Free Recall

today I learned that for a surface with boundary, which I believe we can say a straw is, the genus is equal to that of a 2-manifold obtained from attaching disks to the boundary. hence the straw has genus equal to that of a 2-sphere, which is 0, therefore a straw has 0 holes

also a straw is not homotopic to a torus I think, but rather to S¹, as it's a product of S¹ and a closed interval, which is contractible. a torus has the fundamental group S¹×S¹, thus they cannot be homotopy equivalent. buuut that requires the straw to be infinitely thin so maybe I'm too idealistic for this claim to hold and it is in fact equivalent to a torus

lmao I love math but I can't stop laughing at the fact that it took me two years of university to be able to have this discussion

I’m really into internet discourse but only pointless and stupid internet discourse like how many holes there are in a straw (it’s 2)

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

Very funny when mathematicians call weird edge cases "pathological".

Mathematician looking at a function that's continuous everywhere but differentiable nowhere: yeah that function has something wrong with it.

Real’s Math Ask Meme

What math classes have you taken?

What math classes did you do best in?

What math classes did you like the most?

What math classes did you do worst in?

Are there areas of math that you enjoy? What are they?

Why do you learn math?

What do you like about math?

Least favorite notation you’ve ever seen?

Do you have any favorite theorems?

Better yet, do you have any least favorite theorems?

Tell me a funny math story.

Who actually invented calculus?

Do you have any stories of Mathematical failure you’d like to share?

Do you think you’re good at math? Do you expect more from yourself?

Do other people think you’re good at math?

Do you know anyone who doesn’t think they’re good at math but you look up to anyway? Do you think they are?

Are there any great female Mathematicians (living or dead) you would give a shout-out to?

Can you share a good math problem you’ve solved recently?

How did you solve it?

Can you share any problem solving tips?

Have you ever taken a competitive exam?

Do you have any friends on Tumblr that also do math?

Will P=NP? Why or why not?

Do you feel the riemann zeta function has any non-trivial zeroes off the ½ line?

Who is your favorite Mathematician?

Who is your least favorite Mathematician?

Do you know any good math jokes?

You’re at the club and Andrew Wiles proves your girl’s last theorem. WYD?

You’re at the club and Grigori Perlman brushes his gorgeous locks of hair to the side and then proves your girl’s conjecture. WYD?

Who is/was the most attractive Mathematician, living or dead? (And why is it Grigori Perlman?)

Can you share a math pickup line?

Can you share many math pickup lines?

Can you keep delivering math pickup lines until my pants dissapear?

Have you ever dated a Mathematician?

Would you date someone who dislikes math?

Would you date someone who’s better than you at math?

Have you ever used math in a novel or entertaining way?

Have you learned any math on your own recently?

When’s the last time you computed something without a calculator?

What’s the silliest Mathematical mistake you’ve ever made?

Which is better named? The Chicken McNugget theorem? Or the Hairy Ball theorem?

Is it really the answer to life, the universe, and everything? Was it the answer on an exam ever? If not, did you put it down anyway to be a wise-ass?

Did you ever fail a math class?

Is math a challenge for you?

Are you a Formalist, Logicist, or Platonist?

Are you close with a math professor?

Just how big is a big number?

Has math changed you?

What’s your favorite number system? Integers? Reals? Rationals? Hyper-reals? Surreals? Complex? Natural numbers?

How do you feel about Norman Wildberger?

Favorite casual math book?

Do you have favorite math textbooks? If so, what are they?

Do you collect anything that is math-related?

Do you have a shrine Terence Tao in your bedroom? If not, where is it?

Where is your most favorite place to do math?

Do you have a favorite sequence? Is it in the OEIS?

What inspired you to do math?

Do you have any favorite/cool math websites you’d like to share?

Can you reccomend any online resources for math?

What’s you favorite number? (Wise-ass answers allowed)

Does 6 really *deserve* to be called a perfect number? What the h*ck did it ever do?

Are there any non-interesting numbers?

How many grains of sand are in a heap of sand?

What’s something your followers don’t know that you’d be willing to share?

Have you ever tried to figure out the prime factors of your phone number?

If yes to 65, what are they? If no, will you let me figure them out for you? 😉

Do you have any math tatoos?

Do you want any math tatoos?

Wanna test my theory that symmetry makes everything more fun?

Do you like Mathematical paradoxes?

👀

Are you a fan of algorithms? If so, which are your favorite?

Can you program? What languages do you know?

@dimiclaudeblaigan asked for a tutorial on how to begin drawing. Good news! If you can draw a funky looking stick man, you have already started!

I think that stick people are a great starting point for artists because of the things you can learn from them that will be important later on.

If you are able to draw a circle and a couple of lines, you can easily put together a stick person.

Congratulations! You have started to draw. :)

A stick person is a very minimal artistic representation of a real life person. It is simple yet recognizable, and is widely used in art, media, and signage.

But what can a stick person teach us about drawing people that look more like… well, people? Lets have a look!

By simply adding a few more lines, we can add a pair of eyes and a mouth. Maybe even a little triangle nose! Or half circles for ears. We can now draw a face, which provides a basis for all sorts of expressions.

These simple additions can allow us to explore the wide range of human emotion and individuality.

This may seem like the basics of the basics. But that is what we want! In order to get to the point where we are able to draw complex, elaborate representations of humans and objects, we will need to start with simple shapes like lines and circles and build our understanding from there.

