14 II 2023

types of mathematical terminology

descriptive

honeycomb

gradient

quiver

computable

less descriptive

centroid

chaos

end

flag

not descriptive

ring

allegory

surreal

group

you know this person, right?

euclidean

abelianization

grothendieck

cartesian

took some non-english word and hoped for the best

eigen

algebra

shtuka

nullstellensatz

i made up a word!

ergodic

functor

adele

logarithm

idk, just give it a generic name

regular

well

admissible

well-admissible

like, specifically, it’s a vague thing

flasque

lax

fuzzy

pseudo

one symbol and a word

*-algebra

D-module

K-theory

†-compact (although that’s going to usually be written “dagger compact”)

just random letters

rg

cwf

Fσ

erf

5 IX 2022

maybe once a month is a bit too seldom to post? I kinda want to form a habit of romanticizing my academic life, I see all those studyblr accounts with beautiful photos of their desks and notes and I'm pretty sure those images exist in their minds as well

maybe one day I will be considered studyspo lol

I'm just starting to work on some geometry problems for today, haven't yet decided what I will focus on, but there is this one problem that haunted me when I tried to sleep yestarday:

given a triangle ABC with ∠A = 60°, let P be a point in the interior of ABC such that ∠APB = ∠APC = 120°. prove that ∠APX = 90°, for X being the circumcenter of ABC

it's supposed to be solved using spiral similarity, which is a composition of a rotation and homothety. there was another problem that was listed as "spiral similarity exercise", but I proved it with angle chasing exclusively, creating some nasty drawings in the process

other than geometry I'm studying homology, at the moment the basics of homological algebra, such as the first proofs by diagram chasing and exact sequences

I made some notes for exact sequences induced in homology

my perspective on doing math is slowly changing I think, I feel inspired to search for problems that I would like to solve. I noticed that I have this mental block: before I start doing math for real, I need to learn all the theory. which is absurd, you can never learn all the theory

sure, obtaining truly groundbreaking results requires years of learning theory and mastering tools if you want to specialize in algebraic topology and geometry, but the mindset I have creates the comfort zone of "play safe, just read your textbook, no challenges for now" and I'm starting to see beyond that

right now I'm taking my first steps into understanding that reading textbooks and learning how to solve basic exercises is not enough. they are just methods that are supposed to help my creativity and curiosity do their thing. essentially what I've been doing so far is not math, merely the preparation to do math in the future. no wonder I've been feeling so bored recently, all I'm doing is just learning basic tools. the idealist in me is asking to be unleashed

I feel like I'm about to see something much bigger than me

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!

Me when I need to rotate 720 degrees to return to my original state

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

20 X 2022

the past few days were hectic

my grandma's burthday was nice, but very stressful, because until the very last minute I didn't know if I can go home with my mother or if I would have to take the train that would arrive at my city at 5am

I tried to study on my way to the event but unfortunately I didn't do much

annotating categories for the working mathematician was the peak of my abilities

I really enjoy the course btw, which is a bit surprising, because there are so many negative opinions about the teacher. right now we are at abelian categories and probably soon will move on to homological algebra

I spent a long time studying adjunctions but I can say that I understand them pretty well now

maybe a part of the reason why I like category theory so much is that I like drawing and chasing them diagrams

I started doing the second problem set for the analytic functions course. they are much easier than the first one. I managed to solve a half of them today. last time it took me a whole day to solve one problem

for the next few days I plan to

complete the analytic functions homework

commutative algebra homework

category theory homework

abelian categories

localization

analytic functions, differentiation and integration of complex functions

wish me luck and I wish you a pleasant evening

i've been working on it with my boyfriend for a while now and–

our first video is up!

i hope you enjoy it and subscribe to our channel

https://youtu.be/-X2BBtRI1Xw

tips for studying math

I thought I could share what I learned about studying math so far. it will be very subjective with no scientific sources, pure personal experience, hence one shouldn't expect all of this to work, I merely hope to give some ideas

