Math - Blog Posts
Hey, im second semester math undergrad, do you recomend any book for calculus?
hi, unfortunately for first year analysis/calculus I used mostly the resources given by the professors, however, when I did use textbooks I really liked Walter Rudin:
as far as I know, many people recommend Apostol's book, which looks very good and if I was to choose a textbook for myself right now I would definitely try this one:
other than textbooks, if you like learning math from videos check out this channel:
Michael Penn is a teacher at a university and he's great at explaining theory and solutions of problems
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
30 I 2023
in a fortnight I will have two oral exams and one problem-based exam
the first oral will be for complex analysis and we are supposed to choose three topics from which the professor will pick one and we'll have a chat. I chose meromorphic functions, Weierstrass function and modular function. I have already received my final score from homeworks, which is 73%. combined with 74% and 100% from tests, I am aiming for the top grade
the rest of exams will be for algebraic methods. a friend who already took this course told me that when someone is about to get a passing grade, they get general questions and the professor doesn't demand details of proofs. when I asked him if we are supposed to know the proofs in full detail or if it suffices to just be familiar with the sketch, he told me that if I will only know the sketch I will sit there until I fill in all the details. lmao that sounds like he wants me to get a top grade. ok challenge accepted
so it seems like I have a chance to ace everything. if I achieve this and do it again next semester I can apply for a scholarship. studying for the sole purpose of getting good grades doesn't feel right, the grades should come as a side effect of learning the material. buuut if I can get paid for studying then I might want to try harder, I enjoy being unpoor
the next two weeks will be spent mostly grinding for the algebraic methods exams, this is what I'm doing today
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
19 I 2023
this week is kinda crazy
I have a complex analysis test on saturday and the professor said that it will cover the entire semester. thank god I might get away with not knowing anything about analytic number theory lmao
I had troubles sleeping lately, it takes me about 3-4 hours to fall asleep every day. I sleep a lot during the day and it helps a bit but I still feel half-dead all the time. every time I fall asleep my brain can't shut up about some math problem
for the algebraic methods course we were supposed to state and prove the analogue of Baer criterion for sheaves of rings. I was the only person who claimed to have solved this, so I was sentenced to presenting my solution in front of everyone. the assertion holds and I thought I proved it but the professor said that the proof doesn't work, here is what I got:
he said that we cannot do this on stalks and we have to define a sheaf of ideals instead. when I was showing this I had a migraine so no brain power for me, I couldn't argue why I believe this to be fine. whenever two maps of sheaves agree on each stalk they are equal, so if we show that every extension on stalks is actually B → M on stalks, then doesn't that imply the extension is B → M on sheaves?? probably not, but I don't see where it fails and I'm so pissed that I was unable to ask about it when I was presenting, now it's too late and this shit keeps me up at night
I enjoy sheaf theory very much and I can't wait to have some time to read about schemes, I have a feeling that algebraic geometry and I are gonna be besties
during some interview Eisenbud said that when deciding which speciality to choose one should find a professor that they like and just do what that professor is doing lol. I feel this now that I talked some more to the guy who taught us commutative algebra. since my first year I was sure that I will do algebraic topology but maybe I will actually do AG, because that's what he's doing. is having one brain enough to do both?
