Mathblr - Blog Posts

doing (basic) algebraic topology in this context feels like going to that jungle and saying you know what bring this thing down we are building a city here. everything is a CW complex, everything is euclidean, and compact or paracompact if it must, all of this so that we can forget about sidestepping around topology and do algebra in peace lmao

Measure theory and topology both have this great flavor where you give the most minimal possible definition for the thing you want and then you get all the nice properties, except no, your definition is soft enough to allow crazy nonsense counterexamples hiding behind everything that you have to carefully sidestep around. It's like doing math in a jungle

maybe a littel late for Real’s Math Ask Meme 18, 6 and 3, please?

hi, thanks for the questions!

3: what math classes did you like the most?

tough choice! for the content itself I'd say abstract algebra, commutative algebra, analytic functions and algebraic topology. for the way the class was taught, a course on galois theory I took last semester was probably the best. the pace of the lecture allowed me to learn everything on the spot, not too fast, but not so slow that my mind would wander. the tutorials were also great, because the teacher found the perfect balance between explaining and showing the solutions, and engaging us to think about what should happen next. the courses I mentioned above were also taught well, but the galois theory one was absolutely perfect

6: why do you learn math?

I enjoy the feeling of math in my brain. I can spend hours thinking about a problem and not get bored, which doesn't usually happen with other things. when I finish a study session I feel tired in a good way, like I spent my time and energy doing something valuable and it's very satisfying

18: can you share a good math problem you've solved recently?

given a holomorphic line bundle L over a compact complex manifold, prove that L is trivial iff L and the dual of L both admit a non-zero section

this problem is quite basic, in a sense that you work on it right after getting started with line bundles, but I believe it to be a good problem, because it forces you to analyze the difference between trivial holomorphic bundles and trivial smooth bundles, so it's great for building some intuition

yes, this, but also among other stem courses in a typical school, math is taken the most seriously. idk about other countries, but in poland in highschool people study chemistry, biology, physics and geography only if they decide to take the advanced final exams in these subjects. with math, everyone has take the standard level exam, so it can't be ignored like other subjects

up to highschool everyone has to complete their share of stem courses, but with the subjects other than math, the teachers often allow students to pass by memorizing the theory or by making some extra projects to earn points. with math you can't do that. when someone struggles with physics, the teacher sometimes says "alright, next year you won't have to study physics, so just learn those formulas and definitions and write them down on a test and I will let you pass". in math this is not an option, the student will have to take n more years of math courses

also, math mainly requires learning new skills, not just new information. many people never memorize the "dry theory" in highschool, because you have access to a reference table of formulas during exams and your job is only to know where to use those formulas – no need to memorize anything. but this does not come naturally to everyone and I think a huge part of the problem is teaching people how to work on their problem solving skills. I tutored a few students who believed they were bad at math and their mindset was "I can solve this type of problem because I know how to substitute into this formula, but when the problem is slightly different I panic, because the teacher never showed us how to solve it", which can be fixed by practicing a wider variety of problems and practicing the awareness of one's thinking process

people do not understand that problem solving is a skill on its own and I blame schools for that, because what we are offered is the image of math being about re-using the same kind of thinking processes but with different numbers. heck, when I was in elementary school I thought this is what math is about and I hated it because it's so boring and repetitive. I can imagine, when someone believes that this is what math is supposed to be and then they see the "more real math", which is about creativity, they panic (and rightfully so, they've been lied to)

my unpopular opinion is that not everyone can be good at this, just like I will never be good at understanding literature – my brain just sucks at processing this kind of stuff and I have aphantasia which doesn't help at all. but what makes it even worse for those people is the belief that it should be about repeating the same patterns over and over, so when they see that it's something completely different, it must be very frustrating – the reality is inconsistent with their beliefs

I am sure it doesn't cover the entirety of the "oof I always hated math" phenomenon, but it certainly does explain some of it, especially in the context of the education system in my country

As I said in a previous post, I have deep sympathy for the frustration of people who are good at math when they see math so almost universally hated by children and adults

And again and again, they try to explain that math is very much within everyone's reach and can be fun and, at least in western countries, education was to blame, messing up this very doable and fun thing by teaching it wrong

But I still gotta wonder - why math? If it is really just education messing this up, why does it mess up so much with math, specifically? I'm sorry but I still cannot shake the sense that even if it's just bad teaching, math is especially vulnerable to bad teaching.