1. note taking

some time ago I stopped caring about making my notes pretty and it was a great decision – they are supposed to be useful. moreover, I try to write as little as possible. this way my notes contain only crucial information and I might actually use them later because finding things becomes much easier. there is no point in writing down everything, a lot of the time it suffices to know where to find things in the textbook later. also, I noticed that taking notes doesn't actually help me remember, I use it to process information that I'm reading, and if I write down too many details it becomes very chaotic. when I'm trying to process as much as possible in the spot while reading I'm better at structuring the information. so my suggestion would be to stop caring about the aesthetics and try to write down only what is the most important (such as definitions, statements of theorems, useful facts)

2. active learning

do not write down the proof as is, instead write down general steps and then try to fill in the details. it would be perfect to prove everything from scratch, but that's rarely realistic, especially when the exam is in a few days. breaking the proof down into steps and describing the general idea of each step naturally raises questions such as "why is this part important, what is the goal of this calculation, how to describe this reasoning in one sentence, what are we actually doing here". sometimes it's possible to give the proof purely in words, that's also a good idea. it's also much more engaging and creative than passively writing things down. another thing that makes learning more active is trying to come up with examples for the definitions

3. exercises

many textbooks give exercises between definitions and theorem, doing them right away is generally a good idea, that's another way to make studying more active. I also like to take a look at the exercises at the end of the chapter (if that's the case) once in a while to see which ones I could do with what I already learned and try to do them. sometimes it's really hard to solve problems freshly after studying the theory and that's what worked out examples are for, it helps. mamy textbooks offer solutions of exercises, I like to compare the "official" ones with mine. it's obviously better than reading the solution before solving the problem on my own, but when I'm stuck for a long time I check if my idea for the solution at least makes sense. if it's similar to the solution from the book then I know I should just keep going

4. textbooks and other sources

finding the right book is so important. I don't even want to think about all the time I wasted trying to work with a book that just wasn't it. when I need a textbook for something I google "best textbooks for [topic]" and usually there is already a discussion on MSE where people recommend sources and explain why they think that source is a good one, which also gives the idea of how it's written and what to expect. a lot of professors share their lecture/class notes online, which contain user-friendly explenations, examples, exercises chosen by experienced teachers to do in their class, sometimes you can even find exercises with solutions. using the internet is such an important skill

5. studying for exams

do not study the material in a linear order, instead do it by layers. skim everything to get the general idea of which topics need the most work, which can be skipped, then study by priority. other than that it's usually better to know the sketch of every proof than to know a half of them in great detail and the rest not at all. it's similar when it comes to practice problems, do not spend half of your time on easy stuff that could easily be skipped, it's better to practice a bit of everything than to be an expert in half of the topics and unable to solve easy problems from the rest. if the past papers are available they can be a good tool to take a "mock exam" after studying for some time, it gives an opoortunity to see, again, which topics need the most work

6. examples and counterexamples

there are those theorems with statements that take up half of the page because there are just so many assumptions. finding counterexamples for each assumption usually helps with that. when I have a lot of definitions to learn, thinking of examples for them makes everything more specific therefore easier to remember

7. motivation

and by that I mean motivation of concepts. learning something new is much easier if it's motivated with an interesting example, a question, or application. it's easier to learn something when I know that it will be useful later, it's worth it to try to make things more interesting

8. studying for exams vs studying longterm

oftentimes it is the case that the exam itself requires learning some specific types of problems, which do not really matter in the long run. of course, preparing for exams is important, but keep in mind that what really matters is learning things that will be useful in the future especially when they are relevant to the field of choice. just because "this will not be on the test" doesn't always mean it can be skipped

ok I think that's all I have for now. I hope someone will find these helpful and feel free to share yours

touch grassmannian

touching grass is not enough sometimes. sometimes what you actually need to touch is a math textbook