anyway I'm glad that my interests fall into the category of fashionable stuff to do in math these days. my bachelor's thesis is likely going to be about simply-connected 4-dimensional manifolds, which is a hot research topic I guess. I won't work on any open problem because I'm just a stupid 3-year, not Perelman, but it will be a good opportunity to learn some of the stuff necessary to do research one day
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
25 XII 2022
this chunk of the semester is finally over, sweet jesus I'm so exhausted. I'm getting the well-deserved rest and later catching up with all the things I put on my to-do list that I kinda learned but not really
the test I had last week went fine. frankly I expected more from it after solving more than 50 problems during my prep, but I scored 74%, which is objectively great and more than I predicted after submitting my solutions
here is my math plan for the break:
in algebraic methods I started falling behind a few weeks ago when I missed two lectures while being sick. they were about resolutions, derived functors and group homology and afterwards I wasn't really able to stay on top of my game like before. high time to get back on track. in commutative algebra I was doing ok, but there are some topics I neglected: finite and integral maps and Noether's normalization. for complex analysis everything is great until we introduced the order of growth and recently we've been doing some algebraic number theory, which btw is a huge disappointment. don't get me wrong, I understand the significance of Riemann's ζ, but the problems we did all consisted of subtle inequalities and a lot of technical details. I am doing mainly algebraic stuff to avoid these kind of things lol
when we were doing simplicial sets I stumbled upon some formulas for the simplicial set functor and its geometric realization and I thought it to be a nice exercise to probe them, so here it is:
I won't know if this proof actually works until I attend office hours to find out, but I am satisfied with the work I put into it
I already started making some notes on the derived functors
other than that I have this nice book that will help me prepare for writing my thesis, so I'd like to take a look at that too
as for the non-math plans, I am rewatching good doctor. my brain has this nice property that after a year has passed since finishing a show I no longer remember anything, the exponential distribution is relatable like that. this allows endless recycling of my favourite series, I just need to wait
I wish you all a pleasant break and I hope everyone is getting some rest like I am
12 XII 2022
I have a test at the end of this week so I am mostly grinding for that, kinda ignoring other things along the way, planning to catch up with them during the christmas break
the new update for my tablet's OS brought the option to insert pictures into the notes, so now I can paste the problem statements directly from the book. I am not sure if this is actually efficient but it surely looks better and the notes are more readable
(I can't vouch for the correctness of those tho lol I just started learning about the Rouché's theorem)
I have been trying to keep up with the material discussed in lectures on commutative algebra and agebraic methods. with each lecture there is a set of homework problems to solve and I predefined a standard for myself that this week it's alright if I don't do the homework because grinding for the test is more important
I made some pretty notes on valuation rings
during the break I need to study finite and integral ring maps and valuation rings for commutative algebra course; resolutions, derived functors and universal coefficients theorem for algebraic methods course. I feel pretty good about the test that's coming up. sure, you can never be too prepared but so far I've been able to solve a good part of the problems I tried, so I should be ok
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
7 XI 2022
I think I found an advisor and a topic for the bsc thesis! or rather they found me
one of the teachers that prepares us for writing our theses approached me and started asking about homology I mentioned during our presentation, he wanted to know what courses I took and how familiar I am with that stuff. I told him that I know a bit about homology only from self-study but I enjoyed everything from algebraic topo so far and I would be happy to write about something from that. "ok then I'll find the right topic for you" was his response. then he suggested I read Groups of Homotopy Spheres by Milnor and Kervaire and write about surgery theory. I was sold the moment I heard that name, it's almost as funny as writing about the hairy ball
so there she is, very high level, very complicated. I barely skimmed the first half of that 34-page paper, it's gonna take a lot of work before I learn the basics necessary to even comprehend what is going on. it feels good to be noticed tho, I'm so happy to start writing asap
other than that my mood hasn't been in a great place, because commutative algebra is super hard and I am struggling to find the right resources to study. the last thing we did was tensor product and I've been procrastinating actually studying it by making pretty notes lol
I found a textbook that seems decent. the theory is very thoroughly explained here and there are plenty of exercises ranging from easy to difficult ones
