Me When Čech Cohomology

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Accepting ourselves the way we are means we allow ourselves the things we need to make life a little easier. You don’t have to fight it, it’s ok have different needs to others. You are worthy of kindness, so be kind to you <3

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" 'They' isn't singular!" Oh yeah? Show me its multiplicative inverse matrix then.

when K ⊆ L is a finite extension by one element, say α with the minimal polynomial f, we can write 0 → (f) → K[x] → L → 0, where (f) is the kernel of evaluation at α. this is quite disappointing and very basic, but I haven't found anything better really. when there are finitely many intermediate fields between K and L for an extension L/K, L can be expressed as an extension by one element (Artin's theorem), which is still very specific

I didn't know about the group extensions, it makes the category of fields even more disgusting. I was hoping that the algebraic closure can be expressed as a colimit but of course not, not in the general case at least. but maybe at least some type of extensions can be realized as such? that's a nice thing to ponder. I'm pretty sure it will fail like every other request I had for this abomination of a category

I wonder what is typically done to make working with this category more pleasant. passing to Grp with the Galois group is one idea, the other I guess would be working with vector spaces or algebras? that would make sense considering that integral and finite ring maps are a thing and the field automorphisms play a role in the integral closure of ℤ in ℚ[√d]

on a sidenote, I laughed at the "lower body" and it reminds me how funny it is to talk about kernels in Polish. kernels and testicles are the same word

I've always thought 'splitting field' was a very cool sounding term. The Galois theorists did good with that one

What's the beef between engineers and physicists and even mathematicians.

Why physicists mock mathematicians: Because playing 51 dimensional chess against your own brain seems silly to us when there’s a whole cosmos to explore.

Why mathematicians mock physicists: The universe can only be understood because some nerd spent the time playing 51 dimensional chess and in the process they created some useful stuff for the physicists to steal and abuse the hell out of.

Why everyone mocks the engineers: π=e=3 is an abomination before God and those pencil pushing dorks make more money than us so we feel the need to vindicate our $75000 student debt.

no fuckin way

21 I 2023

so the test I had today, our professor went crazy with grading it and we all had our scores by midnight

I don't think I ever scored 100% before, but here it is

I was insanely lucky. yesterday I was watching some series (and by that I mean Young Royals, not Fourier) and I had a thought you know might as well give them elliptic functions a quick read. today one of the easy problems required to only know the basic definitions and properties, have I not spent those 40 minutes reading I would probably not solve it. the other easy problem was solved by picard's theorems, my favourite, which I tried to use with every given opportunity so now it's as they say: when your only tool is a hammer every problem looks like a nail. and today it actually was a nail. two other problems were just objectively easy and the last one took a lot of my time but it was "my type" of problems, so I enjoyed working on it and I had some good ideas thanks to solving about 20 similar problems before

so that's how it feels to reach above my goals. I dreamt of this moment and it feels exactly like I thought it would. ah feels good man

Every now and then I remember that Malbolge exists and I get to spend the better part of an hour cry-laughing at the world’s worst programming language

already starting off strong, but it gets worse

Wow! Sounds easy and intuitive to use! What’s the “crazy operation” you ask? We’ll get to that later. For now let’s see what a program in this language looks like :)

Thanks! I hate it!

it’s so difficult to work with that the first program was written by another brute force search program

mmmmm delicious base-3 arithmetic, what could go wrong? (For reference, that means this program forgoes the usual “0/1″ values of binary code in favor of a much more fun “0/1/2″ set of values)

ah.

Here’s how the language actually figures out what to do. It’s got 8 “simple” commands that can be executed easily by *checks notes* running the code itself through the modulo operation and taking the result.

As a bonus, on top of all that every single character in your code will now alter what every single other character does. So I hope you’re alright with cracking a cipher every time you add a new letter to your program!

oh god oh fuck.

behold, Malbolge’s primary arithmetic operation and what you’ll be using for most of your math while programming with it :)

This looks specifically designed to be the least logical math operation you could make, and knowing what the rest of Malbolge is I’d wager that’s precisely what happened. I never want to ever use this and it’s my favorite thing I’ve ever seen.

https://en.wikipedia.org/wiki/Malbolge

Anyways here’s the wiki page if you wanna read through it more deeply, I’m gonna sit here holding in my laughter staring at the hello world program again.

this is going to be difficult -> i am capable of doing difficult things -> i have done everything prior to this moment -> this difficulty will soon be proof of capability

I neglected this blog like hell, sorry

I had a lot of work to do, that's kinda what happened. but I would like to go back to posting regularly, so maybe I could write about something people would want to see?

for now my ideas for posts include

more study tips

a quick intro to moduli functors, since a lot of sources are written in a way that requires advanced algebraic geometry. I could explain the basics using (almost) only commutative algebra

updates on my life and what I've been working on

books recommendations

interesting math problems I encountered recently

if you'd like to see any of that, let me know! and feel free to give me more suggestions in the comments

Astronaut sculpture from an ex-physicist (Source/Credit)