I don’t see why China should be terribly friendly to Jews and Israel when Jews, with their media power and verbal gifts, have done so much to distort modern Chinese history in the West and to smear, sabotage, and peacefully evolve a political system that has worked wonders for China, very plausibly with ulterior motives. I have also seen many Jews support Taiwanese independence, including this guy I talk to. Certain American interests might want to mould the political thinking of Chinese who grow up in America like me (most of whom do not read Chinese), but honestly, I feel like I am too intelligent and politically discerning and realistic to fall for it. I value independent, impartial thinking that is reality grounded, that is cognitively empathetic of interests relations wise, which means the American exceptionalist versions of history and politics don’t work on me, and neither would any such form of exceptionalism in favor of any country or system. I don’t think Chinese who grow up in America will be terribly happy once they realize, as more of them are doing, that the American version of the history and culture of where their parents are from is fraught with glaring inaccuracies and falsehoods motivated by political bias and ill-intent, and elite Jews, who have the most prominent voice in America, can be mostly easily blamed for that. One can even go more extreme and say that the Jews are the main culprit for the shitty and grossly dishonest media in America, with their dominance of press and Hollywood in this country, which they unabashed laud as “free media.” I don’t think this facade can last forever.

Often, one, including myself, is met with the dilemma of whether or not to engage in aggressive and ethically questionable behavior that gives one an advantage, at least in the short term, that goes on record either directly or indirectly in the memories of those alienated by the action as well as oneself.  Even it brings me major gains, I know eventually I might look back and feel shame and regret on the dishonorable means I took to earn them. I like it most when I achieve something based on genuine ability and hard work, as opposed to politicking, striverish behavior, which everyone engages in to some extent. I don’t think lies or deceptive talk can be concealed forever regardless of how much power or media control one has, and in some sense, it is the truth that is the most potent. Additionally, as a Chinese, I am somewhat conscious of how my behavior in every way affects microscopically how Chinese are perceived in general. When I see so many Jews spout nonsense about history and politics, especially parts I am familiar with, it sure doesn’t give me a good view of the group in general, character wise, so as to separate from their objective achievements, especially when that group controls so much of the media where I live, though I am careful to disentangle individuals of the group with the group in general. I do believe that one is to some extent responsible for the actions of one’s group at large. The actions of a nation, of an ethnic group, especially against others, are not just the responsibility of the elites in power who made the decisions but also the ones who allowed those people to come in power. When a nation or people, as a collective, chooses some system or leader or development strategy, they should take some responsibility for the outcomes and “dictatorship” or “democracy” is not an excuse. Fundamentally, what I am describing is actual democracy, as opposed any democratic system by name or by election. If Americans want to elect “democratically” their leaders and their leaders make shitty decisions against their interests and country at large, they should take responsibility for it and blame themselves for choosing such people to elect or blame the election system that is the root of it all. On this, I recall how this guy way smarter than me technically (also of Jewish descent), on my mentioning of a guy I know whose parents were from the USSR whose grandfather could only become a theoretical physics professor in some remote university the name I remember not, was like: “his parents helped destroy the Soviet Union,” followed by that if he, who moved to Israel, were still in Russia, he would be working for MacDonalds, with reference to the economic crisis there in the 90s that was statistically far more murderous and damaging than Stalin’s purges. It was then that occurred to me again that as unpleasant and sad as it may be to accept, Soviets and Russians share collective responsibility for promoting certain wrong people to power in the Soviet era that rendered their nation less competitive and especially for their later letting oligarchs, many of them Jewish, wreck their country irrecoverably, a specific of the generality I had just described. That many of those mega civilization and wealth leechers/destroyers were Jewish tells us more that anti-Semitism is not without reason, and Jews should all take some responsibility for it. Pardon any cultural bias, but this brings to mind a famous quote attributed to Chairman Mao which is “世界上没有无缘无故的爱,也没有无缘无故的恨”,that translates to “the world has no love without reason and no hate without reason,” an obvious reality of human nature that I believe one of high moral character ought to always be cognizant of.

Urysohn metrization theorem

The Urysohn metrization theorem gives conditions which guarantee that a topological space is metrizable. A topological space (X, \mathcal{T}) is metrizable is there is a metric that induces a topology that is equivalent to the topological space itself. These conditions are that the space is regular and second-countable. Regular means that any combination of closed subset and point not in it is separable, and second-countable means there is a countable basis.

Metrization is established by embedding the topological space into a metrizable one (every subspace of a metrizable space is metrizable). Here, we construct a metrization of [0,1]^{\mathbb{N}} and use that for the embedding. We first prove that regular and second-countable implies normal, which is a hypothesis of Urysohn’s lemma. We then use Urysohn’s lemma to construct the embedding.

Lemma Every regular, second-countable space is normal.

Proof: Let B_1, B_2 be the sets we want to separate. We can construct a countable open cover of B_1, \{U_i\}, whose closures intersect not B_2 by taking a open neighborhoods of each element of B_1. With second-countability, the union of those can be represented as a union of a countable number of open sets, which yields our desired cover. Do the same for B_2 to get a similar cover \{V_i\}.

Now we wish to minus out from our covers in such a way that their closures are disjoint. We need to modify each of the U_is and V_is such that they do not mutually intersect in their closures. A way to do that would be that for any U_i and V_j, we have the part of \bar{U_i} in V_j subtracted away from it if j \geq i and also the other way round. This would give us U_i' = U_i \setminus \sum_{j=1}^i \bar{V_j} and V_i' = V_i \setminus \sum_{j=1}^i \bar{V_j}.     ▢

Urysohn’s lemma Let A and B be disjoint closed sets in a normal space X. Then, there is a continuous function f : X \to [0,1] such that f(A) = \{0\} and f(B) = \{1\}.

