/sci/ - Science & Math

Could somebody explain to me what topology actually deals with?

>>10778817
It deals with topological spaces and continuous maps between them.
These topological spaces are sometimes pointed, i.e. have a base point, and we sometimes require that maps between pointed spaces preserve the base point.

>>10778817
Topology is about recognizing the symmetry is deformity.
If you have a ball and you deform it into an egg, you still have the same class of surface.
If you then tear the ball apart, that is a different class of surface.

>>10778812
You are right anon. There is nothing wrong with feeling the way you do. Most academics have probably gone through this as well (pic related).
It is better that you ask yourself what you really want now than after two postdocs when you are 30 and possibly realize that you hate your life.

>>10778572
You are right anon. There is nothing wrong with feeling the way you do. Most academics have probably gone through this as well (pic related).
It is better that you ask yourself what you really want now than after two postdocs when you are 30 and possibly realize that you hate your life.

are there any good math yt videos/udemy Courses?
I cant read a book without my mind wandering off after 2 pages

"Given a carousel with 10 seats, how many different seatings are possible for 6 children?"
If I understand this correctly, I need to put 4 equal empty seats somewhere in 10 spaces, then organise the children in the remaining spaces, then divide by 10 because it's circular and therefore gives redundant seatings?
[math]nCr(10,4)*6!/10[/math]

If you guys get this question a lot, then feel free to tell me to fuck off, but I want to get back into Math as a hobby. I studied it at Uni and got an undergrad degree, but that was a solid half decade ago and it was an unfocused course at that so not worth much. I've not touched the stuff since, so some of the knowledge is gone, but I've recently become interested in it on a personal level.

What's a good way to try and make it into a personal hobby, as in learn it just at my own pace? I'm planning to start by looking back at what I studied back then to see if I can bring it all back and then build from there, but any tips or materials you recommend would be cool.

>>10779119
I dunno, depends on what counts as a different seating.
>>10779152
Get really angry at how some people know stuff you don't.
Yukarifag's autism has propelled me into wisdom.

>>10779152
listen to >>10779214 this is excellent advice, perhaps the most intelligent ive ever seen on this site

>>10779214
redpilled

> postfix notation

This is more of a sanity question than anything else, but I am correct in saying that E(X(s) | F_s) = X(s), where X is some stochastic process adapted to the filtration F?

>>10779152
Why don't you go get a masters?

>>10779720
Not him, but a good masters degree is a very expensive gamble

>>10779721
It depends on his country.
Him referring to uni as 'uni' made me think he was a britbong.

Either way, if you're passionate about mathematics, it's worth doing.

>>10779152
>What's a good way to try and make it into a personal hobby, as in learn it just at my own pace? I'm planning to start by looking back at what I studied back then to see if I can bring it all back and then build from there, but any tips or materials you recommend would be cool.

I’m majoring in math, about to start taking Cal 2 and linear algebra this semester, but there’s a ton of math i don’t know, in geometry, algebra, precal, and Cal 1. I want to start all over and learn everything from the ground up, with nothing left out. At this point I’ve just accepted that I don’t really learn much just by getting good grades in my classes. It’s easy for me to get an A and only have a superficial understanding, and possibly forget stuff later. So I just want a nice, clean sequence of books that I can read and teach me everything I need for undergraduate math. If i have to read 2 books on the same topic, I will. I just want to get rid of that awful feeling that I’m missing something, that I don’t know it completely. What books should I read? What comes first? Do I really need to read some introductory book like A Mind for Numbers? Do I need to read logic books? Do I start with set theory and proofs or do i start with geometry and algebra?

>>10780130
I've personally found that letting knowledge stay still in my head for a year lets it gradually soak until I've acquired masterful domain over it.
But I'm fairly sure it changes from person to person.

>>10780130
>What books should I read? What comes first?
see >>10780117

>>10780167
Meme list

>>10780191
>Meme list
What do you mean?

>>10780196
Doesn’t even introduce set theory

>>10780201
>Doesn’t even introduce set theory
Meme theory

>>10780205
You're retarded, got it.

>>10780256
>You're retarded, got it.
Why the ableism?

>>10780261
Question, have you ever actually engaged in tumblr-level debates or are you using tumblr-speak because you need to mock it since you don't understand them?

>>10780276
>Question, have you ever actually engaged in tumblr-level debates or are you using tumblr-speak because you need to mock it since you don't understand them?
We're not familiar with tumblr, may you please elaborate?

>>10778812
>be math major in senior year
>take PDEs as my math elective
>think this is gonna be some tuff stuff
>open book
>it's literally calc 5
>''this is the general IVP wave equation"
>"here, I solved it for you"
>HW problem: What is the answer to this IVP wave equation?
God, I feel like such a brainlet. I wish I hadn't taken a year off of school to smoke weed and be lazy. I could have skipped this bullshit and just taken grad level math if I hadn't poissoned my brain.

>>10780117
I don't think these are realistic goals.

>>10779119
yes that's correct
>>10780276
shit, remember when tumblr was actually relevant?
>>10780302
>literally calc 5
kek

>>10780302

what PDE book?

>>10780302
lol imagine unironically taking PDE for engineers
real PDE is fucking nuts. educate yourself and read Evans.

>>10780302
>>''this is the general IVP wave equation"
>>"here, I solved it for you"
>>HW problem: What is the answer to this IVP wave equation?
Which school for brainlets do you go to?

>>10780533
who are you quoting?

>>10780547
>who are you quoting?
>>10780302

>>10780130
Any actual answers to this?

>>10780659
you could use your high iq to follow the link in the sticky that you should have read before ever posting on this board and there you will find the answer to your question

These mathematics books for engineers, physicists or scientists are good for a quick and superficial review of algebra and early calculus?

Considering that's for an exam in a week. I've procrastinated fuck a lot, and I have some exams which have pre-calculus questions with some integrals and derivative functions.

Some of these books cover elementary algebra to differential calculus (except geometry) in only 200~300 pages. I think this is the best choice for me to go through at the moment, isn't?

>>10780130
Gelfand books & Basic Mathematics by Lang are sufficient. These books will provide a more than sufficient background for calculus.

>>10780682
>Gelfand books & Basic Mathematics by Lang are sufficient.
Lang is a meme.

>>10778817
Me. It deals with me.

Is it a bad idea to take Real Analysis 2, Abstract Algebra and partial diff equations on the same semester? 3.3 gpa brainlet btw

>>10780818
>Is it a bad idea to take Real Analysis 2, Abstract Algebra and partial diff equations on the same semester?
Why don't you try it and find out?

Homotopy thread

>>10778974
Why don't you guys just work in the engineering or banking industry for a few years and then cycle back to academia? Building up a network will help you get industry grants later anyway.

>>10780885
Getting into finance is very hard unless your PhD is at an Ivy or something similar

>>10780856
Homo

What ever happened to Michael Atiyah's claimed proof of the Riemann hypothesis before he died?

>>10781022
Fucking hell, I remember the night before he gave the presentation.
Someone got access to the paper via Google Drive. I wanted to believe that what I was reading was a joke. It turned out to be his actual work.
I still cry every time.