Or is it maybe just that math is the only truly exact science, so there is no margin of error, so unlike every other field where you can sortof weasel around and get away with teaching and retaining half-truths and oversimplifications and purely personal opinions, math is unforgiving with the vague and the incorrect?

I finally tried mindmaps to keep track of the 200 kinds of morphisms and another 100 kinds of sheaves (that one is work in progress)

I kinda like the result and I hope to finally use my notes in the future

30 VIII 2023

aight it's been a while, time for an update

recently I've been doing mostly algebraic geometry, my advisor gave me some stuff to read, so I'm working through that. the goal is to familiarize myself with hilbert schemes – the topic is advanced, so there are many prerequisites coming up when I'm trying to read the book, that's kinda annoying

we are planning for my thesis to be about a certain generalization of the hilbert scheme, so basically the question is "investigate this space" and I've been having second thoughts whether I'm up for the challenge. I'm just getting to know how all that stuff works, so it's quite overwhelming to see how much I need to learn before I can do anything on my own

nevertheless, I'm pushing through as I will have to learn all of that anyway

I am working on finishing the proof from my bsc thesis and honestly I'm kinda losing hope lmao it turns out that what I probably have to do to complete it is a massive amount of extra reading and an even bigger amount of proving lemmas. the thing is that my work is about something like a generalization of results that have been proven by two people (one of which is khovanov, yes, that khovanov) and I feel it in my balls that the case I'm working on should be treated in a similar way. now the problem is that I can barely understand what they wrote for the "easier" case and I just can't see myself doing that for the more complicated one. oh and for my case I should probably use equivariant cohomology. but all I know about it is the definition, I have never even calculated anything for that + I will do a course on it this semester so it feels futile to study it now. idk I need to talk to my former advisor about this and ask him to be honest, does he even believe that this can be done?

other than that I'm applying for a scholarship. I don't think I will get it, but it is worth trying

I moved in with my boyfriend and our cat decided that my desk is way too big for one person, so now it's our desk

uni starts in a month so I should probably spend that time doing something other than math, which I will be doing all the time once uni starts, but I struggle with coming up with things to do that are not math-related. I should complete some tasks for work, but I would also like to have a hobby

there is a number of things that I could try, for instance reading, drawing, singing, grinding metas for geoguessr (apparently I'm a gamer now), but I can't commit to any of those, my interest comes in waves

maybe I could schedule about an hour per day to do one of those things so that my brain gets used to it. it is not like I can focus on math 24/7, I need to take breaks and I have days when my motivation is zero, so I just sit at my desk and watch stupid shit on youtube. but that's the point, days like that could be spent doing something meaningful and refreshing, instead I just procrastinate math lol

tips for studying math part 2:

studying for an exam but the course is super boring and you don't care about it at all, you just want to pass

start by making a list of topics that were covered in lectures and classes. you can try to sort them by priority, maybe the professor said things like "this won't be on the exam" or "this is super important, you all must learn it", but that's not always possible, especially if you never showed up in class. instead, you can make a list of skills that you should acquire, based on what you did in classes and by looking at the past papers. for example, when I was studying for the statistics exam, my list of skills included things such as calculating the maximum likelihood estimators, confidence intervals, p-values, etc.

normally it is recommended to take studying the theory seriously, read the proofs, come up with examples, you name it, but we don't care about this course so obviously we are not going to do that. after familiarizing yourself with the definitions, skim through the lecture notes/slides/your friend's notes and try to classify the theorems into actionable vs non-actionable ones. the actionable ones tell you directly how to calculate something or at least that you can do it. the stokes theorem or the pappus centroid theorem – thore are really good examples of that. they are the most important, because chances are you used them a lot in class and they easily create exam problems. the non-actionable theorems tell you about properties of objects, but they don't really do anything if you don't care about the subject. you should know them of course, sometimes it is expected to say something like "we know that [...] because the assumptions of the theorem [...] are satisfied". but the general rule of thumb is that you should focus on the actionable theorems first.

