Very Funny When Mathematicians Call Weird Edge Cases "pathological".

also a funny thing is happening

my title here on tumblr is "you can't comb a hairy ball" – hairy ball theorem, which says that whenever an n-dimensional sphere admits a continuous field of unit tangent vectors, n must be odd. I love how geometric this is, math is full of memes

anyway when I found out about it I was joking that my thesis will be on it. and now it's actually very likely that my first thesis will be about hairy manifolds, I can't wait till I can start writing

15 V 2022

I have a topology test this friday, not gonna lie I'm kinda stressed. this is my favourite subject and I am dedicating a great deal of time to learn it so if I get a low grade it undermines the efficiency of my work. everyone thinks I'm an "expert", but internally I feel like I lied to them. it's ridiculous, because I can solve all the theoretical problems fairly well but the moment I have to calculate something for a specific example of a space I am clueless. and it's about applying theory to problems, right? so what is it worth

other than that tomorrow is a participation round in the integral competition at my university. I am participating. I don't have any high hopes for this, because it's been a while since I practiced integration and I am not motivated to do so because it's not an important skill – wolfram exists. either way could be fun, that's why I decided to go there

I am dreading the fact that I'll have to sit down and learn all the material from the probability theory until the exams. I've been ignoring it completely so far, because it's boring and complicated. the last homework broke me, it's high time to get my shit together

some experimentation with colors, i tried the inverse:

looks neat tho i think i won't stick to that, as my previous notes are black on white and the inverse is taken by changing the template and the marker color uhh they're gonna be incompatible

Mathematics: What do grad students in math do all day?

Mathematics: What do grad students in math do all day? - gist:4158578

“A lot of math grad school is reading books and papers and trying to understand what’s going on. The difficulty is that reading math is not like reading a mystery thriller, and it’s not even like reading a history book or a New York Times article.

The main issue is that, by the time you get to the frontiers of math, the words to describe the concepts don’t really exist yet. Communicating these ideas is a bit like trying to explain a vacuum cleaner to someone who has never seen one, except you’re only allowed to use words that are four letters long or shorter.

What can you say?

“It is a tool that does suck up dust to make what you walk on in a home tidy.”

That’s certainly better than nothing, but it doesn’t tell you everything you might want to know about a vacuum cleaner. Can you use a vacuum cleaner to clean bookshelves? Can you use a vacuum cleaner to clean a cat? Can you use a vacuum cleaner to clean the outdoors?

The authors of the papers and books are trying to communicate what they’ve understood as best they can under these restrictions, and it’s certainly better than nothing, but if you’re going to have to work with vacuum cleaners, you need to know much more.

Fortunately, math has an incredibly powerful tool that helps bridge the gap. Namely, when we come up with concepts, we also come up with very explicit symbols and notation, along with logical rules for manipulating them. It’s a bit like being handed the technical specifications and diagrams for building a vacuum cleaner out of parts.

The upside is that now you (in theory) can know 100% unambiguously what a vacuum cleaner can or cannot do. The downside is that you still have no clue what the pieces are for or why they are arranged the way they are, except for the cryptic sentence, “It is a tool that does suck up dust to make what you walk on in a home tidy.”

OK, so now you’re a grad student, and your advisor gives you an important paper in the field to read: “A Tool that does Suck Dust.” The introduction tells you that “It is a tool that does suck up dust to make what you walk on in a home tidy,” and a bunch of other reasonable but vague things. The bulk of the paper is technical diagrams and descriptions of a vacuum cleaner. Then there are some references: “How to use air flow to suck up dust.” “How to use many a coil of wire to make a fan spin very fast.” “What you get from the hole in the wall that has wire in it.”

So, what do you do? Technically, you sit at your desk and think. But it’s not that simple. First, you’re like, lol, that title almost sounds like it could be sexual innuendo. Then you read the introduction, which pleasantly tells you what things are generally about, but is completely vague about the important details.

Then you get to the technical diagrams and are totally confused, but you work through them piece by piece. You redo many of the calculations on your own just to double check that you’ve really understood what’s going on. Sometimes, the calculations that you redo come up with something stupid, and then you have to figure out what you’ve understood incorrectly, and then reread that part of the technical manual to figure things out. Except sometimes there was a typo in the paper, so that’s what screwed things up for you.