>>10780885
If you want to work in academia to advance mathemathics and not just to meme away teaching uni students, you should NOT do this.
You reach peak intelligence between ages 20-30, later on you only get dumber. History shows that most mathematicans don't prove/find anything truly amazing after 30, and they completely run out of gas after 40.

Let [math]A[/math] be a [math]n \times n[/math] matrix with entries in [math]\mathbb{Z}[/math] and [math]b \in \mathbb{Z}^n [/math].
Under which conditions does the system
[eqn]A x \equiv b \mod m[/eqn]
have an unique solution modulo [math]m [/math]?


If [math]m[/math] is a prime number then it's clear that it has a unique solution iff [math]\det(A) \not \equiv 0 \mod m[/math].
I suspect that in general there is a unique solution iff [math]\gcd(\det(A),m) = 1[/math] but I don't have a proof for that.

>>10781488
What have you tried?

>>10780885
Why would somebody who is in banking or industry, go back to academia. That's just so completely stupid.
After a few years in industry you can get to be a manager and make bank. Instead you are going to take a MASSIVE pay cut to become some assistant professor without tenure, lol, so that you can teach first years Calculus I to III?
Doesn't make sense at all.

The whole appeal of academia is that you can go through life doing your own thing instead of slaving away for fat cats. If you waste your prime in industry, it makes no sense to go back to academia later on. Worst of both worlds.

>>10781514
I know a former quant who became a "prépa" teacher in France, so basically teach calc and linear algebra. They are very well paid, but it was still a massive step down.
The thing is, the guy was working all the time and barely had time to even spend the money he was making. Now, he only has to teach 15 hours a week and makes more than an engineer. Seems like a pretty comfy deal.

>>10781514
If you go to academia you are just switching fat cats. Now not only coaches and university presidents, it made my blood boil when I saw that now “diversity officers” at colleges make 300k a year. Imagine 300k a year for that guy while your postdocs make 30k a year. Fuck universities.

Also, all the non-retarded researchers come from industry. For example, oil industry engineers (ENGINEERS!) figured out climate change decades before your p hacking retards “academics” even knew what CO2 was. Universities will fall in this century. We are moving back to the old system where academics were tied not to universities but to nobles (now Fortune 500 execs).

>>10781539
I can tell you many people have this exit strategy, even me. Finance is great but the problem is that it is a job that follows you home. Even while you jerk off you are probably moving millions of dollars and if anything bad happens to them, you have to answer. Goal is to become a millionaire and then bounce. Very few last in hedge funds more than 5 years.

>>10781616

>still haven't received my acceptance letter from Germany
>potentially have to wait another three weeks for a reply
This torture. I want to know my future.

>>10781713
>he doesn't know acceptance letter are send first and quick
a-anon...

>>10781728
I'm not German and my case is a bit special, they have to verify some stuff with my previous uni so that back and forth might take some time.
In any case the status of my application is still pending processing by faculty so I don't think they have even got to it yet.
Well, that's what I hppe.

>>10781713
>>still haven't received my acceptance letter from Germany
What are you applying for?
And usually the semester starts in a couple of months, right.

>>10781748
Just a normal math msc. at a semi reputable uni. Semester starts in October I think.

>>10781787
Might just be that right now is a pretty busy time, but I have to say that for me it took a while.
Isn't it usually the case that you have to wait till the end of the registration period?

>>10780553
Who are you quoting?

>>10781491
To be able to solve [math]A x = b[/math] uniquely you normally need the existence of [math]A^{-1} [/math]

Usually there is the formula [math]A^{-1} = \frac{1}{\det(A)} C^T [/math] where [math]C [/math] is the cofactor matrix of [math]A[/math] but since we're in [math]\mathbb{Z} [/math] it can't be used since there is no division in [math]\mathbb{Z} [/math].

Well if [math] \gcd(\det(A),m) = 1 [/math] then Bézout says that the congruence
[eqn] \det(A) \cdot k \equiv 1 \mod m[/eqn]
has a solution [math] k \in \mathbb{Z} [/math].
I thought with it you can get something similiar to [math]A^{-1}[/math] by talking [math] k C^T [/math] instead.

>>10781558
hahahaha this post made me laugh

>>10778974
source?

>>10779152
You completed undergrad, so you should be able to just pick and read a book about whatever interests you.

>>10781945
The product rule det(AB)=det(A)det(B) holds modulo n, because it holds over reals and thus over integers and then you reduce both sides mod n; so you have that if a matrix is invertible mod n, then its determinant is invertible mod n. For the converse implication I think you can still use the cofactor matrix: follow the proof for real matrices and replace nonzero with invertible mod n.

>>10780117
What are the fourth and fifth books pictured here?

>>10782268
Atiyah's K-Theory and Besse's Einstein Manifolds.

>>10778817
Like geometry if everything is made of stretchy rubber.

>>10782748
Thank you anon.

Here is a great youtube channel you might like.
https://www.youtube.com/channel/UCRr9mh4Z9xiW4F9rC59jdEQ

>>10779076
You can try Khan Academy. Honestly If you cant focus on a book, I'd look into developing study habits. You can also try just realizing you are not focusing, and force yourself to get back at the task at hand. Had to do that quite a bit today. I would also see if the space you are studying in makes you unfocused: try switching it if it does.

>>10780117
Can someone name off all of the books? Particularly the last two, they're impossible to read.

>>10782784
Last book

Mathematics is built around the idea of "mathematical rigor" which changes with time in accordance with major crises (impossibilities) over the course of mathematics's development. Said idea of mathematical rigor is effectively an attempt to derive "thus" (consequently) from "then" (subsequently) as a finite set of rules of inference. However the very fact of derivation requires "thus", not "then". In other words, you have to use "thus" in order to explain\explicate what "thus" is (in terms of "then" or otherwise). This is, of course, an irreconcilable circularity.

>>10782784
Penultimate.

>>10782841
P.S. This argument effectively kills mathematics as a branch of knowledge much the same way Hume killed philosophy, as a branch of knowledge.

>>10782841
>>10782865
try >>>/lit/ or >>>/lgbt/

>>10782868
Did you derive this statement in accordance with the currently held standards of rigor, I wonder?

>>10782874
blow it out your ass wildberger

Statistics and CS is applied math. Isn't that true?

>>10780117
Wtf is wrong with dates in that picture

>>10782984
Nothing, it's Arabic.

I'm in a bachelor/master's program with one year left, so I've already taken a significant number of graduate level courses.
I haven't taken an actual topology course yet (math course rotation combined with prerequisites made this impossible) and in spring semester I'll be able to take algebraic topology, but I just had a few questions.
Should I learn general point set topology before algebraic topology? I have some exposure to topology from analysis I and II and a manifolds course, so I'm not totally ignorant to basic concepts.
If I should self-study point-set topology, is it better to use an undergrad text or just jump straight into a graduate text given that most of my course work is graduate level at this point?
Thanks in advance

>>10780533
I go to one of the top math unis in the USA.
>>10780502
I realize now that grad and undergrad level PDE are way different. Unfortunately only one was offered over the summer.
>>10780488
I'm not telling you. Why does it matter?