Proof: Observe that if for all dyadic fractions (those with least common denominator a power of 2) r \in (0,1), we assign open subsets of X U(r) such that

  1. U(r) contains A and is disjoint from B for all r
  2. r < s implies that \overline{U(r)} \subset U(s)

and set f(x) = 1 if x \notin U(r) for any r and f(x) = \inf \{r : x \in U(r)\} otherwise, we are mostly done. Obviously, f(A) = \{0\} and f(B) = \{1\}. To show that it is continuous, it suffices to show that the preimages of [0, a) and (a, 1] are open for any x. For [0, a), the preimage is the union of U(r) over r < a, as for any element to go to a' < a, by being an infimum, there must be a s \in (a', a) such that U(s) contains it. Now, suppose f(x) \in (a, 1] and take s \in (a, f(x)). Then, X \setminus \bar{U(s)} is an open neighborhood of x that maps to a subset of (a, 1]. We see that x \in X \setminus \overline{U(s)}, with if otherwise, s < f(x) and thereby f(x) \leq s' < f(x) for s' > s and U(s') \supset \overline{U(s)}. Moreover, with s > a, we have excluded anything that does not map above a.

Now we proceed with the aforementioned assignment of subsets. In the process, we construct another assignment V. Initialize U(1) = X \setminus B and V(0) = X \setminus A. Let U(1/2) and V(1/2) be disjoint open sets containing A and B respectively (this is where we need our normality hypothesis). Notice how in normality, we have disjoint closed sets B_1 and B_2 with open sets U_1 and U_2 disjoint which contain them respectively, one can complement B_1 to derive a closed set larger than U_2, which we call U_2' and run the same normal separation process on A_1 and U_2'. With this, we can construct U(1/4), U(3/4), V(1/4), V(3/4) and the relations

X \setminus V(0) \subset U(1/4) \subset X \setminus V(1/4) \subset U(1/2),

X \setminus U(1) \subset V(3/4) \subset X \setminus U(3/4) \subset V(1/2).

Inductively, we can show that we can continue this process on X \setminus V(a/2^n) and X \setminus U((a+1)/2^n) for each a = 0,1,\ldots,2^n-1 provided U and V on all dyadics with denominator 2^n to fill in the ones with denominator 2^{n+1}. One can draw a picture to help visualize this process and to see that this satisfies the required aforementioned conditions for U.     ▢

Now we will find a metric for \mathbb{R}^{\mathbb{N}} the product space. Remember that the base for product space is such that all projections are open and a cofinite of them are the full space itself (due to closure under only finite intersection). Thus our metric must be such that every \epsilon-ball contains some open set of the product space where a cofinite number of the indices project to \mathbb{R}. The value of x - y for x,y \in \mathbb{R} as well as its powers is unbounded, so obviously we need to enforce that the distance exceed not some finite value, say 1. We also need that for any \epsilon > 0, the distance contributed by all of the indices but a finite number exceeds it not. For this, we can tighten the upper bound on the ith index to 1/i, and instead of summing (what would be a series), we take a \sup, which allows for all n > N where 1/N < \epsilon, the nth index is \mathbb{R} as desired. We let our metric be

D(\mathbf{x}, \mathbf{y}) = \sup\{\frac{\min(|x_i-y_i|, 1)}{i} : i \in \mathbb{N}\}.

That this satisfies the conditions of metric is very mechanical to verify.

Proposition The metric D induces the product topology on \mathbb{R}^{\mathbb{N}}.

Proof: An \epsilon-ball about some point must be of the form

(x_1 - \epsilon/2, x_1 + \epsilon/2) \times (x_2 - 2\epsilon/2, x_2 + 2\epsilon/2) \times \cdots \times (x_n - n\epsilon/2, x_n + n\epsilon/2) \times \mathbb{R} \times \cdots \times \mathbb{R} \times \cdots,

where n is the largest such that n\epsilon < 1. Clearly, we can fit into that an open set of the product space.

Conversely, take any open set and assume WLOG that it is connected. Then, there must be only a finite set of natural number indices I which project to not the entire space but instead to those with length we can assume to be at most 1. That must have a maximum, which we call n. For this we can simply take the minimum over i \leq n of the length of the interval for i divided by i as our \epsilon.     ▢

Now we need to construct a homeomorphism from our second-countable, regular (and thereby normal) space to some subspace of \mathbb{R}^\mathbb{N}. A homeomorphism is injective as part of definition. How to satisfy that? Provide a countable collection of continuous functions to \mathbb{R} such that at least one of them differs whenever two points differ. Here normal comes in handy. Take any two distinct points. Take two non-intersecting closed sets around them and invoke Urysohn’s lemma to construct a continuous function. That would have to be 0 at one and 1 at the other. Since our space is second-countable, we can do that for each pair of points with only a countable number. For every pair in the basis B_n, B_m where \bar{B_n} \subset B_m, we do this on \bar{B_n} and X \setminus B_m.

Proposition Our above construction is homeomorphic to [0,1]^{\mathbb{R}}.

Proof: Call our function f. Each of its component functions is continuous so the entire Cartesian product is also continuous. It remains to show the other way, that U in the domain open implies the image of U is open. For that it is enough to take z_0 = f(x_0) for any x_0 \in U and find some open neighborhood of it contained in f(U). U contains some basis element of the space and thus, there is a component (call it f_n) that sends X \setminus U to all to 0 and x_0 not to 0. This essentially partitions X by 0 vs not 0, with the latter portion lying inside U, which means that \pi_n^{-1}((0, \infty)) \cap f(X) is strictly inside f(U). The projections in product space are continuous so that set must be open. This suffices to show that f(U) is open.     ▢

With this, we’ve shown our arbitrary regular, second-countable space to be homeomorphic to a space we directly metrized, which means of course that any regular, second-countable space is metrizable, the very statement of the Urysohn metrication theorem.

Jewish pro-Americanism

In America, people often bring up what they view as China’s suppression of free expression. I personally dislike strongly the usage of “free expression,” because it is meaninglessly vague. And there is no such thing as free expression in the strictest sense of it. Especially when you are in a job dealing with a boss who can fire you, which is why politics is generally supposed to be a no-no in the workplace, discussion wise. People avoid it out of prudent protection of their careers. One naturally feels at disease when what one wishes to express is such that is unwelcome or hostile in the environment of one’s residence. In such case, one feels that his or her right of free expression is being beaten down. This is very much the case in America right now, in many places.

I’ll say that overall I would feel that China is actually more free in expression overall. Go on the Chinese internet and people can discuss certain matters honestly in a manner unimaginable on the American internet. It helps much that it is for the most part a ethnically homogeneous society, unlike in America, where you have to often be very sensitive to the background of the person you’re talking to (another peril of our cherished diversity I guess). This excepts a few in some sense politically taboo topics like Tiananmen, which people with some interest in the matter might discuss say eating out, just not publicly online. There are also the other two Ts, Taiwan and Tibet. From what I know, Tibet is seldom on the minds of people in China and neither is Taiwan really. In all honestly, people in China have, for the most part, way more interesting things to think about politically than any of these three Ts.

