Bsdndprplplld - You Can't Comb A Hairy Ball

I hate it when I am blessed with a new tumblr feature without my consent

℘²

september

I decided to start posting monthly, I hope it will help me keep it regular during the semester, it may also bring more structure into my posts

I gave my talk at the conference, I was surprised with the engagement I received, people asked a lot of questions even after the lecture was over. it seemed to be very successful in a sense that so many people found the topic interesting

what I need to do the most in the next 3 weeks is learn the damn geometry. sometimes I take breaks to study algebraic tolology, I did that yesterday

you guys seem to enjoy homology so here is me computing the simplicial homology groups of the projective plane. I tried to take one of these aesthetic photos I sometimes see on other studyblrs but unfortunately this is the best I can do lmao

my idea for mainly reading and taking notes only when it's for something really complicated seems to be working. I focus especially on the problem-solving side of things, because as I learned the hard way, I need to learn the theory and problem-solving separately. what I found is that sitting down and genuinely trying to prove the theorems stated in the textbook is a good way to get a grasp of how the problems related to that topic are generally treated. sometimes making one's own proof is too difficult, well, no wonder, experienced mathematicians spend months trying to get the result, so why would I expect myself to do that in one sitting. then I try to put a lot of effort into reading the proof, so that later I can at least describe how it's done. I find this quite effective when it comes to learning a particular subject. I will never skip the proof again lmao

in a month I'll try to post about the main things I will have managed to do, what I learned, what I solved, and hopefully more art projects

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

I loved the quiz! and the result is quite flattering haha

Well…

Looks like I made a uquiz

Try it here

Me: *Removes my cat from my lap to do something else.*

My cat: Father is…evil? Father is unyielding? Father is incapable of love? I am running away. I am packing my little rucksack and going out to explore the world as a lone vagabond. I can no longer thrive in this household.

[ID: a figure in a textbook that has curved arrows to look like vectors in a field. The figure caption reads, "Is this a vector field? No. It's a picture" /end]

My favourite fucked up math fact™ is the Sharkovskii theorem:

For any continuous function f: [a,b] -> [a,b], if there exists a periodic point of order 3 (i.e. f(f(f(x))) = x for some x in [a,b] and not f(x) = x or f²(x) = x), then there exists a periodic point of ANY order n.¹

Yes you read that right. If you can find a point of order 3 then you can be sure that there is a point of order 4, 5, or even 142857 in your interval. The assumption is so innocent but I cannot understate how ridiculous the result is.²

For a (relatively) self-contained proof, see this document (this downloads a pdf).

(footnotes under read more)

¹ The interval does not have to be closed, but it should be connected. (a,b), (a,b] and [a,b) all work.

² Technically the result is even stronger! The natural numbers admit a certain ordering called the Sharkovskii ordering which starts with the odd primes 3 > 5 > 7 > ... , then doubles of primes, then quadruples of primes and so forth until you get no more primes left, ending the ordering in 2³ > 2² > 2. Sharkovskii's theorem actually says that if you have a periodic point of order k, then you have periodic points of any order less than k in the Sharkovskii ordering. It is frankly ridiculous how somehow prime numbers make their way into this mess.

oh and there is the dual thing: sometimes you just know that the professor hates the subject. like when I was taking one of the analysis courses, where the lecture was with one professor and the tutorials were with a different one

at the lectures we were two months into measure theory while at the tutorials haven't even started doing exercises on that topic, but oh it was fine, still plenty of time, he knows what he's doing – we thought, like fools. then the midterm was announced, two weeks left, we still haven't started measure theory. then it was one week left, so the professor tried to solve some lebesgue integrals with us, but he got so bored with each example that he hasn't finished a single one. at this point we just hoped that maybe measure theory just won't be on the midterm, it was too late to do anything. well, unfortunately, the midterm consisted mostly of measure theory problems, it made sense because that was the main content of the course

the professor was clearly very passionate about hating measure theory

One of the really amusing things about college is that if you pay attention you sometimes can discern some of your professor's favorite pet concepts.

For instance, in my Topology course this semester, the Zariski topology has come up at least once in every single homework set so far, and in multiple lectures.

And okay, that's not that weird. The Zariski topology is a really important object in a LOT of fields, especially algebraic geometry. And discussing it at length is a really pedagogically sound move because the Zariski topology is a good example of a topology with a very well motivated structure (the closed sets are the algebraic sets!) that still very naturally gives rise to a lot of strange features, like the way all open sets in the standard topology are Zariski-dense. It was quite effective at startling me out of the complacency of unconsciously basing my intuition of how topologies behave entirely on the standard topology on the reals. So my professor bringing up Zariski so often doesn't necessarily mean he has any special affection for it.

except...

My professor writes many of the homework problems himself. Not all of them - the less interesting ones he lifts from the textbook- but some. Well, every single Zariski topology question I've encountered so far is an original from this guy. I know because the all the questions he writes personally have paragraphs of commentary contextualizing why he thinks the problem is interesting and where the ideas in the problem are going later in the course. And well- let's just say the asides on the Zariski topology have been copious indeed

AND THEN there's the way he talks about the Zariski topology in class! It's with this blend of enthusiasm and fascination only comparable to the way I've seen tumblrites talk about their blorbos. Like hey! Come behold this sgrungy little guy! Isn't he fucked up? Isn't he marvelous? And I look and I can only conclude YEAH that is indeed a spectacular specimen, he's so strange, I want to put him in a terrarium and study him (and then I get to! In my homeworks!)

Anyways. It makes me really happy picking up on how excited my professor is to share this topology with us. I'm kind of baffled that people assume math is a boring field full of boring people when there exist folks like my professor who get this passionate about a topology!

"numbers don't lie" the real numbers are literally a lie group