>>10782993
If you self study with Munkres, this plus your prior coursework should be sufficient preparation

>>10782993
>Should I learn general point set topology before algebraic topology?
Yeah, but it isn't the strictest of requirements besides the basic concepts.
I've honestly never heard of a graduate text on point-set topology.

>>10782999
I'll get Munkres' text and get started then. Much appreciated.

>>10782841
That's incorrect. There is a quantized difference between any two statements.

>>10782796
>>10782857
Many thanks. Enjoy this my friend https://www.youtube.com/watch?v=RcumNZ4Y3Ck

>>10782776
>>10783158
What sort of shilling is this? Could you stop?

>>10782997
Yeah, a summer course is likely to be a bit flaky on the details. The upshot of this is that there's a ton of more interesting things in PDEs left for you to learn, so that's probably cool.

is suicide the end conclusion to any question?

>>10783343
yes

>>10782857
>GIT

>>10783359

it's torture to know our desire to discover more about math and science is just as trivial as any religion, it's all just people trying to entertain themselves and procrastinate the end

>>10783393
>Homo-things

>>10780695
t. know nothing regurgitator

>>10783557
>t. know nothing regurgitator
Not an argument.

>>10783158
what on earth

>>10783557
hey lang, how's the anti-HIV movement going?

>>10781945
>but since we're in [math]\mathbb Z[/math] it can't be used since there is no division in [math]\mathbb Z[/math].
But you aren't working in [math]\mathbb Z[/math] precisely. You're actually sort of right in everything that comes after this post: you're actually working with matrices whose coefficients live in [math]\mathbb Z / n\mathbb Z[/math], the ring of integers modulo [math]n[/math], and an element of [math]\mathbb Z / n\mathbb Z[/math] is invertible if and only if it is coprime with [math]n[/math] as you have showed. Thus >>10782253 is also right.

>>10781256
In the sense that he was going senile?

>>10781488
Use cramer rule

>>10778812
Anyone here a later bloomer regarding your interest in maths? In my early twenties and shit-the-fuck-off with myself for not noticing for not noticing how fascinating it is when I was younger.

>>10781540
This is unironically a good thing.

>>10778817
balls. big balls. topolgists have the biggest balls of all mathematicians.

>>10780130
>I just want to get rid of that awful feeling that I’m missing something, that I don’t know it completely
>completely
Anon, I...

>>10784514
It's never late enough to learn. The only people who are missing out are the dead. Now get fucking studying.

My paper was accepted for a conference abroad (France), but my visa got fucking denied. I'm suddenly feeling demotivated to do math, but at the same time this makes me want to try harder.

I know my motivation shouldn't be dependent on being able to attend prestigious conferences. But this all makes me feel like months my of work going down the drain (I still get a publication out of it though).

feelbatman.jpg

>>10784574
Thanks, anon.

>>10784745
>he's a frogposter
nevermind, you'll never make it.

Any advice for statistics books?
I really want to derive a understanding of the material rather than just blindly memorizing the usage of different tests and what they’re good for

>>10784599
What kind of shithole country are you from?
Also consider getting a visa for Spain/Germany and just taking the train/bus to France. It's an open border.

>>10785151
http://sgsa.berkeley.edu/current_students/books/

>>10783158

>>10778812

I am interested in the following system of equations:

[math]x_1^2+x_2^2 = bx_3 + x_4[/math]
[math]x_3^2+x_4^2 = bx_5 + x_6[/math]
...
[math]x_{2n-3}^2+y_{2n-2}^2 = bx_{2n-1}+x_{2n}[/math]

Notice each equation has a linear term on the right side.

Say we solve for the linear variable in each equation and substitute for that variable in the next equation.

Does this process create a single equation which has the same set of solutions as the system? or does this change it?

How do you genuinely learn stuff with self-studying? At university, you eventually learnt formulas and proofs just by reading and encountering them over and over again, but I'm finding that when going through a textbook at home, this doesn't happen.

What are the best books for self learning high school level algebra and trigonometry? I know Gelfand has covered both but I don't think those books are as rigorous in exercises as they should be. Or are they enough? Any recommendations?

>>10785099
Hey, take that back. He'll be okay.

>>10785176
Schengen visa

>>10785762
i know, i was only joking! i'm sending all my love your way mathlearninganon :)

I hate mathfags

>>10778817
It studies the category TOP where the objects are topological spaces and the arrows are continuous functions

>>10785795
>mathfags
Why the homophobia?

>>10785795
Why do you hate mathematicians?

>>10785795

>>10785853
All the math you need to solve problems are calculus and linear algebra

>>10786211
Which of these helps me do my taxes?

>>10786265
Oh that's highschool shit.

What are the prerequisites for noncommutative geometry?

>>10786417
What type of noncommutative geometry.

>>10786417
IIRC C*-Algebras and you're golden.
>>10786425
Noncommutative is noncommutative and anabelian is anabelian.

>>10786431
It is enough to understand Conne’s book?

>>10786431
>Noncommutative is noncommutative and anabelian is anabelian.

There are different types of non-commutative geometry. Connes style is the *-algebras stuff.

>>10786446
Thanks anon

Is pic related an actually good foundational approach to maths or am I just getting memed?

>>10786483
>Is pic related an actually good foundational approach to maths or am I just getting memed?
Memed

>>10786483
memed

>>10786483
>Is pic related an actually good foundational approach to maths or am I just getting memed?
Why don't you try it and find out?

>>10786483
100% memed. The picture was either made as a hyperfoundational joke or by a hyperautist.

>>10786417
Yang-Mills theory and Atiyah-Patodi-Singer
http://www.alainconnes.org/docs/bookwebfinal.pdf

>>10785652
bump

>>10779076
Professor leonard

>>10781010
Lol nice

Is there a better book for learning Calculus than Spivak?

>>10786417
Tohomology

>>10779702
Yes.

For a sigma-algebra F, E[X|F] is the expected value of X, given we know everything about F.
If X is measurable with respect to F, then knowing F also means knowing the exact value of X, so the expected value is just X itself.

Similarly, if F was a trivial sigma-algebra, then F won't give us any new knowledge about the expected value of X, so E[X|F]=E[X].

>>10778817
Topology is all about the process of converting (or parsing) x-dimension to y-dimension.

Practical Example:
In the machine language level, everything is one-dimensional (a RAM is just a huge 1D array). But higher level programming languages like C++ need to have three-dimensional arrays to solve real-world problems.

So we need to create a "pseudo" three-dimensional array by folding the "real" one-dimensional array. That's topology in a nutshell.

>>10787263
>converting (or parsing) x-dimension to y-dimension
>machine language level
>a RAM is just a huge 1D array
>higher level programming languages
>That's topology in a nutshell.
Refer to >>>/g/.

>>10787263
Your kind isn't welcome here. Use >>>/g/ and/or >>>/sci/amg/.

>>10787266
>>10787272

Ok, I will explain in a way that's more friendly to this thread.

Physical/Mathematical Example:
In physics, everything is a 3-dimensional object. But manipulating a 3-dimensional object is more complex. So we can use topology to strip down 3d objects to 2d or 1d.

