Today I gave my first attempt at actually learning a bit of toki pona. I already knew it was a language from a while ago watching some youtube videos, but today I decided to see how it looks like. I was pleasantly surprised to know the vocab was based off of multiple languages around the world, which personally already makes me more fond of it than esperanto, which heavily based itself off of European languages, most influenced by romance languages. It was never fit to be a universal language in the first place (my opinion).

Toki pona however, wasn't constructed to be one in the first place. It's mostly a trial to see how minimal of a language can we make, in a way that it's still functional (seems like nowadays it has about ~137 words in overall usage, naturally it has dialects), but that's where it would be a nice auxiliary language. It's good for conveying simple ideas, but I've heard their dictionary contains more than 10 thousand words, so I'd love to see how those words are written using compounds of the root toki pona words.

I've looked up about Munkres in the past and from what I've read, the two things he is known for his books, especially topology, and for some an algorithm I didn't bother to look up at the time.

Now that the uni semester is finally over, I tried tackling some of my small programming projects, and I found myself needing to solve an interesting problem. For a given $m \times n$ matrix, I need to extract a subset of its entries, such that every row and column has at most one chosen entry (and containing as many rows/columns as possible, which means we're choosing $k$ elements, where $k$ is the minimum between $m$ and $n$), while minimizing their total sum.

Turns out the so-called Munkres assignment algorithm does exactly what this problem describes. Like, it has nothing to do with topology, it's pretty much a very crazy coincidence. I don't know how it works though, I just imported a library for now.

To unite topology and group theory, we can define a topological group as a group $(G, \ast)$ with a topology such that $(x, y) \mapsto x \ast y$ and $x \mapsto x\inv$ are continuous functions. Only requiring the first to be continuous is not enough.

Some sources like Munkres additionally requires it to be $T_1$, which seemed arbitrary at first, why not Hausdorff instead? It turns out that (1) being Hausdorff (2) being $T_1$ (3) $\{e\}$ being a closed set; are all equivalent statements for topological groups. So one interesting way to prove some space is Hausdorff is to define a group operation on the space which is continuous and its inverse is continuous too, then prove that $\{e\}$ is closed. This is probably useless.

One of the biggest issues about programming is that coding isn't enough. I love writing code, I love tackling a problem and writing a program that does a task I want, and figuring out how to optimize or have fun with it. However, programming is also about integrating tools together. Maybe you have some sort of backend running your code, or you're making some code which will embed in some other programming language, or you have to balance between multiple different apps.

My biggest issue with all things like that is web dev. Because guess what? You wanna make a REST API? You wanna make a database integrated with that API? You wanna make a dynamic website that makes requests to that API that has access to the database? Guess what, doing all that is the easy part! Having all of that setup and running in your localhost is a smooth process, which follows by the absolute existential dread of figuring out how to make that public.

I just read about quotient spaces, and I'm quite fascinated to have learned that those "images" that I would see online which involve "visualizing" some two-dimensional shape like a torus as a unit square with some funky business going on at the border are actually quite simple to formalize using basic topology and equivalence classes.

The idea is that if we define an equivalence class on a topological space $(X, \tau)$, then there is a unique topology on $X/{\sim}$ such that a subset of equivalence classes is open if and only if its union (which is a subset of $X$) is open in $X$. So when we take a unit square and write "arrows" at the border, what we're really doing is describing visually what the equivalence relation is!

I'm too lazy to put a picture here, I just wanted to share my joy.

Youtube's algorithm recently blessed me with a channel called Zundamon's Theorem. It's a japanese channel about maths where the hosts are actually two characters which are vocal synthesizer characters (basically like Miku, I'm not sure if I can say it's like vocaloid but for speech instead of singing).

It's very cute and the creator actually adds english subs for most videos, which is impressive. Hopefully this will help me learn math terms in japanese such as 定理 (it means theorem and it surprisingly sounds like theorem, probably a coincidence). Hopefully that'll make youtube recommend me more japanese math channels.

I've been studying topology using the famous Munkres book. In fact, if you look up which topology books are good, it's pretty much guaranteed that his name will show up. Not only that, but he's been a professor at MIT for decades. So I got curious to see him talk. Maybe an interview, or even better, a lecture or any sort of speech.

I'm stumped. There is absolutely not a single video anywhere with James Munkres. This guy is probably one of the most famous topology authors in the world, was even elected as a fellow in the AMS in 2018, a professor emeritus at MIT, and still not a single record of him anywhere, except for a single picture of him in his MIT professor profile. Is it because he doesn't want to be recorded anywhere and avoids speaking in conferences? Or did nobody ever think to invite him for any sort of interview?

I'm taking a vector analysis class, and upon reaching the Stokes's theorem, our professor briefly mentioned Stokes did not actually prove his theorem. It's funny, but definitely one of many situations where we name theorems after people just "involved" in the theorem but not actually the provers.

I found this very nice paper¹ describing the history we know. In short, Ostrogradsky proved the first version of what we know as the divergence theorem. Green then proved a different version of it, which when reduced to two dimensions, derives what we know as Green's theorem (except that Green never seems to have done that).

The first version of Stokes' theorem in $\R^3$ was actually in the Smith's Prize exam from 1854 (yes, it was the question to an exam, specifically question #8 found here²), and the one who and the author of the exam was Stokes himself. So we only call the theorem after him for perhaps shedding light on this problem, but from what we know, the first to actually prove it was Hermann Hankel in 1861, using the results of Green.

One of the three writing systems of Japanese, Kanji, originates from Chinese, and so they are basically chinese characters. A fluent reader should be able to read about ~2,000 of them, and thankfully, they're made up of smaller components which helps to memorize and write. For example,

All of these kanji contain 艮 as a subcomponent. I've been using jpdb to memorize these components, and because the kanji for "good" is 良, they call this component "almost good" (get it?). But this produces a very fun pattern. The kanji for gold is 金 and the kanji for silver is 銀, if you think about the olympics, what is "almost as good as gold"?

Back when I was a new undergrad student taking my first analysis course, I was shown many proofs involving the cardinality of sets. I was introduced to the fact that the cardinality of the power set is always strictly greater than the set itself $$\text{card}(X) < \text{card}(\cP(X))$$ But I never bothered with looking into the proof, being scared of it looking complicated. Looking back, that was silly, because I finally got to properly read it, and it couldn't be simpler.