8 V 2022

for the sake of an updates to this, I didn't get 100% on that topology test. I got 85%, which was the third best score. I finally scored the highest possible final grade on that subject, so I'm satisfied. fuck I love algebraic topology so much and I think she loves me

oh and I scored fucking 54% on the analysis test. I think I had a mental orgasm when I found out about that lmao it felt so good. I finished the course with a grade of 4 (idk if it's universal, so 2=the lowest, failed, 5=the highest) which is the best I ever got in the analysis course

28 V 2022

topology and analysis tests are over, both went I think alright

if I don't get 100% from topo I'm going to be very frustrated, because I studied hard and acquired deep understanding of the material – so far as to be able to hold a lecture for my classmate about any topic

analysis ughhh if I get ≥40% I will be overjoyed. but that's just the specifics of this subject, you study super hard and seem to be entirely ready, you solve all of the problems in prep and then best you can do is 40%. my best score so far was 42%, so anything more than that will be my lifetime record lmao, I want this so bad. I solved two problems entirely I think, which should give 40% already, and some pieces from two more, chances are I get 50%, which would be absolutely amazing

here are some pictures from me transforming math into an art project

stokes theorem

topology

I was thinking about how annoying I find what people say to me when I tell them that I'm not happy with how I'm doing at math. their first idea is to tell me how great I am and how all I do is good enough and shit like that. it doesn't help, it just feels like I am not being taken seriously. when I barely pass anything, am I really supposed to believe that everything is actually good? it feels like they skip getting to know my situation and just tell me what they would tell anyone, automatic

when I try to calm myself down and think something that will keep me going I don't try to force myself to be happy, fuck that, not being content with one's achievements is very fine, I believe not being happy all the time is fully natural and all that positivity feels so fake

instead what seems to work is asking myself where the rational threshold of being ok with how I'm doing is. the thing is I will never be satisfied, whatever I have, I always want more. but I can set the limits in advance and that stops me from falling into self-loathing loops

although what has really changed the game for me was getting a few good grades, finally I am achieving something, anything. people tell me that I should learn to be alright without this external reliance on achievements but how am I supposed to do that when the source of my low moods is precisely getting less than I want? I don't understand why I should brainwash myself into thinking that this is actually not what I want. the trick here is to separate the goal-orientedness from the sense of self-worth. the groundbreaking realization of mine was figuring out that I believe I deserve more than I get, that's why I am unhappy. so now that I am getting what I think what I deserve I obviously feel much better

I've been thinking about how different math feels after three years of consistently doing it. it's a sad thought, because I used to get super excited about learning new things and solving problems, whereas now my standards seem to be higher..?

I spent the day doing exercises from galois theory and statistics, in preparation for the tests I have soon. it felt like a chore. sure, the exercises were easy and uninteresting, I decided to start from the basics, so there is that. however, in general practicing like this became a routine and there used to be a sense of mystery around it that is now gone

when I don't have any deadlines but feel like doing some math the obvious choice is to learn something that will be useful in the future. more homological algebra, algebraic geometry, K-theory, or digging deeper into the topics I already am familiar with. all of those are good candidates and I used to be very motivated to just learn something new. but here comes to paradox of choice, where every option is good, but there isn't a great one

I think I might be annoyed with always learning the prerequisites for something not yet defined. it did feel exciting when I was studying the modules of tangles so that I could answer an open question, it doesn't feel as exciting to learn about the galois theory to pass a test. a metaphor comes to mind. doing math without a fulfilling goal feels like taking a walk – it's rather nice, I enjoy going on walks. with a fulfilling goal it feels like walking towards a destination such that the walk itself is a pleasant activity, but I really want to get to said destination. by that I mean that I still enjoy simply learning new stuff and working on exercises, but it doesn't feel as fulfilling as it used to, how much walking without getting anywhere can you do in three years? you can do the same thing in prison

three years is nothing compared to how much knowledge and experience is necessary to do actual research, I know that. I fail to feel it, but I know it. when I am asking myself what state of mind is the most fulfilling I'd say exploration, discovery, getting an idea that is new to me and seemingly comes from nowhere, not just an obvious corollary of what I've seen in lectures, an insight, an act of creating. I suppose all those things are to be found in the future, but god how long do I have to wait

on a more pragmatic and realistic note, I think I'll talk to my professors about what I can do to speed up that process. I'll ask them how the actual research feels and how they went from being a student learning basic concepts to where they are now

a question to those of you who are more experienced than me: does this even sound familiar at all? what were you like as a student and what took you to where you are now? how does math feel after 3, 5, 10 years?

Square is a rhombus, regular hexagon can be tessellated with three equal rhombuses, and every regular polygon with even number of sides can be rhombi-tessellated.

30 VII 2021

did some stuff today. found out my cat would die tomorrow if it wasn't for an operation he had today, that didn't feel good but also oh god was he lucky

sleep: better but still trash. yesterday fell asleep between 3 and 4, today planning to go to bed at 3 so in a few minutes

concentration: not as great. couldn't focus because my cat was fucking dying

bo phone time: decent

did some topo today, i think i managed to understand the idea behind the quotient spaces and i really liked it. can't wait to dive deeper tomorrow. other than that i wrote a method that takes a parametric function and provides a partition dividing the curve into k intervals. also i'm almost done with the art comission

sooo tomorrow i plan to finish the code, i want to achieve the functionality that takes a parametric function and draws it with a dashed line. that's why i needed the partition, it's gonna look fucking beautiful. and i need to complete the comission. i hope there will be some time left for quotient spaces, i am very hyped. oh and i forgot i'm drinking tomorrow. so i guess no topo for that gal. eh

I have just met you and I love you (via)

So the exponential function is given by

which evaluated at a real number x gives you the value eˣ, hence the name. There are various ways of extending the above definition, such as to complex numbers, or matrices, or really any structure in which you have multiplication, summation, and division by the values of the factorial function at whatever your standin for the natural numbers is.