recently I've been trying a new method of tracking, which is instead of writing to-do lists, I write down what I did each day, here is what it looks like for now:
I find it much less anxiety-inducing than the to-do approach because I know damn well what I need to do and writing down what I actually completed feels much better than crossing things off of the list
this week I hope to study the tensor product, representable functors (yoneda is still not done with me) and probably start the complex analysis homework. if I have time I will study the prerequisites for the Milnor's paper
29 X 2022
another exhausting week finally over! fortunately I have two extra weekend days, so I can rest and do my homework without stressing over it
I found another promising youtube channel about learning. and "insanely difficult subjects" sounds about right when it comes to everything that's happening in math
I wish there was more content about learning math specifically. the tips I see, however good and useful for studying memory-based stuff such as biology or history, don't seem to work for math
for now my best method is to study the theory from the textbook, trying to prove everything on my own or if that fails, working through the proofs, coming up with examples of objects and asking (possibly dumb) questions that I then try to answer. afterwards I proceed to solving exercises
recently I've been studying mainly commutative algebra, in particular the localization
we didn't spend much time discussing local rings so I had to find some useful properties on my own. the whole idea of "local properties" is an interesting one and I definitely want to read more about it
I find it to be much more elegant to study localization through its universal property and exact sequences rather than through calculation on elements. it's funny how you can cheat so many of our homework problems by knowing basics of category theory and a little bit of homological algebra
I wonder if it's possible to learn math using mind maps, never actually tried. here is my attempt at doing that for one of the subjects in complex analysis:
other than studying I had to prepare a presentation for one of my courses
the topics were given to us by the professor so I thought it would be boring and technical, but I got lucky to discuss the possible generalizations of the Jordan theorem
now I'm gonna talk about something more personal
this week has been difficult because my brain doesn't enjoy existing. some days I had so many meltdowns and shutdowns, I could barely think and speak, let alone study difficult subjects in math. it's really disappointing, as I thought it got better after introducing new medication, but apparently I still can't handle time pressure and I break very easily when emotions become overwhelming (which they frequently do). one of the most discouraging parts of a neurodivergent brain is that you can't always say "alright then I'll just work harder" when you see that the situation requires it. you can't, because your brain has a certain threshold of "how much can you take before you snap" and no tips for studying when you're tired can change that. if you try, you'll just have a meltdown and your day is over, the rest of it must be spent regaining your strength and all you can do is hoping that tomorrow will be better
I wish I could always simply enjoy math and see it as an escape route from a confusing world of human interaction and unpredictable emotions, but whenever there is a deadline or grading criteria, I can hardly enjoy it anymore. I know that this is not what it's always gonna be, the further I go the less deadlines and exams we have, so I must wait and one day it might be okey
since june I've been trying to discuss accommodations regarding adhd and autism with my university but the process takes forever and I'm slowly losing hope that I will ever have it easier
nonetheless, I'm willing to do everything to achieve the goal of spending my days alone working on developing some new theory. just a few more years and I might start living the dream
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
13 X 2022
I dedicated the weekend to meeting with people from the machine learning club, helping my friend through her analysis homework and studying category theory for one of my subjects. then I did mostly the complex analysis homework
here are some wannabe aesthetic notes
my main goal at the time was to truly understand yoneda's lemma and the main intuition I have is that sometimes we shouldn't study the category C, but thw category of all functors from C to Set
after studying for a few hours I can say that the concept became a bit more intuitive
one of the problems in my "putnam homework" was to calculate the product of all differences of distinct n-th roots of unity – or so I thought. for a few days I believed that my solution doesn't work. I ended up with a disgusting fomula interating cosines of obscure angles but the visual intuition is neat, especially for an odd n. aaand that's no surprise since it turns out I'm fucking illiterate. not distinct roots, just differences of distinct roots, so that the whole thing is symmetric and there is no distinction of n odd vs n even
anyway I finally solved it, so that's nice!
I completed 5 out of 10 problems, which was my goal, so I should stop now and do my commutative algebra homework. there is one more exercise I want to solve:
the complex polynomial P with integer coefficients is such that |P(z)| ≤ 2 ∀z∈S¹. how many non-zero coefficients can P have?