Back to the title of this article, I would say that I am somewhat surprised and also amused at how many highly educated American Jews express openly some diehard belief in American exceptionalism, in particular its “freedom and democracy.” There are plenty of prominent Jewish voices and even actors (like Kissinger) in American foreign policy, one of whom, Amitai Etzioni, I learned about yesterday, seeing that he has written Security First: For a Muscular, Moral Foreign Policy and Avoiding War with China: Two Nations, One World along with many articles on mass media channels like CNN. It’s kind of funny that a guy who fought for Israel against Arabs in 1948 as a teenager (according to Wiki) has such a high position in the BS field of geopolitical strategy in America, as a director and professor in something policy at George Washington University. I won’t name more names but I’ve seen many.

This is not surprising until I think about the situation more carefully. A cynic has a every reason to view Jews in their elite to desire infinite world power and control for America, the country where they exert the most political and economic influence, which their homeland Israel depends on much. On the other hand, they know that anti-Semitism is still very real in Russia, which is not that powerful anymore. I’ll say that I listen to some Soviet songs written by very talented Jewish composers and I admire much the brilliance and work of genius Soviet scientists with Jewish blood too many to name. However, let’s just say that in the Soviet era, Soviet Jews were kept out of political and economic power and more or less confined in the arts and sciences in which they excelled. On that, I’ll say that Trotsky sure left a bad mark for Jews in the Soviet Union. The second most powerful country in the world now, China, Jews basically have zero chance in. Plus, many of them might be aware that in China, one can talk about the grossly disproportionate economic, cultural, and political power possessed by American Jews without any fear of repercussions. That is basically an openly acknowledged fact among Chinese who engage in business. So this highly talented but very small subgroup which has made so many enemies only has America and to a lesser extent its puppet Europe to cling to.

What is rather ironic is that recently, Jews have arguably contributed much to America’s decline. Let’s just say that the Iraq War (which Israel very likely supported) and the financial crisis and recession (Goldman-Sachs is run by Jews) did no good for America, weakening greatly its international position. Those might have put Jews more in favor in America in terms of their control of the economy and their political influence, but of course, that only really counts if America is actually powerful.

Don’t know for sure what Jews were thinking with all that, but if they wanted to play genuine zero-sum games for their own favor, they’d want to strengthen America as much as they can (provided they maintain reasonable level of control over it) and weaken its adversaries, where they have little chance of gaining power without bringing about a coup that replaces the regime with a pro-American one. Of course, support for Israel is always desired, but Israel too small can never sustain itself, which means leeching off America (or some other giant) is an absolute necessity. It’s fair to say that Jews have boosted America to some extent by promoting immigration of high-end talent to work for American companies, whose smart kids will also, by virtue of growing up there, become American culturally and inevitably stay there. It’s also fair to say that Jews have tried hard to bring American culture and products into the rest of the world (to further integrate the rest of world into the American-led world order) with some success. The most glaring failure there is highlighted by that the Chinese government could not be convinced to let in Google and Facebook, which has contributed to a boom of indigenous Chinese tech companies, like Baidu and Tencent. China back a few decades ago seemed puny (with very low GDP per capita and lack of many advanced technologies), but now it is, for the most part, a superpower rivaling the US. With this, China is much more confident and is seeking more create an alternative system that challenges America and thereby Jews. Jewish anti-Chinese (often disguised as anti-communism) sentiment explained. It also hurts that the position of North Korea, which Israel, which itself has nukes, views as a major threat, having survived, whereas Iraq and Libya did not, is more secure the more powerful China becomes. With this, any fantasy of Jewish-led American world domination is ever more a fantasy.

I’ve seen much contempt for China and Chinese among Jews. There are all these stereotypes that Jews are creative and Chinese are not, with Jews 625 times more likely to win a Nobel Prize than an Asian person. It is so much engrained in the culture of stereotypes that I used to sort of believe it myself. Of course, when one looks more closely, one sees that those Nobel Prizes (which may have bias towards certain groups themselves) are mostly awarded to those already in old age, which means it takes not only time but also that a place has been developed and advanced for quite a while. I was rather surprised on seeing how many Nobel Prizes have been awarded to Japanese (mostly working in Japan) in the 21st century. That rate is comparable or close to the rate at which Americans win Nobels if one excludes the BS prizes of peace, literature, and economics and immigrants. Jews can be dismissive of China’s ability to innovate and they even tie it irrelevantly to its political system, in particular its great firewall. They are contradicting themselves. Anyone in the right mind knows that the political system doesn’t affect science research at all so long as the research is adequately economically supported and not disrupted. Ask yourselves why Jews were so successful in science in the “totalitarian” Soviet Union.

I have especially seen contempt among Jews for China’s political system, which some of them see as menacing and threatening. The faked moral superiority will not fool anyone who is not delusional. Everyone acts for his own interests for some degree or another, including China, including America, including Israel, including Russia. To back off from pursuing what is best for oneself under soft pressures and political deception is nothing but a sign of weakness. Anyone strong of heart, including the genuinely loyal Chinese party members, working in all arenas, know the importance of conviction and dedication and not letting it go amidst distraction and enticement.

Anyone with the slightest of political consciousness is aware that most always extraordinary talent is not enough though surely it can overcome initial disadvantages. For instance, being born into a rich, well-educated family is always an advantage. It gives you more material resources to develop your talents and more importantly, the access to connections which often make or break careers. To study and pursue excellence is a privilege that implies that the problem of basic material necessity has already been solved. In this regard, I shall comment that Jewish preeminence in intellectual and artistic pursuits is arguably as much a product of the superior economic, cultural, and social conditions they have accumulated over the past generations as it is of their superior raw talent. It is the former that turns the latter to fruit at higher rates. There is also that Jews, with their verbal talents (and perhaps certain personality characteristics too) combined with their being part of Western culture essentially, are excessively good and willing at self-promotion, which explains why they excel more in softer fields than in hard ones. I don’t want to sound too unpleasantly incisive here, but every time a smart Jew takes a scarce opportunity or position, it deprives a talented white or Asian from developing herself, her gifts, and her career.

