## 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.

## 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.

## 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 $\tilde{x} \in \tilde{X}$, Continue reading “Path lifting lemma and fundamental group of circle”

## Oleg

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?

## More mathematical struggles

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.