I'm almost there with it and it's really cool
ofc the opportunity to include pretty drawings in my homework couldn't be wasted
during my category theory tutorial the professor asked me to show my solution on the blackboard. I was kinda stressed because now is the first time when I have my lectures and tutorials in english and on top of that this is a grad course. that whole morning I was fighting to stay awake, after the blackboard incident I didn't have to anymore
this is what I did
this week is likely to be the hardest out of many proceeding ones, because I won't have the weekend for studying (it's my grandma's birthday) so I need to use the maximum of my time during the week and get as much done as possible. I still need to do two homeworks, and study the theory. I am trying to learn how to prioritize and plan things, this is still a huge problem for me
I found an interesting youtube channel: Justin Sung. he talks about how to study/ how to learn and I like what he says, because it just makes so much sense. it's been a while since I started suspecting that methods such as flash cards or simple note-taking don't work and his content explains very well why they indeed might not work. it's very inspiring to see a professional confirm one's intuition
7 X 2022
my first week is over. I'm tired and I can tell already that it will be a hard semester. I have already spent more than 15 hours on my complex analysis homework and I solved 1 problem out of 10, ugh
this subject is gonna give me major impostor syndrom lmao I know that these problems are putnam level difficulty but it's frustrating to have spent the whole day on something and fail. and I'm not kidding, I have a book on problem solving techinques for putnam and the exercises there are easier than those we do in class
one could say I'm bragging but it doesn't mean anything if I can complete only 1 of 10 problems which is a trivial corollary from Vieta's and took me about 4 hours to realize anyway
algebra homework was relatively easy, I discussed it with a few people who also take the course and together we completed the whole thing
for now I still have the motivation to try to look good so this week I've been pulling off dark academia aesthetic
I am afraid of my brain because it likes to give me meltdowns right when I need my cognitive performance to be reliable. I spent the whole holiday working on coping skills so I could spend less time sitting on the floor and crying
I spend most of the time with my boyfriend studying together. having a body double really helps
1 X 2022
new month huh
yesterday the commutative algebra teacher sent out the first homework assignment. you know, fuck the holiday, we need that grind
I have a week to solve it but I started yesterday as I was so excited
we need to prove some elementary properties of commutative unitary rings and I am enjoying it, I completed a half of the exercises so far. I can tell that the intuition acquired from studying module theory is paying off. many of the requested properties are the special cases of what I encountered during my module venture, so I feel like I understand them quite well. the problem I come across is how to write it down in a rigorous way, but I guess this is why we're supposed to do those exercises
I just got home from the math camp, it was so exhausting. I am not used to being around people all the time, so I my tolerance for interactions is low. I'm glad I went there tho, because I gained some teaching experience – my lecture, choosing contest problems and then grading the solutions
my university offers jobs as graders, older students can make some extra money checking homeworks of younger ones. the requirement is to have a decent GPA, which I don't have so I'm afraid they won't accept me. I don't know how decent exactly tho, so I'm going to try. in particular I might get bonus points for my extracurricular activities, giving talks at conferences and the grading I did at the camp. I'm so done with being poor, I hope I get in. otherwise I might start looking for some programming jobs, not for this academic year but in general, to find out what I could do at all
a few days ago I found a book that I wish I had found sooner: Vector Analysis, Klaus Janich
these are some of the chapters I needed a few months ago for my analysis course. the book is written like a novel and contains many interesting examples. on the bright side there are chapters about riemannian manifolds and other stuff that I haven't yet had an opportunity to study, so I plan to skim through the topics I already know and stay longer at those new to me
well, the sememster starts on tuesday so I don't have much time for that book, but as a sidequest it seems just right
26 IX 2022
I spent the past few days watching good doctor and doing algebra (mostly). I am trying to get used to working in the library
right now I'm at the math camp for the olympiad where I'm giving a lecture on the power of a point and radical axes
I wish I had been in a more math-oriented highschool, I feel like I missed out on so much. my school was focused on literature and philosophy, I switched to math and physics in my last year. on the one hand it's probably a nice achievement that I've managed to get into the university to study math, on the other hand I could have done so much more
I've been struggling to motivate myself to study lately, because the semester starts next week and I cannot really start anything new right now, but I also don't have anything in particular that I could continue. I decided to just read eisenbud and solve some exercises with homology
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
13 IX 2022
my euclidean geometry journey will be over soon and the start of the semester is so close, it's kinda scary
recently I stumbled upon someone's post with a time-lapse video of their study session. I liked it so much that I decided to make mine
this is me learning about the snake lemma and excision
the excision theorem is the hardest one in homology so far btw, I spent about 4 hours on it and I am barely halfway through. I like the idea of the proof tho, it's very intuitive actually: start simple and tangible, then complicate with each step lmao
I realized two things recently. one of them is that deeply studying theorems is important and effective. effective, uh? in what way? in exams we don't need to cite the whole proof, it suffices to say "the assertion follows from the X theorem"
yeah right, but my goal is to be a researcher, not a good test-taker, researchers create their own proofs and what's better than studying how others did it if I am for now unable to produce original content in math?
