Acubed

Math geek.

[busymarina]

15 things I tell myself when I don't want to work/study

1. You are very lucky and privileged to have access to almost unlimited knowledge and you should appreciate that. 2. Be one of those rare people who step over their insecurities and succeed. 3. Only 5 minutes. Only today. (Repeat it 5 minutes later and every day). 4. You will know what to do as soon as you start. Ideas never appear from inactivity. 5. Make yourself proud. 6. One hour every day doesn’t feel much but it’s 365 hours a year. You can’t not succeed after so much work. 7. It’s not supposed to be easy. Nothing good is easy. 8. If you had a child to look after, you’d make them study because you want them to accomplish something. Don’t you love yourself? 9. “Everything you want is on the other side of fear” George Adair 10. Every mistake increases our chance to make progress. 11. If you give up now, you’ll have to return to this later anyway but from the very beginning. 12. Let the process be your result. 13. Every moment you thought your fears would suppress you has become the time you made it. 14. Maybe you think you can never find something to use your skills and mindset for. But if you continue investing in what matters to you, it will find its way out there. 15. I allow you to think globally. You have a right to the boldest dream.

[todayisnot]

If I could add one - there exists a time in the future where you’ve already written this paper/completed this project. So you must know what you want to write. You just need to close that gap.

[busymarina]

I like this one ⬆️

[theravenclawstudent]

“Yes, grad school can be a very lonely period of your life. And it is only going to get worse. But it is in these quiet pockets of my time when I was able to do more textbook readings, sort out my calculations and go deeper into my understanding of the topic. This is how you push the frontier of science and this feeling gave me enormous sense of satisfaction amidst all the dark clouds that were surrounding me. These are the best parts of my PhD thesis as well. So I think it just boils down to your mindset and managing your mental health. This is how you derive your fulfillment and meaning from life amidst all the frenzy.”

- My former supervisor’s advice.

[model-theory]

understanding maths which you did not previously understand is very sexy

[flyingsquidzeez]

me: *wants to live a minimalist life with little to no clutter*

also me: I’m keeping this math assignment from 5th grade I might need it later.

[mathprofessorquotes]

“Probability isn’t all statistics - it’s actually interesting.”

— Topology professor relating unions of open sets with probability.

[argumate]

Mathematician: do I contradict myself? oh shit I contradict myself, oh fuck,

[bayesic-bitch]

Do I contradict myself? Very well, then I contradict myself. This shows our previous supposition was false. QED

[argumate]

Mathematician: hmm, looks like your proof doesn’t work,

Mathematician about to invent proof by contradiction: oh, haven’t you heard,

[mathprofessorquotes]

“This problem’s making me hungry, it’s taking so much glucose from my brain.”

— Differential equations professor, solving the Laplace transform of sin(t)

[justsomeantifas]

If anyone is interested in learning mathematics!

First a great resource to start learning most fields but especially mathematics is Khan Academy It has courses in: Early math, arithmetic, pre-algebra, algebra, geometry, trigonometry, intro to statistics, precalculus, calculus, multivariable calculus, differential equations, and linear algebra. 

Here also gives you a great set of resources for you to learn beginning mathematics Heres a nice calculator to check your work: Wolfram Alpha

However if you learn better by reading books I’ll link a few pdfs for a couple of fields, these are mostly introductory books and notes.

Pre-algebra: here

Algebra: here

Geometry: here

Trigonometry: here, 

Calculus and Multivariable Calculus: here 

Differential Equations: here

Linear Algebra: here, here, 

Intro to Proof: here, here, here, 

Set Theory: here, 

Logic: here

Category Theory: here

Group Theory: here 

Intro to Abstract Algebra: here

Commutative Ring Theory: here

Intro to Combinatorics and Graph Theory: here

Intro to Algebraic Topology: here

Intro to Topology: here, here, 

Convex Geometry: here

Discrete Geometry: here here

Differential Geometry: here

Manifolds: here

Intro to Analysis: here

Intro to Real Analysis: here

Complex Analysis: here

Elementary Number Theory: here

Feel free to add to this with more pdfs for mathematics and more resources to learn!

[the-real-numbers-deactivated202]

[nowhites]

math problem: *begins with “we know that..”*

me: WE dont know SHIT

[mathprofessorquotes]

“Ugly is okay.”

— Differential Equations Professor

[mathprofessorquotes]

“Most of our attempts are failures. Like artists.”

— Quantum Mechanics Professor

[mathprofessorquotes]

“You will never see inertia on any test because…I don’t understand it either.”

— Multivariable calculus teacher

[newtonpermetersquare]

Absolute-ly

[fun-o-physics]

[collegemode]

Okay, so you’ve been called smart all your life. As a kid, you were one of the smartest in your class. Maybe you could read at a much higher level than your peers, or you could fly through multiplication drills like they were nothing. Then, you get to high school and suddenly you’re surrounded by lots of people who were ‘gifted kids’. None of what made you ‘special’ seems all that important now. Your work is actually challenging, and it’s actually requiring effort.

