A Monoid? Oh, You Mean A Monad On A One Point Set In The Bicategory Of Spans Of Sets?

some more animation: how secant-teenager becomes tangent-adult

also! it illustrates a well-known inequality of a differentiable convex function and its tangents & the monotony of slopes

mathematicians, constantly: god I’m so tired of people telling me how much they hate math whenever i mention what im into

me: yeah it’s fucked up. i do probability theory, what about you

them: oh man I hate probability theory

13 IX 2022

my euclidean geometry journey will be over soon and the start of the semester is so close, it's kinda scary

recently I stumbled upon someone's post with a time-lapse video of their study session. I liked it so much that I decided to make mine

this is me learning about the snake lemma and excision

the excision theorem is the hardest one in homology so far btw, I spent about 4 hours on it and I am barely halfway through. I like the idea of the proof tho, it's very intuitive actually: start simple and tangible, then complicate with each step lmao

I realized two things recently. one of them is that deeply studying theorems is important and effective. effective, uh? in what way? in exams we don't need to cite the whole proof, it suffices to say "the assertion follows from the X theorem"

yeah right, but my goal is to be a researcher, not a good test-taker, researchers create their own proofs and what's better than studying how others did it if I am for now unable to produce original content in math?

the second things is that I learned how to pay attention. I know, it sounds crazy, but I've been trying another ✨adhd medication✨ and after a while I realized that paying attention is exhausting, but this is the only way to really learn something new, not just repeat what I already know. it made me see how much energy and effort it takes to make good progress and that it is necessary to invest so much

I am slowly learning to control my attention, which brings a lot of hope, as I believed that I had to rely on random bouts of hyperfocus, before I started treatment. I am becoming more aware or how much I am focusing at the given moment and I'm trying to work on optimizing those levels. for instance, when I'm reading a chapter in a textbook for the first time, it is necessary to remember every single detail, but wanting to do so consumes a lot of energy, because it means paying constant attention. it is ineffective because most likely I will have to repeat the process a few more times before I truly retain everything. being able to actually pay attention at will sure does feel good tho, as if I had a new part of my brain unlocked

I am solving more exercises for algebraic topology, procrastinating my lecture prep lmao. I am supposed to talk about the power of a point and radical axes, I have a week left and I can't force myself to start, because there is so much good stuff to do instead

I have a dream to produce some original results in my bachelor's thesis. it may be very difficult, because I hardly know anything, that's why I'm calling it a dream, not a goal. the plan is to start writing at the end of the semester, submit sometime in june

I spent last week at the seminar on analysis and oh boi, I will have to think twice next time someone asks if I like analysis. the lecturer who taught me at uni had a different approach than the "classic" one. we did a little bit of differential geometry, Lie groups and de Rham cohomology, those are the things I like. meanwhile at the seminar it was mostly about analytic methods of PDEs, the most boring shit I have ever seen

complex analysis will most likely be enjoyable tho, I'm taking the course this semester

for the next few days I need to force myself to prep that damn geometry lecture. other than that I plan to keep solving the AT exercises and maybe learn some more commutative algebra. I wish everyone a pleasant almost-autumn day 🍁

I have a bet going on with a friend. We need a third opinion. Can one find the square root of 2 in pi? And pi in the root of 2

Gut instinct says no. But when you work with infinities, gut instinct is NOT proof. (And such a gut feeling could have easily been the dodgy dinner I cooked myself last night.)

However I cannot even provide a proof.

I have, however tried to give some insight in another post (it should be the one immediately below this one) to perhaps help/provoke a more concrete argument from someone else :)

i’m starting a collection

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.

lmao wtf people seriously say shit like "math is homophobic"? because queer people can't do math… a priori?? isn't it inherently homophobic to claim that homosexuality implies being bad at math? this is the most unhinged thing I've seen this week

It's so funny to me when people reblog math posts on this site and say shit like "this is homophobic" "this is an attack on queer people" like hello this is tumblr. OP is queer. Like we've got a HANDFUL of cishets here on mathblr but most of us are queer. Because it's tumblr. And everyone is queer on tumblr. If you see math on this site, a queer person probably put it there. Stfu about "gays can't do math" if it's on tumblr, gays can do it, that's why it's on tumblr.

And of course there is also a large and brilliant and beloved queer math community off of tumblr but I just think it's extra funny when people don't notice it ON TUMBLR.

Alan Turing didn't kick the Nazis' collective ass laying foundations for the field of computing and then go on to also lay foundations for the field of biomathematics, Leonardo Da Vinci didn't give us fundamental physical and mathematical diagrams used in engineering well beyond his time, Moon Duchin doesn't study the math of fair redistricting, Chad Topaz and Jude Higdon don't analyze criminal sentencing disparities, et cetera et cetera, for y'all to call math homophobic and an attack on the queer community.

Shortest math paper ever.

And with so much impact! It just disproved a widely accepted theorem from the year 1769 in 5 rows!!!

I'll never publish anything even remotely badass like this! But I want it so much!!!

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

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