the second things is that I learned how to pay attention. I know, it sounds crazy, but I've been trying another ✨adhd medication✨ and after a while I realized that paying attention is exhausting, but this is the only way to really learn something new, not just repeat what I already know. it made me see how much energy and effort it takes to make good progress and that it is necessary to invest so much
I am slowly learning to control my attention, which brings a lot of hope, as I believed that I had to rely on random bouts of hyperfocus, before I started treatment. I am becoming more aware or how much I am focusing at the given moment and I'm trying to work on optimizing those levels. for instance, when I'm reading a chapter in a textbook for the first time, it is necessary to remember every single detail, but wanting to do so consumes a lot of energy, because it means paying constant attention. it is ineffective because most likely I will have to repeat the process a few more times before I truly retain everything. being able to actually pay attention at will sure does feel good tho, as if I had a new part of my brain unlocked
I am solving more exercises for algebraic topology, procrastinating my lecture prep lmao. I am supposed to talk about the power of a point and radical axes, I have a week left and I can't force myself to start, because there is so much good stuff to do instead
I have a dream to produce some original results in my bachelor's thesis. it may be very difficult, because I hardly know anything, that's why I'm calling it a dream, not a goal. the plan is to start writing at the end of the semester, submit sometime in june
I spent last week at the seminar on analysis and oh boi, I will have to think twice next time someone asks if I like analysis. the lecturer who taught me at uni had a different approach than the "classic" one. we did a little bit of differential geometry, Lie groups and de Rham cohomology, those are the things I like. meanwhile at the seminar it was mostly about analytic methods of PDEs, the most boring shit I have ever seen
complex analysis will most likely be enjoyable tho, I'm taking the course this semester
for the next few days I need to force myself to prep that damn geometry lecture. other than that I plan to keep solving the AT exercises and maybe learn some more commutative algebra. I wish everyone a pleasant almost-autumn day 🍁
10 IX 2022
today I need some extra motivation to study because I didn't sleep well these past few days and it has drastic effects on my productivity, energy, motivation and what have you
also I am struggling to make the choice as to what I should do today
yesterday I started solving some basic exercises from hatcher's textbook
Δ-complex structures are becoming more intuicitve with time. take my solutions with a grain of salt, I am just starting to learn about these things and won't vouch for them lmao
some more complicated objects (the last one is an example of a lense space)
I decided to study commutative algebra today
so far I'm enjoying it. not as much as algebraic topology (which will always be my number 1) but it has its beauty
right now I'm at hom and tensor functors, the structures are fairly complicated, but pretty, and they look like they need to be studied in stages, with repetition and breaks, to fully grasp what's going on
my sensory issues are terrible today and I'm exhausted and hyperactive at the same time uh
I'll try working through a lecture on commutative algebra and give an update on how it went later
update: I studied for a while, but it wasn't going great so I decided to take a nap instead. god knows I tried
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
this looks so great! I need to check this out as well
25 VIII 2022
I found the most beautiful math book I have ever seen
it covers the basics of algebraic topology: homotopy, homology, spectral sequences and some other stuff
one of the authors (Fomenko) was a student when this book was being published, he made all the drawings. imagine being an artist and a mathematician aaand making math art
just look at them
other than those drawing masterpieces there are illustrations of mathematical concepts
I'm studying homology right now, so it brings me joy to know that this book exists. I don't know how well it's written yet, but from skimming the first few pages it seems fine
I just finished watching a lecture about exact sequences and I find the concept of homology really pretty: it's like measuring to what extent the sequence of abelian groups fails to be exact
I'm trying to find my way of taking notes. time and again I catch myself zoning out and passively writing down the definitions, so right now I avoid taking notes until it's with a goal of using the writing as a tool for acquiring understanding. I'm trying to create the representations of objects and their basic relations in my mind at first, then maybe use the process of note-taking to further analyze less obvious properties and solving some problems
I will post more about it in the future, we'll see how that goes
september
I decided to start posting monthly, I hope it will help me keep it regular during the semester, it may also bring more structure into my posts