If you’re experiencing this, just know that so many students have gone through the same thing. Maybe it happens in high school, maybe college. But a lot of us who were considered gifted as kids suddenly run into this and it challenges our entire identity. It can be paralyzing, but it’s 100% possible to overcome it and succeed! I’ve compiled a few tips for ex-gifted kids dealing with impostor syndrome and self-doubt. I’m not a therapist, psychologist, or any sort of education expert. I’m just speaking to my own experiences, and I welcome any input from others who have insight into this as well!

1. Understand that working hard does not mean you aren’t intelligent. If something doesn’t come naturally to you, that’s not a reason to give up. Believing that people can do things “just because they were born with a talent for it” is only going to hurt you. It’s not true! People may have natural aptitudes for things, but hard work is involved even for the smartest or most talented people. You are capable of learning anything, and you don’t have to be “good at it” right away to do so.

2. Comparison will kill you. You are your only competition. Focusing on how you rank with other students, and comparing yourself to your classmates is going to exhaust you. By focusing on others, you can’t put your full energy into focusing on your work and yourself. You belong. Even if you struggle with your work, you belong. Focus on your own self-improvement and doing your best.

3. Don’t focus on the goal, focus on your current actions. If you’re always thinking about the future, and about whether you’ll get into that school or that program or win that award or get that scholarship, you’re not using that time to get work done. Don’t worry about college applications, just do your homework. Focus on what you are doing now to reach your goals so you can apply to schools with confidence later.

4. Your grades may not reflect intelligence, but they do reflect work ethic. Don’t let others convince you that grades mean nothing. They sure as hell mean a lot to colleges, and thinking that you should “reject the current education system” is not going to harm anyone but yourself. If you don’t feel like you’re learning anything in your high school classes, that’s all the more reason to want to get into a university that will challenge you. If you put effort into your work, it will not let you down. Your hard work will be reflected on your transcript. Don’t lose focus.

5. Talk to someone. Let people know if you’re struggling. It can be hard to feel like you aren’t allowed to identify as “smart” or to feel pressure to constantly compete and improve. I went to a highly competitive high school that pushed kids to cope in dangerous ways. This is not healthy and not okay. If you’re feeling overwhelmed you need to find healthy coping mechanisms. Speak with someone you trust and don’t let yourself spiral. Don’t try to self medicate. Your well being is always more important than your grades. Period.

6. Enjoy yourself. School may seem like hell, and you may feel like it will never end and you’ll always be stressed and worried. But high school is only four years, and you can do things during that time that you probably won’t ever again. Take advantage of things that seem fun, even if people think they’re nerdy or weird. Try and remind yourself that you’re lucky to have your education and you have the power to do great things with it. Don’t lose sight of your own ability and your bright future!

[matan-matika]

A More Intuitive Topology

Topology is defined a bit oddly. Mostly,we know what sort of properties we want a “space” to have, but formalizing them to be defined using set theory is a little trickier.

The mixed metaphor I will use will be maps and lenses. It won’t be perfect, since it’s hard have a good intuition for “infinitely many.” Also, many intuitions generalize from very nice topologies called “manifolds,” and so won’t generalize perfectly.

What we really want to do with space is figure out what is close together, what an individual area looks like, but we don’t want to see individual points.

So, a topology should contain sets of points, and think of those as the areas we want.

Now, it can’t just be any collection of sets, so let’s think of some constraints we can impose on them.

1) If you can see two different areas, then you should be able to see what happens when you put both together.

In set theoretic terms, this means we want it to be closed under union.

There is no reason for it to just be a finite number. If we get infinite areas, when you put them all together, there is no reason for them to not also be like a space.

In fact, infinities let us make different shapes that we also want to be able to use.

So, it should be closed under arbitrary (both finite and infinite) union.

2) If you see two different areas, then you should also be able to only see areas where both are. You could think of putting the two filtering lenses on top of each other, or drawing only where the maps agree.

In set theory terms, we want it to be closed under intersection.

Unlike union, we don’t want it to allow all infinities. Think of if you have circles getting smaller and smaller.

Then, if you could see the point they all share as an “area,” then we can see any individual point too. This is ok for some topologies, but we don’t want it to be a condition for all topologies.

That is, it should be closed under finite intersections (but not necessarily infinite ones).

Note that this makes an asymmetry between the operations, which is fun and interesting.

3) It must contain both the empty set and the entire space.

If we want to see areas that aren’t near each other, for completeness, their intersection is the empty set.

If we take all of what we define as areas, they should cover the whole space, because we don’t want to focus on anything we can’t see.