America has, needless to say, provided Jews with too much opportunity. It is not often that a group can obtain a disproportionate number of positions of money and power in a powerful, advanced country where they are an immigrant minority with a different culture. America has virtually handed much of itself, its vast resources, to this group, and thus, this group must put America above all, so long its control of it is maintained.

From my experience, white Americans are often too nice and too naive. They don’t know how to scheme and deceive and are often oblivious when it is done to them. America is a wealthy, resource-rich country (per capita) that has not had a war at home since 150 years ago, unlike most of the rest of the world, after all, so there is less of a need to. On the other hand, Jews, aside from having a much higher IQ, have been met with some form of persecution for centuries and even millennia as a minority within Gentile society and necessarily developed such instincts, useful in moneylending, for their own survival that eventually enabled them to dominate more and more of the upper echelons of society with their form of shrewdness. Chinese, being from a densely populated place with little arable land and mountains abound, who have for the last century dealt successfully with powerful opponents, in one case the most powerful country in the world, trying virtually everything to make their country fail, can also easily see through shenanigans. Being much better positioned economically now, Chinese are also more equipped to fight back against them when it is in their interest to do so.

When dealing with anyone or any group, always expect them to place their interests first regardless of how they appear on the surface. As a special case of that, Jewish pro-Americanism is not Jewish pro-Americanism but Jewish pro Jewish domination of a powerful America that not necessarily pro-American at heart, as evidenced by the decline of America in aggregate over the past couple decades, especially following the 2008 financial crisis.


Riemann mapping theorem

I am going to make an effort to understand the proof of the Riemann mapping theorem, which states that there exists a conformal map from any simply connected region that is not the entire plane to the unit disk. I learned of its significance that its combination with the Poisson integral formula can be used to solve basically any Dirichlet problem where the region in question in simply connected.

Involved in this is Montel’s theorem, which I will now state and prove.

Definition A normal family of continuous functions is one for which every sequence in it has a uniformly convergent subsequence.

Montel’s theorem A family \mathcal{F} on domain D of holomorphic functions which is locally uniformly bounded is a normal family.

Proof: Turns out holomorphic alongside local uniform boundness is enough for us to establish local equicontinuity via the Cauchy integral formula. On any compact set K \subset D, we can find some r for which for every point z_0 \in K, \overline{B(z_0, 2r)} \subset D. By local boundedness we have some M>0 such that |f(z)| \leq M in all of B(z_0, 2r). Thus, for any w \in K, we can use Cauchy’s integral formula, for any z \in B(w, r). In that, the radius r versus 2r is used to bound the denominator with 2r^2.

\begin{aligned} |f(z) - f(w)| &= \left| \oint_{\partial B(z_0, 2r)} \frac{f(\zeta)}{\zeta - z} - \frac{f(\zeta)}{\zeta - w}d\zeta \right| \\ & \leq  |z-w| \oint_{\partial B(z_0, 2r)} \left| \frac{f(\zeta)}{(\zeta - z)(\zeta - w)} \right| d\zeta \\ & <  \frac{|z-w|2\pi r}{2\pi 2r^2} M. \end{aligned}

This shows it’s locally Lipschitz and thus locally equicontinuous. To choose the \delta we can divide our \epsilon by that Lipschitz constant alongside enforcing less than 2r so as to stay inside the domain.

With this we can finish off with the Arzela-Ascoli theorem.     ▢

Now take the family \mathcal{F} of analytic, injective functions from simply connected region \Omega onto \mathbb{D} the unit disk which take z_0 to 0. On this we have the following.

Proposition If f \in \mathcal{F} is such that for all g \in \mathcal{F}, |f'(z)| \geq g'(z), then f surjects onto \mathbb{D}.

Proof:  We prove the contrapositive. In order to do so, it suffices to find for any f that hits not w \in \mathbb{D}, f = s \circ g, where s, g are analytic with g(z_0) = 0 and s is a self-map on \mathbb{D} that fixes 0 and is not an automorphism. In that case, we can deduce from Schwarz lemma that |s'(0)| < 1 and thereby from the chain rule that g'(z_0) > f'(z_0).

Recall that we have automorphisms on \mathbb{D}, T_w = \frac{z-w}{1-wz}, for all w \in \mathbb{D} and that their inverses are also automorphisms. Let’s try to take 0 to w, then w to w^2 via p(z) = z^2, and finally w^2 to 0. With this, we have a working s = T_{w^2} \circ p \circ T_w^{-1}.     ▢

Nonemptiness of family

It is not difficult to construct an analytic injective self map on \mathbb{D} that sends z_0 to 0. The part of mapping z_0 to 0 is in fact trivial with the T_ws. To do that it suffices to map \mathbb{D} to \mathbb{C} \setminus \overline{\mathbb{D}} as after that, we can invert.

Since D is not the entire complex plane, there is some a \notin D. By translation, we can assume that a = 0. Because the region is simply connected, there is a path from 0 to \infty outside the region, which means there is an analytic branch of the square root. For any w that gets hits by that, -w does not. By the open mapping theorem, we can find a ball centered at -w that is entirely outside the region. With this, we can translate and dilate accordingly to shift that to the unit disk.

Construction of limit to surjection

We can see now that if we can construct a sequence of functions in our family that converges to an analytic one with the same zero at z_0 with maximal derivative (in absolute value) there, we are finished. Specifically, let \{f_n\} be a sequence from \mathcal{F} such that

\lim_n |f'_n(z_0)| = \sup_{f \in \mathcal{F}} \{|f'(z_0)|\}.

This can be done by taking functions with sufficiently increasing derivatives at z_0. With Montel’s theorem on our obviously locally uniformly bounded family, we know that our family is normal, and thus by definition, we can extract some subsequence that is uniformly convergent on compact sets. Now it remains to show that the function converged to is analytic and injective.

The injective part follows from a corollary of Hurwitz’s theorem, which we now state.

Hurwitz’s theorem (corollary of) If f_n is a sequence of injective analytic functions with converge uniformly on compact sets to f, then f is constant or injective.