I gave my talk at the conference, I was surprised with the engagement I received, people asked a lot of questions even after the lecture was over. it seemed to be very successful in a sense that so many people found the topic interesting
what I need to do the most in the next 3 weeks is learn the damn geometry. sometimes I take breaks to study algebraic tolology, I did that yesterday
you guys seem to enjoy homology so here is me computing the simplicial homology groups of the projective plane. I tried to take one of these aesthetic photos I sometimes see on other studyblrs but unfortunately this is the best I can do lmao
my idea for mainly reading and taking notes only when it's for something really complicated seems to be working. I focus especially on the problem-solving side of things, because as I learned the hard way, I need to learn the theory and problem-solving separately. what I found is that sitting down and genuinely trying to prove the theorems stated in the textbook is a good way to get a grasp of how the problems related to that topic are generally treated. sometimes making one's own proof is too difficult, well, no wonder, experienced mathematicians spend months trying to get the result, so why would I expect myself to do that in one sitting. then I try to put a lot of effort into reading the proof, so that later I can at least describe how it's done. I find this quite effective when it comes to learning a particular subject. I will never skip the proof again lmao
in a month I'll try to post about the main things I will have managed to do, what I learned, what I solved, and hopefully more art projects
25 VIII 2022
I found the most beautiful math book I have ever seen
it covers the basics of algebraic topology: homotopy, homology, spectral sequences and some other stuff
one of the authors (Fomenko) was a student when this book was being published, he made all the drawings. imagine being an artist and a mathematician aaand making math art
just look at them
other than those drawing masterpieces there are illustrations of mathematical concepts
I'm studying homology right now, so it brings me joy to know that this book exists. I don't know how well it's written yet, but from skimming the first few pages it seems fine
I just finished watching a lecture about exact sequences and I find the concept of homology really pretty: it's like measuring to what extent the sequence of abelian groups fails to be exact
I'm trying to find my way of taking notes. time and again I catch myself zoning out and passively writing down the definitions, so right now I avoid taking notes until it's with a goal of using the writing as a tool for acquiring understanding. I'm trying to create the representations of objects and their basic relations in my mind at first, then maybe use the process of note-taking to further analyze less obvious properties and solving some problems
I will post more about it in the future, we'll see how that goes
today I learned that for a surface with boundary, which I believe we can say a straw is, the genus is equal to that of a 2-manifold obtained from attaching disks to the boundary. hence the straw has genus equal to that of a 2-sphere, which is 0, therefore a straw has 0 holes
also a straw is not homotopic to a torus I think, but rather to S¹, as it's a product of S¹ and a closed interval, which is contractible. a torus has the fundamental group S¹×S¹, thus they cannot be homotopy equivalent. buuut that requires the straw to be infinitely thin so maybe I'm too idealistic for this claim to hold and it is in fact equivalent to a torus
lmao I love math but I can't stop laughing at the fact that it took me two years of university to be able to have this discussion
I’m really into internet discourse but only pointless and stupid internet discourse like how many holes there are in a straw (it’s 2)
22 VIII 2022
I will have to give a talk soon, in a few days I'll be attending a student conference. I decided to prepare something about my latest interest, which is knot theory. what makes it so cool for me is that the visual representations are super important here, but on top of that there is this huge abstract theory and active research going on
I decided to talk about the Seifert surfaces. this topic allows to turn my whole presentation into an art project
other than that I'm studying euclidean geometry and unfortunately it is not as fun as I thought it'd be
my drawings are pretty, ik. but there is almost no theory
I had a thought that working through a topic with a textbook is a bit like playing a game. doing something like rings and modules, the game has a rich plot (the theory), and quests (exercises) are there to allow me to find out more about the universum. whereas euclidean geometry has almost no plot, consists almost solely of quests. it's funny cause I never played any game aside from chess and mine sweeper
commutative algebra turned out to be very interesting, to my surprise. I was afraid that it would be boring and dry, but actually it feels good, especially when the constructions are motivated by algebraic geometry
commalg and AG answer the question from the first course in abstract algebra: why the fuck am I supposed to care about prime and maximal ideals?