We use topology to broaden the problem (more ways to think, but more complex), or to strip down the problem (simpler, but fewer ways to think).
We use topology to solve problems that are too hard to conceptualize in lower dimensions.

This can be analogous to human knowledge, an average mathematician is now more knowledgeable than Newton, but that's only because we have now more mathematical concepts that connect to previously impossible thoughts.

>>10787302
you don't know what you're talking about

>>10787302
Jesus christ, which pop sci guy told u that garbage?

>>10787302
>Physical
Refer to >>>/toy/.

>>10787307
>>10787323
> t. retards who only follow mathematical jargons

If you understand something in only one way, then you don't really understand it at all. The secret of what anything means to us depends on how we've connected it to all other things we know. Well-connected representations let you turn ideas around in your mind, to envision things from many perspectives until you find one that works for you. And that's what we mean by thinking!

>We use topology
No, you don't use topology. Nobody who uses it has this retarded of an intuition about it.

>>10787324
Kekk

>>10787302
>previously impossible thoughts
Refer to >>>/lit/.
>>10787325
>The secret of what anything means
>what we mean by thinking
See the above message.

>>10787325
>Well-connected
Do you mean "simply-connected"?

>>10787302
>In physics, everything is a 3-dimensional object.

>>10787333
He doesn't mean anything, it's just >>>/lit/ glibberish. There is no way this anon has any kind of mathematical education, much less in topology

>>10787325
>you don't really understand it at all
Maybe try posting at >>>/lit/, you're really good at writing "subtle" and "meta" "self-satire".

Guys you don't need to fake having read (and understood) a single page of topology to post here.I for one don't even know trigonometry and everyone treats me very well.

>>10787302
>>10787263
>>10787325
Please refrain from answering questions if your knowledge of that subject is zero.
Either you are trolling or trying to helpwith something you have no knowledge about, in both cases, stop

if grisha hadn't declined the million bucks and the fields medal none of this popsci smear would've happened to topology

Anons here are stuck in thinking in rigid "formal mathematics" (whatever that means).
They mock me because I have my own informal ways to explain things, and that makes them uncomfortable.
I just want to explain topology in the simplest way that doesn't use circular mathematical terms (which for me is cringy).
Because that's the only way to test if you really understand something.

>>10787382
I'm just interested in "understanding". The "meta" aspect of knowledge.
/sci/ wants to be an expert in science and mathematics,
/lit/ wants to be an expert in philosophy, religion, and literature.
I think mathematics and philosophy are *not enough* to proclaim that you "understand" everything.
And calling yourself an expert in one of those fields is like saying "I'm so smart, I don't need to understand everything else".
But that's not smart, isn't it?
Maybe this is why self-proclaimed experts are so smug.
For me, they are not experts at all. Just people "stuck" in a very specific way of thinking.
That's not smart, that's like a magic trick tricking people you are smart.
For me; to be smart, you need to be childlike, to be adventurous and "find out" everything your own way.
The trick is not to get stuck with mathematics and philosophy (or whatever subject that is) and call it done.
My heuristic is to be comfortable with the idea of being uncomfortable.
For a child, everything is different, everything requires an effort of understanding, but he does not get uncomfortable,
he sees it as an adventure.

>>10787459
>I'm just interested in "understanding". The "meta" aspect of knowledge.
Fuck off to >>>/lit/ and discuss that there then.

>>10787459
>They mock me because I have my own informal ways to explain things
We have our own informal ways too, they're just not cringy pop-sci-tier garbage with no basis in the actual subject (you know, the thing you claim to want to """"understand"""").
>which for me is cringy
Literally every single post of yours in this thread is also cringy, yet that doesn't stop you from making them.

>to be smart
>"understand" everything.
see >>>/lit/ for pseudo-inellectual retardation.
>I just want to explain topology
someone who doesn't understand topology on a basic level has no business trying to "explain" topology.

>>10787459
>I'm just interested in "understanding". The "meta" aspect of knowledge.
No need to say it explicitly, we can tell by your complete lack of mathematical intuitions.

>>10787459
Can we get some mods in here, trolling is a bannable offense

remember back in 1600 when one measely little volcano cooled the entire planet?

>calling yourself an expert in one of those fields is like saying "I'm so smart, I don't need to understand everything else".
No, you fucking moron, it just means that you understand a particular field really well. This whole post is just your retarded >>>/lit/ mind projecting and trying to cope with the fact that you can't really call yourself an expert in any particular field even though you really want to seem "intelligent".

>>10787459
>If you understand something in only one way, then you don't really understand it at all.
>Because that's the only way to test if you really understand something.
>like a magic trick tricking people you are smart
>be comfortable with the idea of being uncomfortable
It's amusing how these "new-agey" popsci-tards always seem to ironically talk about themselves in all of their ramblings. I can't even tell satire from reality anymore.

>>10787496
>mathematical intuitions
>>>/x/

>>10787459
>They mock me because I have my own informal ways to explain things
No, they mock you because you are retarded and you clearly know very little on the subject...

>>10787671
>mathematicial intuitions are "para"normal to him
This thread is for mathematicians only.

>>10787701
>>mathematicial intuitions are "para"normal to him
We're not a "him".

>>10787701
>mathematicial intuitions
see >>10787671

>>10787812
Dear him/her, see >>10787701

>>10785572

bump

>>10786417
https://www.uwo.ca/math/faculty/khalkhali/files/Beginner.pdf

>>10787333
No, he obviously means path connected. Simply connected is far too strong.

>>10788002
No, think about it in terms of the fundamental groupoid, not the group. You can understand C*-Algebras in functional analytic and algebraic terms, but unless these two comprehensions can be continuously deformed into each other, you didn't truly understand anything.
Truly a 21st century genius.

>>10786417
>>10786445
Horrendous idea desu. Try just reading Nikolaev, it should be doable if you have a solid basis in functional analysis.

>>10782215
Felix Klein - Development of mathematics in the 19th century

Since this thread is just people being mean to one guy (as they should) I'll try to give a simple idea of what topology actually is.
Topology is the study of one of the weakest notions of structure on a space. Essentially, a topology on a given set just tells you roughly what points are close to one another. This is enough to do things like deform the space without changing its overall sort of "shape" (holes and such) or do continuous things like take limits (really nets). You can also talk about how connected your space is, and about how spread-out it can be made (compactness, locally compact, ...). There are types of topology in which you relate certain algebraic structures like groups to your space, or where you give your space an extra differential sort of structure. Some types of topological spaces include manifolds (any space that looks like n-dimensional space if you zoom in, like a curve or a torus) and graphs (edges between vertices help determine closeness). In topology we can determine facts from very basic "analytic" assumptions which tell us things about a wide variety of "analytic" objects, just like how groups can tell us things about a huge amount of "algebraic" objects from just some very basic "algebraic" assumptions.
>>10787511
While they're at it perhaps they can do something about the "mathematicians say we" poster and the "refer to >>>/lit/" poster

>>10788016
you know i want to find a way to disagree with your post but i like it way too much to do that. thanks for making me smile.