After a while, things finally click, and you finally understand what a vacuum cleaner is. In fact, you actually know much more: You’ve now become one of the experts on vacuum cleaners, or at least on this particular kind of vacuum cleaner, and you know a good fraction of the details on how it works. You’re feeling pretty proud of yourself, even though you’re still a far shot from your advisor: They understand all sorts of other kinds of vacuum cleaners, even Roombas, and, in addition to their work on vacuum cleaners, they’re also working on a related but completely different project about air conditioning systems.

You are filled with joy that you can finally talk on par with your advisor, at least on this topic, but there is a looming dark cloud on the horizon: You still need to write a thesis.

So, you think about new things that you can do with vacuum cleaners. So, first, you’re like: I can use a vacuum cleaner to clean bookshelves! That’d be super-useful! But then you do a Google Scholar search and it turns out that someone else did that like ten years ago.

OK, your next idea: I can use a vacuum cleaner to clean cats! That’d also be super-useful. But, alas, a bit more searching in the literature reveals that someone tried that, too, but they didn’t get good results. You’re a confident young grad student, so you decide that, armed with some additional techniques that you happen to know, you might fix the problems that the other researcher had and get vacuuming cats to work. You spend several months on it, but, alas, it doesn’t get you any further.

OK, so then, after more thinking and doing some research on extension cords, you think it would be feasible to use a vacuum cleaner to clean the outdoors. You look in the literature, and it turns out that nobody’s ever thought of doing that! You proudly tell this idea to your advisor, but they do some back of the envelope calculations that you don’t really understand and tell you that vacuuming the outdoors is unlikely to be very useful. Something about how a vacuum cleaner is too small to handle the outdoors and that we already know about other tools that are much better equipped for cleaning streets and such.

This goes on for several years, and finally you write a thesis about how if you turn a vacuum cleaner upside-down and submerge the top end in water, you can make bubbles!

Your thesis committee is unsure of how this could ever be useful, but it seems pretty cool and bubbles are pretty, so they think that maybe something useful could come out of it eventually. Maybe.

And, indeed, you are lucky! After a hundred years or so, your idea (along with a bunch of other ideas) leads to the development of aquarium air pumps, an essential tool in the rapidly growing field of research on artificial goldfish habitats. Yay!”

Complex Number Grapher

https://jutanium.github.io/ComplexNumberGrapher/

This grapher is really fun to play around with! 

A normal function takes in a number, x, and outputs another number, y. But a complex function takes in a complex point on a plane (a+bi) and outputs another complex point. Without 4 dimensions, it would be impossible to graph a complex function :(

The creator of this project instead uses complex domain coloring which they explain much better than I have here so you should 100% go and check it out!

Look at this cool function I got:

f(z)=(sin(z^3))^((cos(z))/2)

you know girls can tell when you look at their boobs

i don’t care how quickly you glance, 1 second is like 5 seconds in boob time

so, for relativistic boobtime, where t is the observer, and t’ is the time measured at the boob. t=t’/sqrt(1-(v/c)^2) solving for t=1, and t’=5, we get that the boobspeed, v, is represented by v=+/- (6*10^8)sqrt(6)i m/s

boobs travel at 1.5 gigametres per second in the complex direction.

when a pelican bites you there's no malice in their eyes. they aren't upset at you. they are just hungry and want to see if you fit in their mouths. and if you don't then it's no problem and everything is fine. and if you do then well i guess your fate is sealed but that's ok it's a beautiful animal

did that to me

We need books that affect us like a disaster, that grieve us deeply, like being banished into forests far from everyone. A book must be the axe for the frozen sea within us. That is my belief.

Franz Kafka

foolproof plan

thinking about the time a prof told us that in real research mathematics it's fine to be slow, speed itself is not essential, as long as you can find it within yourself to make consistent unyielding inexorable forward progress, like the time some guy stole an M60A3 tank and terrorized a suburban neighborhood with it, said guy wasn't going that fast but plowed through cars and telephone poles and shit no problem. i'm not kidding that's what he said, that's the metaphor he used, he told us that the act of mathematics is like the 1995 san diego tank rampage