now the problem practice. if you did a lot of problems in class and you have access to past papers, then it is pretty easy to determine how similar those two are. if the exercises covered in class are similar to those from past exam papers, then the next step is obvious: solve the exercises first, then work on the past papers, and you should be fine. but this is not always the case, sometimes the classes do not sufficiently prepare you for the exam and then what you do is google "[subject] exercises/problems with solutions pdf". there is a lof of stuff like this online, especially if the course is on something that everybody has to go through, for instance linear algebra, real and complex analysis, group theory, or general topology. if your university offers free access to textbooks (mine does, we have online access to some books from springer for example) then you can search again "[subject] exercises/problems with solutions". of course there is the unethical option, but I do not recommend stealing books from libgen by searching the same phrase there. once you got your pdfs and books, solve the problems that kinda look like those from the past papers.

if there is a topic that you just don't get and it would take you hours to go through it, skip it. learn the basics, study the solutions of some exercises related to it, but if it doesn't go well, you can go back to it after you finish the easy stuff. it is more efficient to learn five topics during that time than to get stuck on one. the same goes for topics that were covered in lectures but do not show up on the past papers. if you don't have access to the past papers you gotta trust your intuition on whether the topic looks examable or not. sometimes it can go wrong, in particular when you completely ignored the course's existence, but if you cannot find any exercises that would match that topic, then you can skip it and possibly come back later. always start with what comes up the most frequently on exams and go towards what seems the most obscure. if your professor is a nice person, you can ask them what you should focus on and what to do to prepare, that can save a lot of time and stress.

talk to the people who already took the course. ask them what to expect – does the professor expect your solutions to be super precise and cuts your points in half for computation errors or maybe saying that the answer follows from the theorem X gets the job done? normally this wouldn't be necessary (although it is always useful to know these things) because when you care about the course you are probably able to give very nice solutions to everything or at least that's your goal. but this time, if many people tell you that the professor accepts hand-wavey answers, during the exam your tactic is to write something for every question and maybe you'll score some extra points from the topics you didn't have time to study in depth.

alright, that should do it, this is the strategy that worked for me. of course some of those work also in courses that one does care about, but the key here is to reduce effort and time put into studying while still maximizing the chances of success. this is how I passed statistics and differential equations after studying for maybe two days before each exam and not attending any lectures before. hope this helps and of course, feel free to add yours!

I have a bunch of followers and mutuals that I never even talked to and I know some of you guys are very into math too, so let's get to know each other, shall we?

if you feel like you'd enjoy talking to me then go ahead, write me a message! I just realized I never said something like this and I would really love to have conversations with like-minded people

if this feels familiar, you can reblog this post to invite people to talk to you

I know your thesis was about something to do with algebraic topology, may I ask what exactly it was about?

(and congrats to you getting your bachelors degree and into a masters program)

(thank you!)

my thesis was about an open question regarding a certain skein module of tangles on 2n nodes. the conjecture is that the module is free and in my thesis I constructed a generating set that is free for n=2,3 (direct calculation) but I have yet to prove that for a general n. if you are interested I can send you the paper in which the question was posed, all the details are explained there and would be hard to write down here without tex lol

21 VII 2023

oh god I haven't posted anything personal in a very long time

I've been super busy with exams, essays and then my thesis, all I did was sleeping and studying

I defended my thesis 40 minutes ago! it's done! in two months I am starting the master's degree program

this was probably the most brutal exam session I ever had lol it started a month ago and I had no day off since. after finishing my normal exams I've been working 12 hours per day to complete my thesis and thanks to my advisor who was working just as hard as me, we did it