Proof: Recall that Hurwitz’s theorem states that if f has a zero of some multiplicity m at some point z_0, then for any \epsilon > 0, we will, past some N in the index of the sequence, have m zeros within B(z_0, \epsilon) for all f_n, n > N, provided f is not constantly 0. For any point to see that a non-constant f can hit it only once, it suffices to do a translation by that point on all the f_ns to turn it into a zero, so that the hypothesis of Hurwitz’s theorem, which in this case, bounds the number of zeros above by 1, with the f_ns being injective, can be applied.     ▢

To show analyticity, we can use Weierstrass’s theorem.

Weierstrass’s theorem Take \{f_n\} and supposed it converges uniformly on compact sets to f. Then the following hold:
    a. f is analytic.
    b. \{f'_n\} converges to f' uniformly on compact sets.

Proof: This is a more standard theorem, so I will only sketch the proof. Recall the definition of compact as possessing the every cover has finite subcover property. This is so powerful, because we can for any collection of balls centered at every point of the cover, find a finite of them that covers the entire space, and finiteness allows us to take a maximum or minimum of finite Ns or \deltas to uniformize some limit.

We can do the same here. For every z on a compact set, express f_n as integral of \frac{f_n}{\zeta - z} via Cauchy’s integral formula on some ball centered at z. Uniform convergence of \frac{f_n}{\zeta - z} on the boundary to \frac{f}{\zeta-z} allows us to put the limit inside the integral to give us f, as represented via Cauchy’s integral formula. The same can be done for the \{f'_n\}

Again we can use two radii as done in the proof of Montel’s theorem to impose uniform convergence on a smaller ball.     ▢

Finally, our candidate conformal map to \mathbb{D} satisfies that f(z_0) = 0. If not, convergence would be naught at z_0 since f_n(z_0) = 0 for all n.

This gives us existence. There is also a uniqueness aspect of the Riemann mapping theorem that comes when one imposes f'(z_0) \in \mathbb{R}. This is very elementary to prove and will be left to the reader.


昨天,我再次与那位犹太国际数学奥赛金牌聊天,当初讨论的有不少与犹太人和以色列相关的话题。他既然会直接的说他觉得若少了些现有犹太人掌握的经济和政治权利,会是对世界不利的事情,也承认自己是个犹太复国主义者。之后,他又对我表示了他对中共和大陆的鄙视以及对台湾的偏向。不仅是KMT > GCD,还有若无日本侵略,则国民党将赢天下之荒谬绝对无任意限定的典型反华之扯话。



在媒体里,我们所得到的感觉是犹太人比较愿意靠自己的天分,自己独一无二的想象力,而不善于勉强手段。相反,华人经常被主流媒体描述为刻苦而缺乏创造性。我承认犹太人绝对有不少很有想象力并且敢于挑战权威的人,不过此非那么一致如所刻板化,而甚至可能是稍微相反。想起Steve Hsu曾在他博客写到在他上学的时候,所有他认识的人为SAT做过准备的都是犹太人,连亚裔家长都真的相信此考试是准备无意无效的智商测试。我所提的这位现在的做法非常符合Steve所说的。昨天,他却跟我说他母亲竟然还催他做GRE的试卷,做了两个还叫他继续做,直到做完第五个,让我非常吃惊。我当时想:“都上了大学犹太父母还会这样管孩子,这种事情我还真没听说过,甚至有点过分了。犹太父母是这么会保护孩子,为他们争取利益啊!”同样,他说高中时母亲还给他请了SAT家教,而像我这华人都从未上过任何SAT课(当然考的也不错)。不过这是高中孩子,还是孩子,此算寻常,但是真无法想象孩子上了大学了父母还会这样管。

我们都知道现在在那些顶级学生竞赛,如USAMO,如Putnam,如Intel STS,犹太人已经比不过华人了,可能他们真的没有那么聪明,所以我那位竞赛出类拔萃的白人朋友觉得事实上犹太人是不如东亚人的,有一定道理。可是他们之所以能够混的更好,尤其在商业界,是因为他们无所顾忌的互相提拔与支持以及不惜任何的追求金钱名利。若事实上在学术界和其他工作场合,犹太人更可能将别人的功劳称为己出,不会令我惊奇,毕竟我们都看到像Wolfram那样过分的人。我的那位朋友还怀疑不少大奖的委员会如诺贝尔委员会都有一定的偏向犹太人的偏见,受到太多犹太人的政治压力和说服。他说”白人和亚洲人基本公平竞争”(而犹太人并非如此),故与已经经济上更或远远更富裕的犹太人相比,优秀的白人和亚洲人失去了不少发展自己的机会。这位犹太人经常对我表示他对丘成桐为人处事的极其反感,将他形容为一个asshole。我也想过的确丘成桐有目中无人炙手可热之行为,但是看到犹太人的任人唯亲之流氓作为,不得不稍微狠一点。



这些话在我已所描述的任意不利于犹太人,即使百分之百客观正确,的话都可以被划为无可容忍并且有一定后果的反犹太主义,美国的政治气氛是很难被公开提到的,而相反,中文是在很多方面远远更自由的语言。再次说,犹太人出了不少绝顶聪明的人为人类做出了伟大的牢牢载入史册的贡献,突破了不少关键而长期悬而未决的科学障碍,创造了不少推进人类的伟大思想和科学潮流,这一点无人可否认。这周末,我在学习Urysohn的引理和度量化定理,觉得它们的证明实在是太原创巧妙了,绝对是天才的产物,而后在维基百科看到Urysohn是二十世纪初在俄罗斯的犹裔数学家,可惜英年夭折,如伽罗华(只不过伽罗华此度远远更高)。毫无疑问,这是一个伟大的民族,有许多非常值得学习的地方。不过,犹太人的卓越也是非常近代的,在这一点那位朋友也明确跟我说过从长远来看,人类文明主要还是欧洲和中国创造的,而不是犹太人的。同时,他们居住在别人的故乡而所为的无耻不羁的经济政治贪婪的现状和历史也是非常可藐视的。与往不同,现在犹太人收回了他们所谓的神圣领土,以色列,可是这一点没有减轻他们的弊端,而是使之加重,恨不得要将西方国家变成以色列的傀儡,一看到被父亲买进哈弗的伊万卡老公加强美国对以色列的支援真的让人感到非常恶心,甚至可以说在这一点,犹太人是人类文明的破坏者,是改造消灭当务之急的寄生虫。我经常读的一位知识渊博而话语真实的历史政治评论家Gwydion Madawc Williams却把以色列的表现形容为自杀式的军事主义,并且提到以色列的时间不会太长,甚至提到澳大利亚是将来最可能接受以色列难民的地方。这一点我感到不可思议,因为在我眼中,以色列拥有先进的武器装备和科技,包括核武器,非常聪明的人,加以西方强国的支持,现已基本保证了自己的生存。不过Gwydion的判断我还是非常尊重,他的理由是以色列不肯和他们敌人做出任何可被敌人接受的妥协,也同时在不断得罪支持他们的西方朋友,而且长期言来,千万余犹太人是无法抵抗十亿多穆斯林人的,也有一定道理。当然,以色列和犹太人的未来,只有时间会告诉我们。