What is this projective transformation they are talking about?

n in this situation is a point of a hypersurface in P^5 whose points correspond to lines in P^3. (By their plucker coordinates.) They don't explain anything else about what exactly u_0 and u_1 are.

Is this just a change of basis?

>>10786603
What would you suggest as an alternative roadmap?

>>10788151
Yes, a projective transformation is induced by a linear transformation of the corresponding vector space (ie. a change of basis).
I am guessing that the u_i’s are a system of homogeneous coordinates on P^3.
Translating this in linear algebra terms, what it says is that, given a plane in K^4, you can always find a basis (e_0, e_1, e_2, e_3) in K^4 such that the plane is given by the equation u_0 = u_1 = 0, which is very easy: start with any basis e_2, e_3 for the plane and complete it to a basis of K^4.

>>10786417
real analysis, dynamical systems, topology, functional analysis, ring theory, some basic algebraic geometry

>>10781324
Can you give examples or cite a study?

Some mathematicians are productive until the day they die.

>>10781324
>>10788241
I never understood where this meme came from. Most first-rate mathematicians are productive until close to their retirement. The only ones I can think of who did notable work in their 20s and not older are the ones who died before they could get old.

>>10788181
>What would you suggest as an alternative roadmap?
see >>10780117

I was trying to derive a combinatorics formula but got bored/mindfogged midway. Does your endurance get better with practice or am I fucked?

Anyone willing to explain matrix multiplication to me as if I was a brainlet?
Following the Strang MIT videos on Linear Algebra currently finished lect 2. I understood what he did but its still kinda confusing. Theres rows and colums and individual cells and the order of multiplicants matters and u cant just swap them wtf...

>>10788374
It gets better, when I started my undergrad I barely could stay focused for 40 minutes but now I can truly focus on a problem for more than 2 hours, easily.
Go for a walk, take a sip of water and sit down again

>>10788383
>as if I was a brainlet
do you really need the "as if" here?

>>10785830
Based Category Theory poster

>>10788383
This is going to be long and autistic, but books usually don't explain it for good reason.

A linear transformation is a map [math]T: V \rightarrow V[/math] such that [math]T( \lambda u+v)= \lambda T(u) + T(v)[/math], where [math]V[/math] is a vector space, [math]u[/math] and [/math]v[/math] are vectors, and [math]\lambda[/math] is an element of the base field (aka a scalar).
We know that we can consider elements of the vector space to be sequences of scalars (their decomposition by some base). We wish to also transplant this to linear transformations.
We fix a base, and order it. It's easy to notice that a linear transformation is entirely determined by its values for the base. So we take the vector that it maps the first element of the base to, and set it as the first collumn, and so on. You can check without much hassle that this works by the definition in Strang.
But now we have a new problem: we can apply linear transformations (matrices) to vectors. But we don't know how to compose them.
So we define matrix multiplication as it is in Strang, and check that it coincides with composition: consider the first element of the base, and check to what it is sent by composing twice: does that coincide with the first collumn of the multiplication? And so on.

>>10788303
It came from Hardy.
I don't agree with it at all. Everyone's different.
Either way, who cares?

>>10788054
>While they're at it perhaps they can do something about the "mathematicians say we" poster and the "refer to >>>/lit/" poster
I suspect they're the same person, and I expect they will tell me to refer to >>>/lit/:

I am interested in the philosophy of science.
Can anyone recommend any introductory texts?
Something from a mathematical perspective might be nice.

>>10788468
>I expect they will tell me to refer to >>>/lit/
And they'd be right. This is just a self-fulfilling prophecy, talking about off-topic subjects will obviously prompt a redirect to the appropriate board.
>philosophy of science
Why would you ask about that here? Ask in a philosophy of science thread on you know where.

Thinking about taking a course in integrable systems next semester.
From the course description it will be "A rigorous treatment of
Manifolds, Lie groups, Lie algebras, Poisson structures, Lie-Poisson structures, Completely integrable systems, R-brackets, r-brackets".
Besides the usual analysis/algebra courses I have taken courses in differential geometry, algebraic topology, probability theory and an intro level course on dynamical systems.
Any anons with experiences in this direction? Is it interesting? Do you regret studying it?

>>10788468
>philosophy
Use >>>/lit/.
>>10788496
>rigorous treatment
See the above message.

>>10788496
I think it's very interesting. Integrability plays a large role in exactly solvable models and hence ergodic/catastrophe theory. There's also a correspondence between integrability conditions and quantum cohomological algebras, where some ODE on the former side leads to flatness condition of connections on the latter side. This leads to an interplay between Yang-Baxter and Gromov-Witten theory where crossing conditions in some current algebra corresponds to point counting in some moduli stack.

>>10788483
Fuck you and fuck off

>>10788530
>Fuck you and fuck off
Do you really need to swear?

>>10788530
Great argument.

>>10788530
Please watch the profanity.

>>10788530
Mathematicians say "I respect your opinion but do not believe it is relevant to the current discussion".

>>10788532
Yes, because this general is shat up by posters like this one, and the ones referred to here >>10788054, and the ones going on endless algebra vs. analysis bickering (and know neither), and the ones posting the shitty curriculum thread after thread, and the ones showing off by giving overly complex answers to people who obviously do not have the background to understand (worst offender is anime guy, but many people are guilty of this) and the general asshole-ish vibe of it really.
I know I’m saying shit a lot, but there is really no better way to put it. This place is a shithole filled with larpers and self aggrandizing assholes, with very little mathematical content or exchange.

>>10788562
*we

>>10787302
>>10787266
why are all machine learning fags so self absorbed that they think everybody else also cares about their faggy subject.
also see >>>/lgbt/ or >>>/hm/

>>10788564
>with very little mathematical content or exchange.
Feel free to start contributing.

>>10788564
Why don't you just close this tab and never fucking open it? There, I've solved your problem. Or you could just shut the fuck up (I know I'm saying fuck a lot) and generate some mathematical content you claim to care so much about.

>>10788578
>>10788580
I already have, but sure. Let’s start with an open problem I have found in a journal for undergrads a year ago that, to my knowledge, has not been solved:
Classify the isomorphism pairs (G,f) where G is a finite group of odd order and f an automorphism of G of order 2. I have an idea of how to treat the abelian case (the classification of finite abelian groups should reduce this to a problem about matrices (why?)).
Does anyone have any idea how to proceed in the general case ?

>>10788636
>a classifying problem
f(f(a)a^-1)=af(a^-1)=af(a)^-1 is my contribution. I'll fuck off now.

>>10788679
>a for both
f(f(a)b)=af(b) is actually cleaner.
Anyhow, since the order is odd, it isn't an inner automorphism.
I get the impression you want to classify something that doesn't exist.

>>10788690
Yes it isn’t inner, that is a good remark, but there are definitely examples.
In the abelian case for example the inversion is an automorphism of order 2 (it is actually an equivalence). And there are many more (see the example of (Z/pZ)^n).

Does your ability to think "outside of the box" develop as you go through undergrad, or am I doomed to be good only at symbol-pushing?