I was so close to failing differential geometry. the exam was really bad, probably my worst ever. the questions were mostly about this one topic covered during the last class – we discussed maybe 3 problems and the professor decided that this is good enough lol basically we were supposed to read his mind and guess what else there is to learn. I scored 35% and apparently that's more than enough to pass – the grades go from 3 to 5 and I got 3.5, so that's literally "more than enough to pass". there were only 3 people who scored 50% or more, so yeah, that seems fair

that week of studying differential geometry was the most stressful week in the last 3 years, I fucking hate it when it's unclear what I'm supposed to learn and I have no idea how to do it. thank god I passed, I don't know how I would do it again before taking the september exam

anyway, I passed algebraic topology, number theory and algebra 2 with flying colors and the reviewers really loved my thesis! they strongly suggest publishing it, but I think I will try to finish the second part of the proof before I do that

I already found the advisor for my master's thesis, of course I don't know what it's gonna be about, but since I had some algebraic topology this year, I am thinking it's time to learn algebraic geometry now

sweet jesus it's finally over, I can't believe it. and something new is starting

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?

2 IV 2023

oh god the programming task for today was so annoying. I was supposed to process the MIT database with ECG records, and the annotation part of it was hell. after three hours I finally did it but the anger I felt at that time put me seconds away from throwing my laptop out of the window lmao

a recent success is that I calculated the rank of the module that I am working with, the problem is almost solved! when I told my advisor about it he looked so happy, he said that maybe he should start looking for another problem for me to ponder, it was so satisfying. I have a thing for mentors. at each point in my life for which I had a mentor who would teach me my special interest the progress I was making improved significantly and those were always the happiest times of my life. I am not sure if my advisor will stay with me to further show me a way into the research, but it certainly feels like a possibility

today I did some algebraic topology and differential geometry, I'm trying not to fall behind with the material even when I don't feel like studying

next week the easter starts, so I will probably have to visit my family. it's an interesting feeling to see my sister all grown up, there is still the image in my head of when she was barely a teenager and we didn't have much to talk about. now she is almost 18 and the significance of the age difference is nearly gone. when she start university it will be even less noticeable as she will understand what I mean by "fuck my life it's exam session season" lol

for about a week I've been trying to eat more healthy food, it's going fine so far. my biggest problem is that I'm eating way too much sugar but undereating in the general sense at the same time. I'm trying to incorporate more fruits and vegetables into my diet, as well as different kinds of nuts. it's so important to be properly nourished for math and yet I neglect it so much

yesterday I had a conversation with my friend and he said that his vision for doing math is working on some huge open problem such as RH. obviously you do you, but this sounds like such a depressive idea to me lol. chances of solving something like this are almost non-existent, that's such a waste of time to work on something like this for 10, 20, 50 years and make no progress. I mean, it certainly would feel nice to prove or disprove something like RH, but I'm perfectly fine with reading papers and answering all the questions I can anwer, which might not be huge and famous but I'm pretty sure creating those small pieces of theory will be useful to somebody one day

free recall

here I am sitting and trying to learn something from a textbook by making notes and ugh I don't think this is gonna work

what I'm writing down will probably leave my head the second I switch tasks

today I found a cool video about taking notes during lectures and a method called free recall is mentioned there:

to summarize: taking notes during the lecture is ineffective, because it requires dividing attention into writing and processing the auditory input. instead of doing that one should just listen and then try to write down the contents of the lecture from memory. I can believe that – this is how I studied for my commutative algebra exam and the whole process went really fast. I highly recommens this guy's channel, he is a neuroscientist and bases his videos off of research findings

I will try to do this with textbooks and after a while I'll share how it felt and if I plan to keep doing it. the immediate advantage of this approach is that it gives raw information for what needs the most work and what can be skipped, which is often hard to see when trying to evaluate one's knowledge just by thinking about it. another thing that comes to mind is the accountability component – it is much easier to focus on the text while knowing that one is supposed to write down as much as possible after. kinda like the "gamify" trick I saw in the context of surviving boring tasks with adhd

I'll use this method to study differential geometry, algebraic topology, galois theory and statistics. let's see how it goes

26 III 2023

I had a lot of headaches recently, idk why. probably something to do with muscle tension, because my back, neck and jaw just lock up sometimes to the point that every movement hurts. I need to see a doctor about it, maybe I injured something or there is some other underlying cause

I wasn't very strict with studying this week, because a lot of stuff we did was a review of what I already knew but obviously it needs a refresher. if I keep ignoring it, I will end up in a situation where I won't know what's going on at all