Arzela-Ascoli theorem and delta epsilons

I always like to think of understanding of the delta epsilon definition of limit as somewhat of an ideal dividing line on the cognitive hierarchy, between actually smart and pseudo smart. I still remember vividly struggling to grok that back in high school when I first saw it junior year, though summer after, it made sense, as for why it was reasonable to define it that way. That such was only established in the 19th century goes to show how unnatural such abstract precise definitions are for the human brain (more reason to use cognitive genomics to enhance it 😉 ). At that time, I would not have imagined easily that this limit definition could be generalized further, discarding the deltas and epsilons, which presumes and restricts to real numbers, as it already felt abstract enough. Don’t even get me started on topological spaces, nets, filters, and ultrafilters; my understanding of them is still cursory at best, but someday I will fully internalize them.

Fortunately, I have improved significantly since then, both in terms of experience and in terms of my biological intelligence, that last night, I was able to reconstruct in my head the proof of the Arzela-Ascoli theorem, which also had been once upon a time mind-boggling. Remarkably, that proof, viewed appropriately, comes naturally out of just a few simple, critical observations.

The statement of the theorem is as follows.

Arzela-Ascoli theorem Let F be a family of functions from I = [a,b] to \mathbb{R} that are uniformly bounded and equicontinuous. Then there is a sequence of f_n of elements in F that converges uniformly in I.

The rationals are dense in the reals and make an excellent starting point. Uniform boundedness enables us employ Bolzano-Weierstrass to construct a sequence of functions convergent at any rational in I. With a countable number of such rationals, we can iteratively filter this sequence to construct one that converges at every rational in I. Essentially, we have an enumeration of the rationals in I and a chain of subsequences based on that, where in the nth subsequence is convergent at the first n rationals in the enumeration, and Bolzano-Weierstrass is applied onto the results of the application of the functions of that subsequence on the n+1th rational to yield another subsequence. Take, diagonally, the nth function in the nth subsequence, to derive our desired uniformly convergent sequence, which we call \{f_i\}.

To show uniform convergence, it suffices to show uniform Cauchyness, namely that for any \epsilon > 0, there is an N such that n,m > N implies |f_m(x) - f_n(x)| < \epsilon for all x \in I. By compactness, open neighborhoods of all rationals of I, as an open cover, has a finite subcover. Each element of the subcover comes from some rational of I and across that finite subset of I we can for any \epsilon > 0 take the max of all the Ns for convergence. This means that so long as our neighborhoods are sufficiently small, we can for any point x \in I have some x' that is the point of focus of the neighborhood of our finite subcover containing x and thereby connect f_m(x) to f_m(x') by equicontinuity and use our maximum N (over finite elements) to connect that to f_n(x') and use equicontinuity again to connect that to f_n(x). Thus, triangle inequality over three of \epsilon/3 suffices.

More explicitly, equicontinuity-wise, we have for every x some open neighborhood of U_x such that s,t \in U_x implies that |f(s) - f(t)| < \epsilon.




前几天,我看了徐道辉(Steve Hsu)与美国极右派Stefan Molyneux的讨论,有了深刻的感受。可以回想到徐提到在首尔或北京,一个女人可以在半夜到街上而对安全无所担忧,在美国的大城市这是无可想象的。虽未直言,可我们都知道是因为智商与罪犯行为的反相关关系应用在智商分布稍高的东亚国家之特例。徐也说道我们都有一点尼安德特人的血统,可是其占我们整个基因组很小一部分。智人所能创造的好多是尼安德特人无能的,故逐渐后者被前者覆盖而代替。他说我们可以想象他们创造物理学家或诗人的几率会比我们小很多。徐又漏出了他直截了当,对政治正确毫无在意的幽默,说:“我觉得我不会愿我的女儿嫁给一个尼安德特人。”



徐道辉对东亚国家所做的成功的地方显然已有认识。谈到这儿,我想起我的那位俄罗斯朋友曾经还开玩笑将他叫做“你的(东)亚裔优越主义朋友”。当然,徐也提到普遍被认为的东亚的过于顺从的文化不利于出做出革命性科学贡献的孤胆怪才,甚至东亚人天生就天才性格少出的可能性。毕竟人类文明最跨越性的时代显明是西方白种人创造的,是西方人创造了文艺复兴,科学革命,工业革命,周游并且占领殖民了几乎整个寰球,而在十九世纪中旬,西方白种人与其他人几乎是人夷之别。十九世纪末期,日本人和中国人都要想西洋人学习,尤其是学习他们的先进科学和技术。在那个时候,东方人都怀疑自己脑子本质上就是不如西洋人的,此在西洋遥遥领先横扫全球的情况是所预料的自然心理反应。不过,日本以飞快的速度吸收了大多西洋科技,成了第一个非西方现代化国家,此由1905年俄日战争之胜利所标志。中国人现代化的比日本晚的多,二十世纪上半中国所处于的内忧外患以及军阀内战对此有大大阻碍,可是中国派出去的留学生在理工科学的很好,逐渐把这些更先进的知识带回了他们的祖国。中国人和日本人打进近代科学的绝对一流的成果也都是从数学然后理论物理开始的,日本是第一世界大战时的高木贞治(Teiji Takagi)然后三十年代时的汤川秀树(Hideki Yukawa),中国是二战时期左右的华罗庚和陈省身,然后五十年代的杨振宁和李政道,这些都是在最需要智商的学科,表示了东亚民族极端的科学聪明才智。之后,中国人和日本人出的这样的人越来越多,现在已到频繁,不过在最顶级比西方还是要差一点或一些,尤其是中国。所以或许还是西方人最能出最天才的种子。

