(I'm Reblogging This For Later To Really Look Into All The Amazing Accounts I Follow)

13 I 2023

two days ago I went to the 0th term exam for commutative algebra and received the highest possible grade!

the thing I noticed when studying for it was that the topics that used to be fairly ok but not very clear became completely intuitive. the best example of this would be fibers of maps induced on spectra. it feels so good when after trying to understand something for two months everything finally clicks and I obtain a deeper level of understanding

also I realized that making pretty notes actually doesn't help at all, so I switched to making more messy, natural ones. maybe I can no longer look at them and admire the work of art, but I think the principle behind it is that the more I focus on making my notes pretty the less attention I pay to actual information processing

so maybe these ^ don't look as good as they could and they are probably hardly useful for anyone other than me lol but the benefit is that I started learning really fast compared to how it was going when my notes were a work of art

currently I am studying sheaf cohomology and preparing for a complex analysis test (it's next week). I have two courses left to pass and I would like to ace them too, although that's rather unrealistic

the second batch of topics for complex analysis includes: order of growth of entire functions, analytic continuation, gamma, zeta, theta functions and probably elliptic functions. significantly more sophisticated than the first part of the material. for the course on algebraic methods, everything is hard lol I am waiting for the moment when homological algebra and sheaf theory become intuitive

next semester I am going to take algebraic topology (fucking finally), differential geometry, number theory, statistics and algebra 2 (mostly galois theory). I have never taken 5 courses in one semester so I'm very scared

i'm conducting an experiment on how to study the theory effectively

there are i guess two main ways:

(1) read and take notes simultaneously

(2) read first, then take notes

so for the first one, there is the risk of going passive with the note-taking, writing down the symbols without focusing on their meaning. for the second one there is the risk of zoning out and just reading the symbols, again, losing their meaning

the problem seems to be that the processing of sheer symbols and processing their meanings might be disjoint and their natural tendency seems to be so

from my recent actions i noticed that (1) doesn't work for me as effectively as (2)

it might be that when i don't plan to write something down right away, i am more inclined to remember these things short-term as "i won't be able to check it later so remember it now in order to understand what comes next", and when i'm taking notes simultaneously it's "i have it written down anyway so i can take a peek anytime"

so now i'm testing the strategy of

read → try to understand the idea and memorize the elements → why all the elements are important → understand the construction in more detail and write it down

this is how i imagine my mind working:

it means that at first i start to remember the elements as points of its own but simultaneously my brain builds its idea on how they interact and then i notice the inner structure of how the elements are connected with each other in less obvious ways

this idea is cool to visualize how i imagine my thinking, because it shows how learning the topic reduces possible permutations and paths. i have this problem that when i start learning something new i see so many possibilities of what can happen to the elements that i can't discern between crucial and additional stuff. in order to use the knowledge i need to provide some structure

thus the main goal of optimized learning is to take the leap from "i memorized the elements" to "i understand their structure" as fast as possible

and so the strategy (2) might be more effective as it forces the memorization of the elements first and then it is easier to provide structure for them, where i would be defining order on something that's already in my mind. whereas (1) strikes at memorization and structuring simultaneously, it is too difficult for me to see at first in which direction the topic is going, i must know the next point

in a few days i will focus on how "the point" can be defined in this and how to characterize the connections

honestly tho this is some sorta pseudo graph theory and pseudo topology and i don't believe this could be as straightforward. otherwise nobody would ever post any study tips and we would have a field of study called "learning optimazation", this would be too big to go unnoticed. i wish it was so easy to just know how brain works and be able to build such an algorithm that would optimize the desired processes lmao

i wish i was a σ-field or something

side not is, i love this kind of thinking and i love to analyze how the thinking works, especially when it can me algorithmized or structured in some ways. the moment i see something is structured or algorithmic it becomes interesting to me

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

8 V 2022

I am on my way home from a math conference, the first one in which I participated actively – I prepaired the talk about the Borsuk-Ulam theorem

my lecture was centered around the connection between the classic "continuous" BUT and its combinatorial analog: Tucker's lemma

I wanted to talk about this because I was amazed at how cool and "versatile" this theorem is. there is a whole book about its applications and generalizations, which is btw very well-written, I highly encourage everyone to read it:

my presentation went well, although after practicing it for about a week the topic seemed really fucking boring to me, no wonder

other than that I have another recommendation to make. do you also hate how messy multivariable calculus is? I do. calculations and technical definitions everywhere, and at the end everything comes down to calculating the determinant of some jacobian. bluh. I stumbled upon a book that describes everything from a sort of algebraic perspective, smells a little bit like category theory too. very clean, very satisfying to read:

I have been studying covering spaces recently and I can give some dope motivation for learning about the structure induced by the covering mapping:

I will never forget that the homomorphism induced by the covering projection is injective

that would be it for my mathemathical life. my personal life, which is still closely connected to math, brings me some psychological progress. I no longer get stuck in loops of "oh I'm so bad at math. maybe I'm not? I got a good grade from X. ah but I got a shit grade afterwards". it might be because I didn't fall on my face for a while now, only decent grades, good ideas, a good presentation, this is correct. but I also do not negotiate with myself that this is supposed to be proof that I'm good enough, I just stopped paying attention to these and focused on math instead. and paradoxically when I stopped caring about being good at math I was rewarded with getting better at math???

a coincidence,

a pleasant one, nonetheless.

anyway I will have to take a fall at some point, unavoidable. and it will be the final test of my progress, becauase I used to get very elevated in my sense of self-worth after receiving a single good grade among trash ones and now I'm just ok. not the god, just ok. but back then, at some point I would no longer be god, I would get smacked in the face by some "proof that I'm actually trash" and that would be a fall from a significant altitude. so I'm hoping that the fall will also be less painful now

