Math - Blog Posts
im sorry
remind me to rewrite this in the latin alphabet (sitelen Lasina) tomorrow
edit:
lon pi nanpa kipisi la [ante K ala la ante P ala] la [ante P la ante K]
( discrete math theorem: (~K => ~P) => (P => K) )
nasin lon
proof
(wan) [ante K ala la ante P ala] la [[ante K ala la ante P] la ante K]
(1) ((~K) => (~P)) => (((~K) => P) => K)
(tu) [ante K ala la ante P] la [[ante K ala la ante P ala] la ante K]
(2) ((~K) => P) => (((~K) => (~P)) => K)
(tu wan) ante P la [ante K ala la ante P]
(3) P => ((~K) => P)
(tu tu) ante P la [[ante K ala la ante P ala] la ante K]
(4) P => (((~K) => (~P)) => K)
(luka) [ante K ala la ante P ala] la [ante P la ante K]
(5) ((~K) => (~P)) => (P => K)
tan seme
why
(wan) lawa tu wan
(1) Axiom 3*
(tu) lon pi nanpa kipisi: ante P la [ante K la ante L]. ni la ante K la [ante P la ante L]
(2) discrete math theorem: ( P => (K => L) ) => ( K => (P => L))
(tu wan) lawa wan
(3) Axiom 1
(tu tu) lon pi nanpa kipisi: ante P la ante K. ante K la ante L. ni la ante P la ante L.
(4) discrete math theorem: P => K, K => L ⊢ P => L
(luka) lon pi nanpa kipisi: ante P la [ante K la ante L]. ni la ante K la [ante P la ante L]
(5) discrete math theorem: ( P => (K => L) ) => ( K => (P => L))
*so i went on Wikipedia to see if axiom numbers used in my class match up with what people usually use, and i found out that thing i was proving (i.e. contrapositive) is axiom 3 according to Wikipedia. however, in my class, axiom 3 is
"((~p)=>(~q)) => (((~p)=>q)=>p)"
so uh... yeah for the purposes of this post, ^ is axiom 3
I hope this makes it more obvious that you can get any number from a diverging sequence if you rearrange it enough.
wanna see some cursed math?
people are always saying that 1+2+3+4+5+6+7+... is equal to -1/12. that's stupid. why would adding positive numbers add up to a negative? so i decided to take matters in my own hands.
The sum of all natural numbers is 1.
(added some random stuff to make it even harder to understand)
explanation of last step
the sum (as n starts from 1 and approaches infinity) of 3n-1
= 2+5+8+11+14+17+20+... = infinity
the sum (as n starts from 1 and approaches infinity) of 2n
= 2+4+6+8+10+12+14+... = infinity
these sums are the same. so we divide both sides by
∞
Σ
n=1
to get 3n-1= 2n, so n=1
STEM fields all mapped out!
Physicist and science writer Dominic Walliman produces the YouTube channel Domain of Science where he shares fantastic animations that map out different STEM fields.
These videos are perfect to share with students to give them a comprehensive view with information about different types of sub-disciplines within fields and how they relate to each other.
I've embedded the Map of Mathematics video below for you but if you click on it and watch the video on YouTube you'll find the playlist which shares other videos on the topics of engineering, biology, physics, chemistry, and more.
And of course he has shared a lot of other great content on his channel. You should view, subscribe, and share ASAP!
Dominic has made the images of maps available via Flickr for educational use and you can purchase Domain of Science posters as well.
Watch on YouTube to view the whole playlist
3/14 Pi Day, School Walkout, Stephen Hawking, and more
A lot happening today! I just wanted to post a couple of updates and thoughts...
Before I had breakfast this morning I heard the word that Stephen Hawking had passed away. What an amazing human being, such an incredible loss for the world. And what an amazing coincidence for him to die on Einstein’s birthday...
Stephen Hawking
Stephen Hawking Taught Us a Lot About How to Live (NYT)
Stephen Hawking Dies at 76; His Mind Roamed the Cosmos (NYT Obituary)
Stephen Hawking: Visionary physicist dies aged 76 (BBC)
This is the most dangerous time for our planet (Stephen Hawking 12/16)
Stephen Hawking Was Right To Worry About Our Impending Doom (io9)
Stephen Hawking’s most mind-blowing discovery: black holes can shrink: Hawking radiation, explained by a physicist. (Vox)
Stephen Hawking’s 5 best and nerdiest pop culture cameos: When Hawking wasn’t changing the world, he played himself on TV. He was hilarious. (Vox)
Pi Day
I always look forward to Pi Day... friends usually send me Pi Day pictures and animated gifs... I wore my new Pi shirt to PT today and I’ve finally updated my Pi Page :) You can also check out my new Flickr post on Pi Day...
School Walkout
These two photos from the 17 photos from today's National School Walkout for gun control that should terrify the NRA are my favorite:
#NationalWalkoutDay on Twitter
Previous post on tumblr about gun violence
Fantasy Sociology (what would it do to agriculture if there was dragons)
Fantasy Psychology (the mental effects of having certain patterns of thoughts that generate fireballs)
Fantasy Biology (what if u had lighting sacks in yr cheeks)
Fantasy Chemistry (these r the elements and what u can do with them)
Fantasy Physics (orbital mechanics and magical floating rocks: a guide)
Fantasy Mathematics (its just normal mathematics)
I neglected this blog like hell, sorry
I had a lot of work to do, that's kinda what happened. but I would like to go back to posting regularly, so maybe I could write about something people would want to see?
for now my ideas for posts include
more study tips
a quick intro to moduli functors, since a lot of sources are written in a way that requires advanced algebraic geometry. I could explain the basics using (almost) only commutative algebra
updates on my life and what I've been working on
books recommendations
interesting math problems I encountered recently
if you'd like to see any of that, let me know! and feel free to give me more suggestions in the comments
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?
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
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