我总是觉得最最聪明的人大多还是犹太人,可以说二十世纪是没有一个,至少得以广泛认可的,与John von Neumann齐智的人了。同样,即使在科学深度和眼光也是犹太人处于巅峰。但是,这一点不是完全没有异议的。我的一位非犹裔国际数学奥赛金牌白人朋友却觉得东亚人比犹太人聪明,令我吃惊。不过,或许今天在年青一代还真的是这样,以中国学生为主的东亚学生常是精英数学竞赛的佼佼者,甚至占其主部为据,加上今年也有越来越多东亚数学家做出的精彩结果,以张益唐的孪生素为代表。徐道辉也跟我说,东亚人和犹太人是两个很不同的分布,前者多广泛,后者少儿精。对此,我想到了类似的比喻,那就是犹太人如斯坦福或哈佛,而东亚人如伯克利。此人口分布之差依然会给以最精犹多之结果,在这一点,我记得一位华裔国际数学奥赛金牌曾跟我说,犹太人虽然平均更聪明,但是东亚裔可以由数量弥补,照样可以出陶哲轩或张益唐这样的人。当然,智力难以作绝对的比较,因为每个人都有他自己的风格和特点,有长有短,而我感觉东亚人与犹太人,作为集体,表现出他们才华也是各有各的“民族特色”,是上千上万年分开进化所导致的基因和文化差异的必然结果。

诸多西方右派学者会谈到当代西方劣生的趋势。在此,已逝世的加拿大心理学家Philippe Rushton曾提到黑死病大大提升欧洲人智商而促使西方和人类文明大爆发的设想,并且猜测从此,白种人一直在逐渐退化到他们所有的“自然水平”,将此事件划为一个彻底改变人类走向的大偶然,并且对东方社会,尤其是中国,具有在西方主流极少有的乐观,并早在2006年就大胆说“他们足有脑力与我们同步或比我们更高。”现在看来,他是一位极其有远见的敢于纯粹真实的挑战主流错误观点的孤胆西方心理学家。他的研究发表曾经引起过轩然大波,不过我相信历史会证明他为类似于伽利略的科学烈士。Rushton的研究既科学又透彻,将智商和性格,在种族之间,与大脑和整个身体结构提出了整体的带有生理发育和进化缘故的描述与结论。Rushton的一位同派对偶学者Richard Lynn甚至觉得东方人会是西方文明的继承人,认为中国有更先进的,更高智商性质的,可以做出更有效决定及决策的”专制“制度可促进超越似的腾飞。





My awesome roommate

I recently met this cool guy because we live in the same place. Though he’s not that nerdy (by that, I mean super mathy), we still share many common interests. For instance, he expressed interest when I told him a bit about 艾思奇(Ai Siqi). Additionally, he told me about his appreciation for André Weil and Simone Weil, particularly her mysticism, which I found quite pleasing as I was reading about them not long ago. He also told me about this guy who is trying to understand Mochizuki’s “proof” of the abc conjecture despite being not long out of undergrad, who has plenty of other quirks and eccentric behaviors. Like, that guy joined some Marxist collective, and goes on drunken rants at 3 am, and is in general “aspie af,” something that he described me as too when messaging that guy himself. There is also: “he would literally kill himself if he had to do a tech job.” (laughter) That guy’s dad happens to be a (tenured) math professor from mainland China, more evidence that madness runs in families.

The guy that is the topic of this post himself did up to high school, as far as I know, in Hong Kong, so we have some more in common than usual culturally I guess. He was just telling me about how he had read 矛盾论, which I haven’t even read, at least not in detail, myself. He was saying, on the putative connection between scientific talent and Marxism, perhaps how dialectical materialism is inherently a very scientific way of thinking. I myself know basically nothing about dialectical materialism and even think it’s kind of high verbal low math bullshit, but I can tell that the materialist side of it is very scientific in its very nature, and similarly, dialectics is a very analogies/relationships way of thinking, which is something that high IQ people are by definition good at. Surely, there is much more I can learn from this guy, especially about Chinese language and culture and politics.

On this, I am reminded of another amateur (but professional, or better, level for sure) Marxist scholar, who is genuinely encyclopedic in his historical and cultural knowledge, in particularly a perceptive quote of him that made a deep impression on me:

Europe has always been in rebellion against itself, and continues to be so.  There was nothing but futility in the attempt by superficially Westernised Chinese to be authentically Westernised Chinese by being imitative and reverential of the current embodiment of those values.  You could only be an authentically Westernised Chinese by being a rebel against the current embodiments of Western values, at least in as far as they hampered China or seemed to be irrelevant.  And that’s why Mao was China’s best Westerniser to date, despite his very limited experience of the mundanities of Western life.

As I’ll detail in a future article, visitors to the Chinese Communist bases at Bao’an and later Yen’an noticed that these were the only Chinese in China who behaved more or less as Westerners would have behaved in a similar situation.  Other Chinese might speak good English, wear Western suits and sometimes show considerable knowledge of Western culture: but it was all imitation and the inner core was different and ineffective.  Western-trained engineers and geologists who returned to China kept their distance from hands-on practical work, because anything resembling manual labour would have lost them status in the eyes of Chinese intellectuals.  They were imprisoned by a tradition stretching back to Confucius and beyond.  Only a few broke these ancient taboos, mostly the Communists and some scattered left-wingers in the weak middle ground.  And it was the modernised Chinese in the Communist Party who chose to raise up Mao as the prime teacher of this new understanding.

I remember when my obsessively talented Russian friend once said to me that sometimes he feels like he’s another Pavel Korchagin, I thought he was ridiculous. Well, I’ll be equally ridiculous and say that I feel like I very much exhibit what Gwydion described in Mao that is “authentically Westernized Chinese,” which is very much the antithesis of what I see in most ABCs, despite being half an ABC myself.

If only more people could be like me…

Path lifting lemma and fundamental group of circle

I’ve been reading some algebraic topology lately. It is horrendously abstract, at least for me at my current stage. Nonetheless, I’ve managed to make a little progress. On that, I’ll say that the path lifting lemma, a beautiful fundamental result in the field, makes more sense to me now at the formal level, where as perceived by me right now, the difficulty lies largely in the formalisms.