>>10788755
Sometimes being good at symbol pushing in areas that others don’t know can seem like thinking out of the box (and in a way, it is)

>>10788636
I’ll also add a funny exercise that I do know how to solve so as not to kill all discussion:
What can be said of a finite group of order > 1 all or whose non-identity elements are conjugate ?

>>10788852
Isomorphic to inner automorphism group, and simple.

Here is a problem for you anons.
The picture shows the correct answer. According to this the end result is 2ℏ^2
According to my own maths it is just ℏ^2
Where does the extra phi come from? What did I do wrong?

>>10788889
>What did I do wrong?
Show your work.

>>10788857
>Isomorphic to inner automorphism group
explain

>>10788896
>explain
It's trivial, what have you tried?

>>10788894
Here's how I did it

>>10788896
We set a map *G->Inn G which we just denote a->a*.
Define the map b(a*) by a*(b). It's trivially bijective.
But a(a^2*)=a(a*), so every element has order two.
Then, we just google the classification of simple groups and exhaust the possible cases.

>>10788907
>sideways picture
>not latexed
didn't read

>>10788952
Thanks for your help :)

>>10788899
I’m asking because it’s wrong
>>10788948
>Define the map b(a*) by a*(b).
What is the domain and codomain and what is fixed here ?
>It’s trivially bijective
Is it ?

>>10788997
>I’m asking because it’s wrong
But it's true, what have you tried?

>>10788997
So this... is the power of lay mathematicians.

>>10788564
you quoted me but i kind of disagree
if you want a welcoming and kind math community, just as if you want a healthy community for any topic, go to reddit.
/mg/ is a place for us to be silly and self aggrandizing and to have fun. this is not a place for people to come ask their homework problems (SQT exists) nor is it a place for people to come ask whether they should take linear algebra or multivariable calculus first.
however, the /lit/ and "we" posts are literally just pure worthless shitposts which reproduce themselves to the point where we have had threads which are 50% plays on that post. i think there's a difference between giving someone a convoluted answer to a simple problem which should have been asked somewhere else as a joke

>>10788889
>>10788907
I don't remember this stuff of shit anymore but I can say that you did the mistake, it's not multiplication but a chain function if you have d/dx and something right next to it.

>>10789009
Saying things are trivial only impresses edgy undergrads.
To give a hint, the only solution to the problem above is Z/2Z, which is not isomorphic to its inner automorphism group

>>10789018
For some reason my head read "non-zero" and also disconsidered conjugation by zero.

>>10789018
>Saying things are trivial only impresses edgy undergrads.
So more than half of this thread? THe other half being randomly saying non-trivial shit with zero context.

>>10789046
if you're going to be a pissy little bitch about everything on /mg/ you should find another general

>>10788383
check out the 3blue1brown essence of linear algebra series on youtube; one interpretation of matrix multiplication is that matrices represent transformations of space and matrix multiplication represents doing a few transformations in some order. another way of many to think about it is when you're changing variables in a system of linear equations---say [eqn] \begin{cases} X = ax + by \\ Y = cx+dy \end{cases} [/eqn] and [eqn] \begin{cases} Z = AX + BY \\ W = CX + DY \end{cases} [/eqn] for some numbers a, b, c, d, A, B, C, D and vectors x, y, X, Y, Z, W. then the coefficients for expressing Z and W in terms of x and y arise from matrix multiplication: [eqn] \begin{pmatrix}A&B\\C&D\end{pmatrix}\begin{pmatrix}a&b\\c&d\end{pmatrix} = \begin{pmatrix}Aa+Bc&Ab+Bd\\Ca+Dc&Cb+Dd\end{pmatrix}, [/eqn] so Z = (Aa+Bc)X + (Ab+Bd)y as an example

>tfw got into university for a stem degree without even knowing time tables and what
commutative,associative and distributive meant

fuck popsci for baiting me into this

>>10787459
Yes you can understand some of topology in a informal way, but at least read something. Reading physics is not reading topology. Maybe try reading wikipedia, don't underestimate it. Start learnign about axiomatic systems, logic and set theory on one hand and on the other hand read about the different mathematical structures and learn differentiate them. Tell apart from the simplist thing like a non-emtpy set and the algebraic structures, the metric spaces and topology spaces for example. The best of all is to learn the connections between this mathematical concepts, for instance: a vector space (this is some algebraic abstraction of geometrical objects) with inner product (the possibility to measure lenghts and angles within that space), can define a norm, with this norm the formal notion of distance is defined and this distance induces metric structure and even topological structure. But this topology is not strong enough to do analysis in that mathematical realm. There is missing something more, this sets already have a structure, but another condition is missing, a thing that would induce a additional structure on the set. But this is not the only structure that you can add to what we started. You can add a more algebraic structure, like Lie algebra o real algebra. Or you can try to see all the geometrical insights of what you have. And so on. Eventually unexpected connections bewteen mathematical fields of study arise and that is one of the reasons of why math worth getting into.

>>10788428
This is indeed autistic, because matrix multiplication is almost never taught after vector spaces and linear transformations. It is always defined without motivation, and the insight coming from a further develop of the theory (one that often is the first encounter with fully abstract math) is not of much help for a begginer.

I'm trying to determine if, given [eqn]f(x) = \begin{cases}
x sin(\frac{1}{x}) & \text{ if } x\neq 0 \\
0 & \text{ if } x = 0
\end{cases}[/eqn]
f'(0) exists or not.

I know that [math]\lim_{x\rightarrow 0}sin(\frac{1}{x})[/math] doesn't exist, but since the range of the function is [math]-1 \leq sin(\frac{1}{x}) \leq 1[/math], multiplying it by x as x approaches 0 would make the limit 0.

Also, the original function is of course 0 AT 0, and since [math]\frac{\mathrm{d} }{\mathrm{d} x}\left ( 0 \right ) = 0[/math], f'(0) would exist and be 0. But apparently this isn't the case?

I'm wondering where exactly I went wrong.

>>10789748
What you are proving is f is continuous. To establish that it is differentiable at 0, you would then need to prove that its rate of change about 0 -- in this case f(x)/x -- approaches a finite limit as x approaches 0.

>>10789197
>popsci
you are a brainlet and deserve this pain

>>10789887
rude

>>10789197
Yeah, I am in a masters degree and keep forgetting what injective and surjective means.

>>10789865
>in this case f(x)/x -- approaches a finite limit as x approaches 0.
Not him, but why?
sin(1/x) doesn't seem particularly convergent to me...

>>10789748
>Also, the original function is of course 0 AT 0, and since ddx(0)=0, f'(0) would exist and be 0.
NO! Think of the line with slope 1 going through the origin, it is 0 at 0 as well, but definitely has NOT derivative 0 at 0.

>I know that limx0sin(1x) doesn't exist
And, I think, that is your answer.

>>10789903
hes saying that you need to prove sin(1/x) approaches 0 to say that f(x) is differentiable at 0--- since it doesn't, f(x) is not differentiable at 0

>>10789913
Phrasing it as "in this case f(x)/x -- approaches a finite limit as x approaches 0." seems entirely wrong though.
f(x)/x doesn't approach any limit as x -> 0.

>you need to prove sin(1/x) approaches 0 to say that f(x) is differentiable at 0
No, you need to show that the limit exists, there is no reason why it has to be zero.