I picked up some side hustles along the way, one of which is reading the extra topics from hatcher. one of the lecturers recommended a book to me, about galois theory in the context of covering spaces, I'm reading it right now, seems pretty good

tomorrow I'm seeing my advisor to discuss my progress with solving the problem for my thesis. I think I found the basis for the module, at least I proved that the set I chose generates all the other elements, remains to show that it's linearly independent. the second part of the question is the rank of the module, which is how an algebraic topology problem turned into a nasty cominatorics problem eh

today I completed the first "serious" task for my IT job, which was translating the code from java to python. I have never seen java before, but it looks a lot like c++, so I managed. I wrote 500 lines of code but I haven't tested it yet so debugging might be very painful. lol I guess that means I shouldn't say I completed the task

I am wondering if I should go to a conference, I have until the end of the month to submit a presentation. I am not sure if I can handle a trip to another city, it would be in a month, so there is no way to predict how I'll be feeling. this week I am giving a presentation about some knot theory (skein modules, bracket and jones polynomial) and it's a good pick for the conference too, which makes it a really touch choice as the hardest part will already be done. idk I guess I'll toss a coin, like I did about the IT job lmao

other than that, big thanks to everyone who interacted with my post about book recommendations! there are many great suggestions, it turned out much better than I expected tbh, I thought I would get like 2 or 3 notes. I will post a list of the books mentioned in that post, so it will be easier to find for anyone interested

I have a possibly unusual question. since I left high school and was no longer being forced to read I completely stopped looking at fiction and other books that are not math textbooks. this is partly because I fucking hated reading as an activity – I have adhd and asd, so not only my attention regulation causes a lot of problems, I also struggle with visualizing what's happening and imagining/undestanding the intentions and mindsets of the characters. now, I am properly medicated so attention isn't as much of an issue anymore and I remember enjoying some of the books I've been forced to go through. I miss reading, overall it had a positive impact on my thinking process, I miss analyzing the human experience, so to speak

I liked Dostoyevsky, Vonnegut, Philip K. Dick, Huxley, Kafka and some other "philosophified" authors. I also enjoyed war-themed books in general such as all quiet on the western front, and everything related to the soviet union

finally, my question is, what would you recommend for someone with this taste, who also likes math¹, has a hard time with visual imagination and people but enjoys analyzing human experience, who would possibly like to see a represenation of themselves in the characters of the novel? I would love to rebuild my reading habits, but I am lost as to where to start so I don't get discouraged

¹ I included math in here because maybe there is a book about what it's like to be a mathematician (or a scientist of some kind) that give some existentialist vibes or something, if it exists I want to read it lol

13 III 2023

I remember putting it in my bio a while ago that I dream of doing actual research one day. well this is already happening, as I mentioned in some post, my advisor found an open question for me to write my thesis about

the progress for now is that I'm done with most of the reading I need to do to tackle it and I'm slowly moving forward with thinking of ideas for the solution (or at least a partial one)

this is what I want to do for the rest of my life: reading papers and trying to write my own ones

ofc I don't know if I manage to solve the problem or achieve anything at all with it but the process itself is fun

other than that I've been catching up with homeworks and assignments from work. fortunately I found an MIT lecture recordings for statistics so hopefully I might not die from boredom

watching probability and stats lectures from MIT has been my relationship's idea of netflix and chill for a while now, gotta cultivate the tradition

the algtop professor asked us to write down a full detailed solution for an exercises we did in class, because the person presenting was unable to explain it so I sent him mine

I don't know yet if it's correct but I'm pretty sure it is. I wrote this down partly because who doesn't want extra points and partly because I didn't have a chance to present it, the person who did was faster