oh and I became the president of the machine learning club. this is an honor but I'm understandably aftaid that I won't do well enough
I'm stressed about the amount of responsibilities, that's what I wanted to run away from by having the holiday. good thing is I gathered so many study resources for this year that I probably won't have to worry about it anytime soon, or at least I hope so
31 VII 2022
finally posting after the exams are over, it was the longest session I have ever experienced, a month of exams. I passed everything and it was a good semester, actually my grades are better than ever before, which comes off as a surprise, I can't believe that it's anything other than luck
now what am I going to do for the holiday huh
next semester I am going to take three courses: analytic functions, commutative algebra and a mix-course of category theory, sheaf theory and homological algebra. then I plan to take algebraic topology, algebraic geometry, number theory and some more abstract algebra, along with writing a bachelor's thesis. this is probably going to be the hardest year so far, I don't know how I am going to survive this, I'm so scared
I was asked to give some lectures on geometry during a math summer camp for people who want to participate in the math olympiad. it's a great opportunity for me to practice giving lectures, as that's what I plan my job to be. moreover, it is my dream to be so good at math that I could prep people for the olympiad, hence that's a fraction of that dream coming true
the problem is I don't know geometry lol last time I did any was like four years ago in high school
thus I play with triangles everyday
other than that I must prepare a talk for a conference, I chose to do one on the knot theory, Seifert surfaces specifically. I started reading about it some time ago and it seems super cool
untangling knots is a perfect thing to do for fun
my plan for the holiday outside of these side-quests is to learn as much as possible for the courses that I'll be taking. the problem with them (besides analytic functions) is that they will be quite technical, detailed and dry, as they are supposed to give the tools necessary to study algebraic topology and geometry. that does sound dreadfully boring, no? that's what scares me, because when I am not interested in what I'm trying to learn everything becomes twice as hard. I asked here and there for advice and people told me to read about algebraic geometry in tandem with commutative algebra, since many constructions have beautiful interpretations and motivations there. sounds like exactly what I need
my bachelor's thesis will be on algebraic or differential topology probably, but I don't know exactly what I want to write about. I was thinking about vector fields on manifolds or de rham cohomology, but the thing with the proseminar on geometric topology (mine) is that it's been planned to give the introduction to the currently researched topics and offer opportunities to work with fresh conjectures and theorems. at least that's how it was described. allegedly geometric topology has this property that undergrads can contribute to the development of new theory, which is very surprising to me ngl, I would guess that this is highly unlikely with any kind of math nowadays and yet here we are
in conclusion, I'm excited but scared
also a funny thing is happening
my title here on tumblr is "you can't comb a hairy ball" – hairy ball theorem, which says that whenever an n-dimensional sphere admits a continuous field of unit tangent vectors, n must be odd. I love how geometric this is, math is full of memes
anyway when I found out about it I was joking that my thesis will be on it. and now it's actually very likely that my first thesis will be about hairy manifolds, I can't wait till I can start writing
15 V 2022
I have a topology test this friday, not gonna lie I'm kinda stressed. this is my favourite subject and I am dedicating a great deal of time to learn it so if I get a low grade it undermines the efficiency of my work. everyone thinks I'm an "expert", but internally I feel like I lied to them. it's ridiculous, because I can solve all the theoretical problems fairly well but the moment I have to calculate something for a specific example of a space I am clueless. and it's about applying theory to problems, right? so what is it worth
other than that tomorrow is a participation round in the integral competition at my university. I am participating. I don't have any high hopes for this, because it's been a while since I practiced integration and I am not motivated to do so because it's not an important skill – wolfram exists. either way could be fun, that's why I decided to go there
I am dreading the fact that I'll have to sit down and learn all the material from the probability theory until the exams. I've been ignoring it completely so far, because it's boring and complicated. the last homework broke me, it's high time to get my shit together
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