I think the biggest change I made was giving up, I abandoned all hope. nooow here is the moment when people interrupt me with "nooo that's horrible don't give up you're a great person you just have to notice that"

fuck off you don't understand shit

I'm doing better now precisely because I stopped hoping that one day I'll stop feeling worthless, that one day something great will happen that will prove once and for all that I'm meant for something great. I can't stand this anymore, I am disgusted by the fact that deep down I still believe that I'm supposed to be the best and that I can't enjoy anything unless I am winning. I want to puke when I'm reminded that everything I do serves the purpose of winning the negotiations I have with myself about what my actual value is

my self-hatred runs much deeper now than ever before and I have no more patience for self-victimization, no more room for "allowing myself to feel". fuck off, all I feel is rage. I want to be able to do things without the prospect of a reward, my goal is to enjoy things, not the sense of being good at doing things

so that's what I'm doing, I made peace with the fact that I will probably never feel good about myself and that I have no chance at achieving the greatness I crave. and I must say I started respecting myself more, turns out I am actually able to do things without the promise of being the best at them, the vision of bringing value to the world motivates me. and fuck the western culture with its oh you must love yourself you are a great person. no, you don't have to do that and you have no way of knowing what kind of person you are, nobody has ever defined it in a strict formal sense, people just use this phrase to trigger the feel-good in others

I am aware that all of this sounds really bad, but I don't care, it works. and my math will be better like that because now that I stopped crying over being trash I have more time to study

I just hope that the fall won't be as painful

2-3 VIII 2021

it's 4am currently, i woke up after a 5hour nap and i don't plan to sleep anymore, time for topo

sleep: weird but going in the right direction i guess

concentration: fine

phone time: good

i am currently dragging myself through some of the most important theorems in multivar calculus i believe. inverse function theorem, implicit function theorem, diffeomorphisms and stuff. the proofs are quite simple but very long hence exhausting, my least fav kind of proofs. right now i'm doing topo

tomorrow (or rather today) i'm planning to do more topo and possibly finish my notes from that calculus chapter

me : I love learning new things

Me when it’s time to learn anything new that I’m not instantly good at:

17 IX 2022

for the past few days life was treating me quite aggressively. today I had a terrible migraine, I feel weak and tired in general. doing math in a state like that isn't as pleasant so obviously I didn't do much, prioritized my health instead

during the semester I used The introduction to manifolds by Loring Tu to study analysis and I forgot that there were many nice exercises there that I didn't have time for but promised myself I would try them eventually

so tonight was the night and I studied grassmannians

I had some "results" done on my own, which later confirmed to be true, namely that the grassmannian over ℝⁿ for a 1-dim subspace is equivalent to a projective space of dimension n-1. I'm pretty sure that we are getting the projective of the same dimension for n-1 dim subspaces but I didn't calculate anything for n>3 so I might go back to that one day

it's fun to get hunches like that even if they turn out to be completely obvious to the authors of textbooks lmao

I am finally in the place with studying the theory for homology, commutative algebra and apparently differential topology (as it turned out today), where I have a variety of exercises I can try and that's the good part for me, always helps to get deeper insights and allows me to be more active

a friend asked me for a talk about the zariski topology in the context of algebraic sets and spectra of rings, so I'll see her soon for that. she will give me a personalized lecture about her thesis, which is about general topology. I am not a big fun of general topo but I'm always a slut for lectures about math so am excited for that

I hope my body will get its shit together because I still have to prep my lecture on euclidean geometry and when I don't feel good it's super difficult to motivate myself to do things that are not super exciting. I will never see productivity as a value on its own for this very reason lol I can barely do anything I don't find interesting

25 XI 2022

I neglected this blog a little, a lot is going on right now

I have a lot of work and I'm barely keeping up, I was sick for two weeks because not going to school would result in even more problems, so the cold didn't want to go away. I'm fine now but the lack of sleep is still fucking with my cognitive performance and I'm in general very exhausted both physically and mentally

today I had a meeting with the dean to talk about the accommodations for adhd and asd and it went very well, he is such a nice guy. we discussed extended time on tests, getting more specific instructions from professors and just a bit of extra care so I don't get overwhelmed. we also talked about a mentor who would help me with organizing my studying and the dean said that he will find someone who would help me with progressing in my field of interest, which sounds very promising. I don't know yet what that's gonna be, maybe algebraic topology, maybe something leaning more towards algebraic geometry, we'll see

when it comes to what I'm doing right now, we did some more stuff from homological algebra (projective and injective objects, derived functors and group homology) and the topics from commutative algebra have more geometric motivations, so the course becomes more and more enjoyable. learning complex analysis is much easier than those two other courses because there is significantly less theory and even if the problems are super difficult, it doesn't require as much brain power

other than doing homework I'm trying to find some time to read Introduction to Differential Topology by Jänich, although recently time is a scarce resource. the book is great tho

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

when I (fucking finally) finish this semester I plan to do a deep dive into TQFT and frobenius algebras with this book recommended by my supervisor:

I find the concept to be very elegant. loosely speaking, take a commutative ring R and an algebra A over this ring that satisfies the axioms of the frobenius algebra. it turns out that for any such algebra there is an R-module associated to a certain 3-manifold, in which there are operations (induced by the algebra) on cobordisms between the systems of curves embedded in the boundary of the manifold. this is related to knot theory and apparently to some quantum blah blah, which I don't know much about yet

rb this with your favorite math concepts/books/videos... things u enjoy and that make you excited! (or reply but i want to hear about it and if you rb it then i hear more cool stuff from more people)

my favorite books are the grapes of math and things to make and do in the fourth dimension. i'm also reallyyyy wanting to read number freak and godel, escher, bach. concepts i love are chaos theory, non-euclidean geometry, and dimensions beyond 3rd!