I like how my life is right now, I want to keep it that way

7 III 2023

it's the second week of the semester and I must say that it's easier than I predicted

statistical data analysis is boring but easy, algebra 2 is easy but probably interesting, so is differential geometry

algebraic topology was funny because ⅓ of the group completed the algebraic methods course, so at first we told the professor to skip half of the lecture (we all know the required part of category theory) and then with every new piece of information he would say "ok maybe this will be the first thing today that you don't know", to which we would reply "naaah we've seen this" lmao. but the course overall will be fun and maybe it's even better that the level of difficulty won't be as high as I though, that would leave more time for my other stuff

the tutorial part of number theory was scary, because the professor wanted us to work in pairs. my autistic ass hates working in groups and the noise in the room was unbearable (everyone was talking about the exercises we were given to solve), so I was on the verge of a meltdown after 30 minutes of this despite ANC headphones. next time I will work by myself from the start. maybe without the requirement of communication it won't be as bad. the course itself will be easy, when it comes to the material. I know nothing about number theory, so the novelty will make it more enjoyable. a few people said that they would prefer the tutorial in the standard form, maybe I won't have to worry about surviving it if there are enough people who want to change it

my birthday is tomorrow and as a gift my parents gave me enough money to buy an ipad, I was saving for it since november. for a few days now I've been testing different apps for note taking, pdf readers and other tools useful for studying. I must say, this is a game changer, I absolutely love it

taking notes itself is less comfortable than on an e-ink tablet, which gives very paperlike experience, but it's better than traditional ones. the upside is that I can use different colors and the whole process is less rigid than on an e-ink

two apps that seem the best for now are MarginNote 3 and GoodNotes

the first one is good for studying something from multiple sources. the app allows to open many pdfs, take pieces from them and then arrange them in a mindmap. it's possible to add handwritten notes, typed notes, photos and probably more that I don't know yet. all of this seems to be particularly useful when studying for exams or in other situations when it's necessary to review a huge chunk of material

the second app is for regular handwritten notes. it doesn't have any special advantages other than I just like the interface lol what I like about taking notes on ipad is that I can take photos and insert them directly into the notebook, which I can't do on the e-ink. it's great for lectures and classes because I don't usually write everything down (otherwise I can't listen, too busy with writing) and even if I do, I don't trust myself with it so I take photos anyway. being able to merge the photos with notes reduces chaos

oh god this is going to be a long post! other news from life is that yesterday I had a meeting with my thesis advisor and we finally picked a topic. some time ago he sent me a paper to try and said, very mysteriously, to let him know if it's not too hard before he reveals more details about his idea. the paper is about symmetric bilinear forms on finite abelian groups, pure algebra, and I was supposed to write about algebraic topology, so I tried to search where this topics comes up, but didn't find anything. it turns out that it's used to define some knot invariant, which I would use to write about the classification of singularities of algebraic curves. in the meantime my advisor had another idea, which is an open problem in knot theory. we decided to try the second one, because there is less theory to learn before I could start writing the paper

to summarize what I'm about to do: there is a knot invariant called Jones polynomial, which then inspires a construction of a certain R-module on tangles and the question asks whether that module is free, if so, what is its rank. now I'm reading the book he gave me to learn the basics and I can't wait till I start working on the problem

25 II 2023

I had an exam yesterday, one more to go. it was the written part, so 12+ hours of solving problems, exhausting just like before. I completed all of them, but of course I am not sure if my solutions are correct, I will find out on monday. I'm proud of the progress I've made

right now I'm studying for the second part, so the theory-oriented one, I can barely focus because I've already learned those things and now I have to relearn them again

I'm trying to prove all the theorems on my own. partly to see how much I remember, partly to see how much I'm willing to improvize. as they say, if you're using too much memory then you're doing something wrong so I'm hoping to be able to come up with the proofs without memorizing anything new

my technique for studying the theory for the exam is to first test myself on how much I remember by trying to write everything down and note where I'm unsure or don't remember at all. then I read the textbooks starting from the worst topics up to the better ones. when I encounter a long complicated proof I am trying to break it down into steps and give each step a "title" or a short description

for instance, the Baer criterion featured in the photo has the following steps:

only do "extenstions on ideals to R→M ⇒ M injective"

define the poset of extenstions of A → M, A ⊆ B and a contrario suppose there is a maximal element ≠B

use the assumption to define an ideal and a submodule that contradicts the maximality of the extension

it is much easier to fill out the details than to remember the whole thing. this is probably the biggest skill I acquired this semester, next to downloading lecture notes pdfs of random professors I find online lmao

a friend suggested that I could make a post about tips for reading math textbooks and papers. as for papers, I don't have enough experience to give any tips, but I can share how I approach reading the books

a big news in my life is that I got a job. I will be a programmer and I start in march. at first I am going to use mostly python, but in the long run they will have me learn java. I'm excited and terrified at the same time, this semester is gonna kill me