For instance, lets give our stick person some cool new features, such as hands and feet. I chose little squiggly circles to represent hands, and triangles to represent feet.

We can go a step further and modify the body of the stick person to include shoulders, hips, elbows and knees. These parts of the human body are quite complex in real life But here, all we need to do is add a few simple lines and dots to our stick person.

The lines provide some additional structural elements to our stick person's body, which are the shoulders and the hips. The dots indicate the points of articulation - elbows and knees, the places where the arms and legs bend!

Now we can use our stick person to show us an even wider range of human movement, action, and expression.

Our little drawing of a human being is evolving! All it took was adding a few more lines and shapes here and there.

By elongating some of the existing lines and making the head an oval instead of a circle, we can give our stick person proportions that resemble that of a real life human.

By this point, we have managed to add more complexity to our stick person simply by using our ability to draw lines, circles, and other basic shapes!

These basic ideas are the building blocks that will enable us to create more complex shapes.

The next part may be a considerable step up if you are absolutely new to drawing, but I have decided to include it in order to show you how complex objects like the human body can be built from shapes that are a bit more complex than circles and lines.

For example. Two ovals and a rectangle can be combined to create a cylinder.

Six squares can be combined to create a cube, or a box. Here, each square is distorted slightly depending on which way the cube is facing.

Note that the back faces of the cube and the bottom of the cylinder are hidden. These shapes allow us to visualize that which should not normally visible.

A sphere from all perspectives can be represented by a circle. But we can make it more like a sphere by adding lighting and shadow if we so desire.

Cubes, cylinders, and spheres are examples of 'solid shapes' because they consist of 3 dimensions.

Lets see how these solid shapes can be used to compose the human body.

By stacking three cylindrical objects, we can create a torso. Two spheres have been added to form shoulders, while a smaller cylinder forms the neck.

An arm is an alternating sequence of spheres and cylinders connected together. Note that the hand has been simplified for this example.

We can apply these solid shapes to the rest of the body to give us a more recognizable representation of the human form. It doesn't even have to be perfect. And just like that, our stick figure now has a silhouette that is unmistakably a person!

In the above examples, notice that we kept the stick person at the beginning while building up the shapes and solids around it. This is because the stick person serves as a guide for positioning the body and its various parts -> also known as posing.

You can do the same thing to everyday objects! Here, I drew a wine glass by stacking these three dimensional solid shapes.

The cup and its contents are two ovoid shapes that were cut in half. The stem is a very thin cylinder shape. The base is a cylinder with a slightly wider bottom.

Solid shapes help inform us how objects and parts of the human body may appear from different perspectives.

For example, a sphere can be used to demonstrate how the human head appears when looking up or down, turned to the side, or tilted at an angle.

With these examples, I hope I have managed to convinced you that if you can draw a circle and a couple of lines, you can draw a person! You just have to train your eye to recognize the simple shapes within complex objects. Try it with everyday objects as well! Or even your favourite media! A drawing subject can be as simple or as complex as you envision it to be.

Once you have mastered that, there are many aspects of drawing you can explore from here that may require you to seek additional resources or a fellow artist's advice.

Last of all, remember that drawing is an iterative process. Even if you draw something correct the first time, you will need to draw it again and again to get it right all times! And by making small changes like the ones we explored in this tutorial, your drawings will gradually transform!

I hope what I've demonstrated here are enough to provide the basics of how to get started with drawing objects and people, and also to help refresh more experienced artists. :) Hopefully I didn't go too off topic with what was requested, and let me know if there are any more questions I can answer.

Cheers :3

why is deciding on a title for my thesis so hard

when K ⊆ L is a finite extension by one element, say α with the minimal polynomial f, we can write 0 → (f) → K[x] → L → 0, where (f) is the kernel of evaluation at α. this is quite disappointing and very basic, but I haven't found anything better really. when there are finitely many intermediate fields between K and L for an extension L/K, L can be expressed as an extension by one element (Artin's theorem), which is still very specific

I didn't know about the group extensions, it makes the category of fields even more disgusting. I was hoping that the algebraic closure can be expressed as a colimit but of course not, not in the general case at least. but maybe at least some type of extensions can be realized as such? that's a nice thing to ponder. I'm pretty sure it will fail like every other request I had for this abomination of a category

I wonder what is typically done to make working with this category more pleasant. passing to Grp with the Galois group is one idea, the other I guess would be working with vector spaces or algebras? that would make sense considering that integral and finite ring maps are a thing and the field automorphisms play a role in the integral closure of ℤ in ℚ[√d]

on a sidenote, I laughed at the "lower body" and it reminds me how funny it is to talk about kernels in Polish. kernels and testicles are the same word

I've always thought 'splitting field' was a very cool sounding term. The Galois theorists did good with that one

parents got a new cat they named lord montague and this morning i heard my dad in the other room say "i would have to advise against that decision, my lord" followed by a crashing sound

I got 55 and it seems to me that the majority of my answers were heavily influenced by asd

I took a test on like where you are on the ‘nonverbal intimacy scale’ and the average female score is 102 and male is 93.8 and I got 56 lolololol

here it is if ya want (reblog/reply w/ what you get!!)

but how is this related to adhd? I'd say my ability to have fun is diminished because of the dopamine deficiency, I'm basically constantly bored, and afaik adhd increases the risk of substance abuse disorders, so I don't see the point of bringing up adhd in this context

Me: I don't take alcohol, smoke weed or do any other substances.

Everyone: Omg!! then how do you have fun??

Me: I have ADHD.