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.
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。
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.
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. Continue reading “My awesome roommate”
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 be a covering projection and be a path such that for some and , Continue reading “Path lifting lemma and fundamental group of circle”
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?
Math is hard. It wrecks my self-esteem, and at times, it makes me feel an utter loser, who simply isn’t smart enough, who is a league if not multiple away from the big name mathematicians, who come up with much if not most of the most original results in mathematics. There are times when the formalism within the mathematics looks, perhaps superficially out of lack of perception no the part of its viewer, so excruciatingly complex and dry, and that one is inclined to simply go: this is too hard, give up. I’ve felt that, and I think just about everyone, no matter how smart, has, to some extent. Over time, I’ve come to realize that the dirty details tend to be a natural product of a few main ideas behind the proof, and once such ideas as grasped, every detail can easily be seen to have its rightful place within the entire construction. There was a time when I felt demoralized or slightly baffled upon seeing this answer of Ron Maimon that can totally come across as intellectually too presumptuous, from a guy too smart who never had to struggle like all us ordinary folks, from a guy who takes for granted as routine what is a slog for most, without being metacognitively aware enough to appreciate that he is of a totally different beast. In this, stood out the following quote:
You need to learn to “unpack” proofs into the construction that is involved, to know what the proof is saying really. It is no good to memorize the proof, you need to understand the construction, and this will motivate the proof.
今天网上搜奖牌榜预测，从而得知了Lindsay Vonn，因为那里的视频竟然有这位美国滑雪运动员为禁俄罗斯之决定张正嘴。好奇这种运动员的背景的我扫了扫此维基百科页，看到这位挪威裔姑娘出生于冬天满雪盖地的Minnesota，两岁就开始滑雪，并且在这方面有曾经赢过junior title的父亲的强烈鞭策。不用说，滑雪是有钱人的运动，所需要的设施是必可避免的昂贵，加上又有受伤的风险，想起前几年赛车手Michael Schumacher滑雪时头撞上了石头，导致达到长期不省人事的重伤，现在基本处于永久脑损伤状态。就是在美国，也有人笑话加以藐视的把冬奥会当做一个rich white person的聚会。
Yesterday, I met this kid who just entered UC Berkeley for undergraduate. I had known him when I was a kid through friends of parents, though of course, when you’re a kid that kind of age gap disallows much meaningful social interaction. I vividly remember senior year of high school in my Spanish class, we were in the school library computer lab doing some project with a partner, and on the front page of the school district, there was news that this kid won a state math competition. I immediately said to my partner: “I know that kid!” And yesterday, he told me that he made AIME in 8th grade, and also later the USAJMO, which shows that he is highly gifted, perhaps more so than I am.