Path lifting lemma:    Let p : \tilde{X} \to X be a covering projection and \gamma : [0,1] \to X be a path such that for some x_0 \in X and \tilde{x} \in \tilde{X},

\gamma(0) = x_0 = p(\tilde{x_0}). \ \ \ \ (1)

Then there exists a unique path \tilde{\gamma} : [0,1] \to \tilde{X} such that

p \circ \tilde{\gamma} = \gamma, \qquad \tilde{y}(0) = \tilde{x_0}. \ \ \ \ (2)

How to prove this at a high level? First, we use the Lebesgue number lemma on an open cover of X by evenly covered open sets to partition [0,1] into intervals of length 1/n < \eta, with \eta the Lebesgue number, to induce n pieces of the path in X which all lie in some open set of the cover. Because every open set is evenly covered, we for each piece have a uniquely determined continuous map (by the homeomorphism of the covering map plus boundary condition). Glue them together to get the lifted path, via the gluing lemma.

Let \mathcal{O} be our cover of X by evenly covered open sets. Let \eta > 0 be a Lebesgue number for \gamma^{-1}(\mathcal{O}), with n such that 1/n < \eta.

Let \gamma_j be \gamma restricted to [\frac{j}{n}, \frac{j+1}{n}]. At j = 0, we have that p^{-1}(\gamma_0([0,\frac{1}{n}])) consists of disjoint sets each of which is homeomorphic to \gamma_0([0, \frac{1}{n}]), and we pick the one that contains \tilde{x_0}, letting q_0 denote the associated map for that, to \tilde{X}, so that p \circ (q_0 \circ \gamma_0) = \gamma_0, with \tilde{\gamma_0} = q_0 \circ \gamma_0.

We continue like this for j up to n-1, using the value imposed on the boundary, which we have by induction to determine the homeomorphism associated with the covering projection that keeps the path continuous, which we call q_j. With this, we have

\tilde{\gamma_j} = q_j \circ \gamma_j.

A continuous path \tilde{\gamma} is obtained by applying to the gluing lemma to these. That

p \circ \tilde{\gamma} = \gamma

is satisfied because it is satisfied on sets the union of which is the entire domain, namely \{[\frac{j}{n}, \frac{j+1}{n}] : j = 0,1,\ldots,n-1\}.

A canonical example of path lifting is that of lifting a path on the unit circle to a path on the real line. To every point on the unit circle is associated its preimage under the map t \mapsto (\cos t, \sin t). It is not hard to verify that this is in fact a covering space. By the path lifting lemma, there is some unique path on the real line that projects to our path on the circle that ends at some integer multiple of 2\pi, call it 2\pi n, and that path is homotopic to the direct path from 0 to 2\pi n via the linear homotopy. Application of the projection onto that homotopy yields that our path on the circle, which we call f, is homotopic to the path where one winds around the circle n times counterclockwise, which we call \omega_n.

Homotopy between f and \omega_n is unique. If on the other hand, \omega_n were homotopic to \omega_m for m \neq n, they we could lift the homotopy onto the real line, thereby yielding a contradiction as there the endpoints would not be the same.

This requires a homotopy lifting lemma. The proof of that is similar to that of path lifting, but it is more complicated, since there is an additional homotopy parameter, by convention, within [0,1], alongside the path parameter. Again, we use the Lebesgue number lemma, but this time on grid [0,1] \times [0,1], and again for each grid component there is a unique way to select the local homeomorphism such that there is agreement with its neighboring components, with the parameter space in common here an edge common to two adjacent grid components.

With that every path on the circle is uniquely equivalent by homotopy to some unique \omega_n, we have that its fundamental group is \mathbb{Z}, since clearly, \omega_m * \omega_n = \omega_{m+n}, where here, * is the path concatenation operation.


Oleg is one of my ubermensch Soviet (and also part Jewish) friends. He has placed at (or at least near) the top on the most elite of math contests. He is now a math PhD student with an advisor even crazier than he is, who he says sometimes makes him feel bad, because he has done too little math research wise. However, this persona alone is not that rare. Oleg’s sheer impressiveness largely stems from that on top of this, he is a terrific athlete, extremely buff and coordinated, enough that he can do handstand pushups, to the extent that he regards such as routine. Yes, it is routine for a guy contending for a spot on a legit gymnastics team, but you wouldn’t expect this from a math nerd huh?

Today, I was talking to him and some others about gym. In particular, I was saying how I could at one point do 10 pullups but dropped down to 2 after a long hiatus. The conversation went as follows:

Me: Oleg I’m back to 5 pull-ups now
Oleg: that’s good although make sure you’re doing them for real
i still don’t believe you could do 10 but then dropped down to 2
Me: Oh I’m very sure they’re full pullups
Okay maybe it was 8
Oleg: i’d like to see evidence
Me: Alright I’ll have someone videotape me do pullups today in gym

And so I did.

Later, Oleg suggested something pretty funny:

i still think you should get tattoos and gain 25 lb of muscle, that would be hilarious
then walk up to girls and ask about their SAT scores
and say “oh, that’s too low, i don’t want to breed babies with you”
followed by a cackle
i’d watch that show

Not surprisingly, Oleg, as buff as he is, has had some success with girls, though he regards himself as shy and struggling in that regard. I keep telling him that he needs to marry a girl who’s both super smart and attractive like he is, so that he can optimize his chance of making superhuman babies. His only disadvantage now is that he’s a poor math PhD student, but he can easily change that by, say, joining DE Shaw, from what I’ve read is full of uber nerdy macho Eastern European men. He’s not very interested in money though, and expresses content with his graduate student stipend, which I find laughable.

I find it regrettable that most ubermensch men smart enough for legit doctoral programs in math and physics are unable to find a mate who is commensurate with them, ability wise, even with some adjustments, even when they’re well-rounded like Oleg is. Why is this? Excessive Aspergers? On that, I know someone who will say along the lines of

in an actual long-term relationship you have to share most of your life with the person, and if they don’t understand the way you look at the world then it creates friction
sure, the girl doesn’t need to understand high energy physics, I have other friends for that

Maybe some females could give us some advice, other than the cliche “hit the gym” that you’ll often hear from males. Such would be much appreciated! 😉