>>10789920
whoops yeah i made a mistake there it doesn't have to go to zero

also, i think he means "in the case of f, to prove that it is differentiable at 0, you would have to prove that f(x)/x has a finite limit as x tends to 0"

>>10789940
>"in the case of f, to prove that it is differentiable at 0, you would have to prove that f(x)/x has a finite limit as x tends to 0"
Yeah, that is of course correct, and pretty much answers the question.

>>10789940
yes, that is what I meant

>>10789903
OP here.

Is there any way to prove the limit of x*sin(1/x) as x approaches 0 doesn't exist? Why doesn't my original argument work? Something like the limit as x approaches infinity for e^-x * sin(x) we know approaches 0, even though sin doesn't have a limit at infinity; so I guess I don't understand why, in my case, a limit that does exist at 0 (x) multiplied by a limit that doesn't exist at 0 (sin(1/x)) doesn't make the resulting limit exist.

Something like x^2 * sin(1/x) is easy with the squeeze theorem, but it doesn't look like I can do that with x*sin(1/x)

Phoneposting atm and don't feel like typing out latex rn

>>10790074
>Is there any way to prove the limit of x*sin(1/x) as x approaches 0 doesn't exist?
No, the limit is 0, else the function wouldn't be continuous at zero, thus not differentiable at zero.

But you should be looking at f(x)/x, that is the difference quotient, you need for the definition of the derivative.
So the relevant question is sin(1/x) as x goes to zero and you correctly pointed out that that limit does not exist, thus there is no derivative there.

>Something like x^2 * sin(1/x) is easy with the squeeze theorem, but it doesn't look like I can do that with x*sin(1/x)
You can, if you have a bounded sequence, times a sequence going to zero the resulting sequence is going to zero as well, you can estimate from above and below.

I think I have a problem and need guidance.
>Be college student.
>Can code the most intricate stuff, employers are impressed at what I can code.
>Want to understand mathematics and try to read the book that is required by my class: Norman l Biggs discrete math.
>Only halfway understand stuff, try to write examples of things and can do the examples.
>I can't connect the theory with the examples.
>Forget Important sentences and try to learn them with Anki.
>Still three months till class starts.
Is this normal?
I passed calc and linear algebra and thought I was good.

Doesn't the statement

"[math]k(X)[/math] is a finite extension of the field [math]f*(k(\mathbb{A}^m))\cong k(t_1,...,t_m)[/math], hence [math]k(X)[/math] has transcendence degree m over k"

imply that the transcendence degree of [math]k(X)[/math] is AT LEAST m? how do we know that the finite extension that they mention doesn't introduce any algebraically independent elements?

Is it because "A finite extension of B" implies "A integral over B" implies "A is algebraically dependent on B"?

Also, how do we know that the [math]t_1,...,t_m[/math] from this statement (which I believe are the image of each coordinate function [math]x_i[/math] of [math]\mathbb{A}^m[/math] under the homomorphism [math]f^*[/math]) are algebraically independent?

My best explanation is that it's because the image of [math]k(\mathbb{A}^m)[/math] to be an ideal of [math]k(X)[/math], hence implies a subvariety of X, which can't happen since X is irreducible.

>>10790776
>Forget Important sentences and try to learn them with Anki.
This line is pretty disconcerting. If you're literally just trying to memorize the textbook verbatim, it's no wonder you don't understand anything.
Your textbook is not presenting you a big jumble of meaningless facts for you to memorize. Mathematical theory is something that is supposed to make sense, something that you can explain logically step-by-step and could in theory think up yourself given enough time (because somebody did have to invent this shit for the first time at some point in history).

>>10790776
Yes. You have no idea how hard I struggled with abstract algebra, before completely dropping it and coming back three years later.
I started at fourteen, tho.
>>10790858
>at least m
No, lad. It's also an "at most m" due to ACC.

>>10790884
Ok, then how do I learn to learn the subject the better way?

>>10790776
>discrete math
https://courses.csail.mit.edu/6.042/spring17/mcs.pdf

Do induction exercises, discrete math for computer science are just induction over structures.

>>10787302
what

>>10787459
>The trick is not to get stuck with mathematics and philosophy (or whatever subject that is) and call it done.
math is everything though, the philosophy is negligible

>My heuristic is to be comfortable with the idea of being uncomfortable.
>For a child, everything is different, everything requires an effort of understanding, but he does not get uncomfortable,
he sees it as an adventure.
this is admirable and in the spirit of discovery but judging from your posts it doesnt seem like you do any real math or physics at all, sure there were people like feynman who created their own notation but they still worked within the traditional system until they were advanced enough to expand on it.

>>10790858
>how do we know that the finite extension that they mention doesn't introduce any algebraically independent elements?
Because then it would not be finite

>>10790933
>math is everything though, the philosophy is negligible
sounds like a philosophical stand

Is there a theory of nested functions or is it just analysis.

Something that could say concrete things, for example, about the series of functions in the attached image.

>>10779119
Try a pattern inventory by Ploya Enumeration formula where you pick 7 colors (1 for each Child and one for empty SEAT) IF children are assumed to be distinguishable, otherwise 2 colors (empty or not). And pick the symmetry Group to be a cyclic Group of 10. InvestigAte the appropriate term in the pattern inventory (one with a factors represemting each Child color in the first power and a factor represemting empty color in the 4th power in the first case, in the other case one with factor of an empty SEAT in the 4th power and full SEAT in 6th power}

>>10791039
I've never heard of a term for it, but unironically, there might be some niche papers in computer science related to compiler optimizations. Maybe related to optimizing recursive functions or something like that.

>>10790933
>>10790991
>the philosophy
Refer to >>>/lit/.

>>10791051
That seems unlikely to be quite frank.

>>10791051
>computer science
>compiler optimizations
>optimizing recursive functions
see >>10787266

>>10791039
Convergence of functions is a classic topic of analysis, I wouldn't know of any other relevant disciplines, it may depend on the question though...

>>10791039
It's the limit along the operator [math]T(f)=\sqrt{x+ \sqrt{f}}[/math], so I'd guess some functional analysis and dynamical systems, besides what other anons already mentioned.

>not doing research in computational arithmetic geometry
how does it feel to be so inferior to me? dummies

>>10791182
What is your research about?

>>10791059
>/lit/
disgusting

>>10791185
Broadly speaking, computing zeta functions for certain arithmetical objects in char p (not going to say much to avoid getting doxxed)
Also, during my PhD I did some things in arithmetic statistics

>>10791039
As another example, here's
[math]
\begin{align}
f_1(x) &= \cos(x) \\
f_{n+1}(x) &= \cos(f_{n}(x) + \cos(f_{n}(x)))
\end{align}
[/math]

It seems to converge to a constant function with a value of around 1.25965

Sorry: cos(1.25965)

>>10791244
If it converges against a constant function f(x) = c then
c = cos(c + cos(c))

Consider the function
g(x) = cos(x + cos(x)) - x
It's root should be equal to c. You can get arbitary good approximations of it with Newton's method.