14 II 2023

so yesterday would be the last of my exams but I decided to retake both the written and the oral part. the grade I would get is 4, so not the highest possible, still pretty good especially for the standards of that course (it's one of the most difficult), but I am not satisfied

it was the professor who suggested I retake the exams, which surprised me, I was mentally prepared to finish being only half-happy about my results and his reactions, strangely enough, inspired me to try harder. he wouldn't offer it if he didn't think I could do better, right?

if he gave me a 5 with my written exam points I would feel like an impostor, because I don't think I am fluent enough with the topics to receive the best grade. to be graded 4 and not being effered the chance to try again would make me feel that it's done, I was just too slow and I can't do anything else to fix it (at least on paper, but we're talking symbolics now) and him giving me a second chance meant to me that he believes in my potential yet doesn't want to give me a participation trophy, instead he made it about earning the reward that I know I deserve

he achieved the aurea mediocritas with this and the most absurd part of it all is that he of all people was to give me this inspiration. half of the students I talk to think that he is pure evil, the majority of the other half think he is an inconsiderate asshole lmao

so in two weeks I'm trying the exam again. in the meantime I will have a party with friends (small – 5 people + my boyfriend's cat) and then I will be grading the math olympiad. afterwards my another grind of algebraic methods shall commence and this time please let me not fuck it up

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:

YouTube

Michael Penn

Math videos covering Calculus, Differential Equations, Number Theory and more

Michael Penn is a teacher at a university and he's great at explaining theory and solutions of problems

11 II 2023

in two days I have my last exam and I have absolutely zero motivation to study for it

yesterday I had an oral complex analysis exam and I did very well, the professor said that I will most likely receive the top grade. my partial scores from this course add up to 80%, so if the oral one was for 100%, it yields 84% total. that sounds like a top grade to me although we haven't received the official report yet

I also had an algebraic methods exam a few days ago and it went ok, I completed 4.5 out of 6 problems. I probably have no chance for a top grade from this course because the professor is very strict with how many points qualify for that and I am not even close to what the best people had. this is why I have zero motivation to study for the oral exam from this course, if there was a chance to score a 5 (the top grade) then I would care, but if my options are 3.5, 4 or 4.5, I don't really see the difference

well, the difference lies in maybe applying for a scholarship after this academic year, but honestly that "goal" is just here to distract myself from feeling judged all the time. somehow I don't care about money as much as an abstract number supposedly rating my abilities so thinking of it as "try harder so you might get paid for it" feels less pressing than "try harder so you'll have higher abstract numbers and you can feel good about yourself"

jesus I fucking hate grades, I wish it was kept secret from me how much points I actually have, only receive feedback on the correctness of my solutions and the information if I am passing or not. I can never be satisfied with I am doing. last year I would see it as a success to score 4's at everything, now it feels like a failure because I already scored some 5's, so that's my new bottom line. and I know that if I did ace everything, I would be happy for about 5 minutes and then move on to picking up twice as many courses for the next semester because "it would be too easy otherwise"

grades, no matter what I'm getting, fuck with my self esteem so deeply. it brings out the worst insecurities, fears and memories, this is when I am thinking my darkest thoughts. I have no one to talk to about this and I am angry at myself for perceiving it this way. I wish these things didn't matter to me but they do, I don't even know why, it feels like a trap

I don't want people to tell me that "I'm great no matter what grades I'm getting" or that "I will do it, because I'm smart". I actually don't know what I want, and it sucks to put my friends into the situation where no matter what they say it's "the wrong line". ughhh I want this semester to be over so I can go back to only caring about learning as much as possible

my thesis advisor (I think that's what you call the thesis boss) sent me a paper to read and I'm curious what topic he picked for me. I will gladly read it right after I'm done with exams

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