For a set A we can do some of these quite naturally. The product of two sets is their Cartesian product, the sum of two sets is their disjoint union. Division and factorial get a little tricky, but in this case they happen to coexist naturally. Given a natural number n, a set that has n! elements may be given by Sym(n), the symmetric group on n points. This is the set of all permutations of {1,...,n}, i.e. invertible functions from {1,...,n} to itself. How do we divide Aⁿ, the set of all n-tuples of elements of A, by Sym(n) in a natural way?

Often when a division-like thing with sets is written like A/E, it is the case that E is an equivalence relation on A. The set of equivalence classes of A under E is then denoted A/E, and called the quotient set of A by E. Another common occurence is when G is a group that acts on A. In this case A/G denotes the set of orbits of elements of A under G. This is a special case of the earlier one, where the equivalence relation is given by 'having the same orbit'. It just so happens that the group Sym(n) acts on naturally on any Aⁿ.

An element of Aⁿ looks like (a[1],a[2],...,a[n]), and a permutation σ: {1,...,n} -> {1,...,n} acts on this tuple by mapping it onto (a[σ(1)],a[σ(2)],...,a[σ(n)]). That is, it changes the order of the entries according to σ. An orbit of such a tuple under the action of Sym(n) is therefore the set of all tuples that have the same elements with multiplicity. We can identify this with the multiset of those elements.

We find that Aⁿ/Sym(n) is the set of all multisubsets of A with exactly n elements with multiplicity. So,

is the set of all finite multisubsets of A. Interestingly, some of the identities that the exponential function satisfies in other contexts still hold. For example, exp ∅ is the set of all finite multisubsets of ∅, so it's {∅}. This is because ∅⁰ has an element, but ∅ⁿ does not for any n > 0. In other words, exp 0 = 1 for sets. Additionally, consider exp(A ⊕ B). Any finite multisubset of A ⊕ B can be uniquely identified with an ordered pair consisting of a multisubset of A and a multisubset of B. So, exp(A + B) = exp(A) ⨯ exp(B) holds as well.

For A = {∗} being any one point set, the set Aⁿ will always have one element: the n-tuple (∗,...,∗). Sym(n) acts trivially on this, so exp({∗}) = {∅} ⊕ {{∗}} ⊕ {{∗∗}} ⊕ {{∗∗∗}} ⊕ ... may be naturally identified with the set of natural numbers. This is the set equivalent of the real number e.

oh and there is the dual thing: sometimes you just know that the professor hates the subject. like when I was taking one of the analysis courses, where the lecture was with one professor and the tutorials were with a different one

at the lectures we were two months into measure theory while at the tutorials haven't even started doing exercises on that topic, but oh it was fine, still plenty of time, he knows what he's doing – we thought, like fools. then the midterm was announced, two weeks left, we still haven't started measure theory. then it was one week left, so the professor tried to solve some lebesgue integrals with us, but he got so bored with each example that he hasn't finished a single one. at this point we just hoped that maybe measure theory just won't be on the midterm, it was too late to do anything. well, unfortunately, the midterm consisted mostly of measure theory problems, it made sense because that was the main content of the course

the professor was clearly very passionate about hating measure theory

One of the really amusing things about college is that if you pay attention you sometimes can discern some of your professor's favorite pet concepts.

For instance, in my Topology course this semester, the Zariski topology has come up at least once in every single homework set so far, and in multiple lectures.

And okay, that's not that weird. The Zariski topology is a really important object in a LOT of fields, especially algebraic geometry. And discussing it at length is a really pedagogically sound move because the Zariski topology is a good example of a topology with a very well motivated structure (the closed sets are the algebraic sets!) that still very naturally gives rise to a lot of strange features, like the way all open sets in the standard topology are Zariski-dense. It was quite effective at startling me out of the complacency of unconsciously basing my intuition of how topologies behave entirely on the standard topology on the reals. So my professor bringing up Zariski so often doesn't necessarily mean he has any special affection for it.

except...

My professor writes many of the homework problems himself. Not all of them - the less interesting ones he lifts from the textbook- but some. Well, every single Zariski topology question I've encountered so far is an original from this guy. I know because the all the questions he writes personally have paragraphs of commentary contextualizing why he thinks the problem is interesting and where the ideas in the problem are going later in the course. And well- let's just say the asides on the Zariski topology have been copious indeed

AND THEN there's the way he talks about the Zariski topology in class! It's with this blend of enthusiasm and fascination only comparable to the way I've seen tumblrites talk about their blorbos. Like hey! Come behold this sgrungy little guy! Isn't he fucked up? Isn't he marvelous? And I look and I can only conclude YEAH that is indeed a spectacular specimen, he's so strange, I want to put him in a terrarium and study him (and then I get to! In my homeworks!)

Anyways. It makes me really happy picking up on how excited my professor is to share this topology with us. I'm kind of baffled that people assume math is a boring field full of boring people when there exist folks like my professor who get this passionate about a topology!

I got 55 and it seems to me that the majority of my answers were heavily influenced by asd

I took a test on like where you are on the ‘nonverbal intimacy scale’ and the average female score is 102 and male is 93.8 and I got 56 lolololol

here it is if ya want (reblog/reply w/ what you get!!)

chaotic good

Pro-tip: You can use paper twice if you take your notes in pencil first and then write over it in pen. 

@shitstudyblr please validate me

Balance