Anyone has a copy of this book?
https://bookstore.ams.org/amstext-35/506
I've already searched in libgen but I can't find it

>>10791594
>arithmetic geometry at the junior/senior level
What the fuck.
Is this what Mochi meant with theory of heights at the late undergraduate level?

>>10791613
There's other book called an invitation to arithmetic geometry and basically you only need a course in commutative algebra (Matsomura/ AM) and a bit of AG its nice but not mandatory

>>10791626
Just learn AG properly and go for Bombieri's book on heights.

>>10791651
I was trying to get my hands on some Arithmetic Goemetry because in my uni there isn't going to be an algebraic geometry course until August 2020 (idk why it didn't open up this year :/) and I don't think that I'm that motivated to self study it

>>10778812
I'll be a freshman in college next year. This past year, I took AP Calc AB, but self-studied for the BC exam and got a 5. Should I skip Calc I and Calc II, or just Calc I?

>>10791892
depends
is there an honors sequence for calc 2?
would you take multivariable calculus next?
does your school have a tougher calc 2 or a less tough one?
how confident do you feel about stuff like series convergence/divergence, taylor series, and various methods of integration?
if you are feeling good and there's no honors sequence, just skip it for multivar.
is there a "shopping period" during which you can add/drop classes for the first week or so? you should check out multivar and see if it feels okay.

>>10791899
>is there an honors sequence for calc 2?
Not for calc 2, but there is one for multivariable (the honors sequence is called calc 3 and 4).
>would you take multivariable calculus next?
Either that or ordinary differential equations and linear algebra.
>does your school have a tougher calc 2 or a less tough one?
I think it's supposed to be fairly tough, it's a T20 school.
>how confident do you feel about stuff like series convergence/divergence, taylor series, and various methods of integration?
At this point I feel mediocre, but my plan is to review over the summer considerably.
>is there a "shopping period" during which you can add/drop classes for the first week or so?
There is! I'll probably try a harder class and drop down if I'm in over my head.

>>10791039
>Is there a theory of nested functions
Yes, this is actually part of the study of dynamical systems. Basically you iterate some initial distribution under some dynamics and see what happens. That whole business with the mandelbrot set is actually a manifestation of this
https://en.wikipedia.org/wiki/Mandelbrot_set
It's actually really sad people just talk about it for "the cool fractal bro" and not the actually interesting properties and it's relation to some pretty cool mathematics. Check out Milnor's book on complex dynamics

>>10791039
funny, I stumbled upon this a while ago: https://en.wikipedia.org/wiki/Infinite_compositions_of_analytic_functions

it's especially beautiful within the context of complex functions as >>10792145 already pointed out

>>10778817
After getting feet wet with Topology, where should I go from there? Algebraic Topology? Differential Topology? Topological Topology?

>>10792331
IMO Algebraic Topology is way more interesting than Differential Topology

>>10791244
I've been doing the same with iterating sine with itself, as in sin(sin(...sin(x)...)). At first it still looks sinusoidal but eventually I think it converges to zero, but not uniformly.

>>10792567
It does converge to 0, and the convergence is uniform. There is actually a fairly simple explanation for this.
Denote by [math]f_n[/math] the composition [math]\sin \circ \dots \circ \sin[/math] with n factors.
First of all, for each fixed real x, we have [math]|f_{n+1}(x)| = |\sin f_n(x)| = \sin |f_n(x)| \le |f_n(x)|[/math], so the sequence [math]|f_n(x)|[/math] is nonnegative nonincreasing and therefore converges to the unique fixed point of sin, ie. 0. Hence the sequence converges pointwise.
Second of all, you have to note that, for each [math]n \ge 1[/math], [math]0 \le |f_n| \le 1[/math] and [math]\sin [/math] is increasing on [math][0,1][/math], hence [math]\sup_{x \in \mathbb R}|f_{n+1}(x)| = \sin \left(\sup_{x \in \mathbb R}|f_n(x)|\right)[/math] and, by induction, [math]\sup_{x \in \mathbb R} |f_n(x)| \le f_{n-1}(1)[/math], which converges to 0.
In general, these questions can get very tricky. As >>10792145 points out, it is a whole field of mathematics in itself with ties to real and complex analysis, number theory, probability and all sorts of cool and strange phenomena.

>>10791922
Sounds like you have a fine plan then. Don't worry so much about what advisors / friends tell you, remember that these people have no fucking clue how to do math. Best thing you can do is to always go with your gut.

>>10792331
Do you like groups and shit? Do you want to understand how to discern holes in spaces? Try algebraic topology.
Do you like analysis and shit? Do you want to understand how manifolds work? Try differential topology.
They are both great to learn.
Alternatively, you could go through some measure theory and functional analysis if you enjoyed your real analysis and linear algebra.

>>10792331
There is no fine distinction between all these fields. Most topologists study objects that happen to be smooth manifolds. You can study them using algebraic tools and differential tools.
There is no reason to learn one and not the other.
You do what you find interesting, but keep that in mind

>>10778812
Did I get Euclid's Axioms right? I hate the way they are worded.

First Axiom: If a = c and b = c, then a = b
Second Axiom: If a = b, then a + a = b + b = c
Third Axiom: If a = b, then a - a = b - b = c
Fourth Axiom: If a < b and a > b are false, then a = b
Fifth Axiom: ∞ > a

>>10792824
euclid was a hack

>>10792797
>most topologists study objects that happen to be smooth manifolds
not real topologists

>>10790923
>>10787637
>>10787683
>>10787477
>>10787469
>>10787461
>>10787432
>>10787334
>>10787333
Haha, this is typical cope of people who don't actually understand a subject on a visual level, they can only push around formulas. Real mathematicians only need intuition (this is how Gauss, Einstein, Tesla did it).
Back on topic.
Topology: You have a multidimensional array (also called vector) which makes makes a space called a vector space. Now a set of vectors makes an n-dimensional object (sometimes called manifold). Topology takes a three dimensional object (i.e. all objects in physics) and maps it into a one dimensional object by deforming it (like rubber) to a one dimensional point (called an open set).
We can now conceptualize those open set one dimensional objects. This reduces the complexity of the problems tremendously, which can be useful.
Now deforming this object without lowering its dimension is the subject of 'Category Theory'.
Because we want to be able to do calculus on those objects, the deformations need to be integrated.
Hence the category theorist would say: Morphisms in the category of topological spaces are the integrable maps.

>>10793153
>Tesla
LOL

>>10793153

boi

>>10793153
A good troll would wait until the beginning of the next thread for such a reply, just to let you know...

>>10793153
if you make it this obvious that you are trolling then it isn't funny

>>10793153
>Real mathematicians
>Einstein, Tesla
stop being popsci

>>10792761
>Do you want to understand how to discern holes in spaces? Try algebraic topology
End this meme. (Co)homology is for proving theorems, not counting "n-dimensional holes" like every first year grad student is led to believe.

>>10794495
>(Co)homology is for proving theorems, not counting "n-dimensional holes" like every first year grad student is led to believe.
It isn't "for" counting holes, but that's what it fucking does. To an extent, depending on how pathological the space is, but still.

>>10793153
i fucking EXPLAINED TOPOLOGY TO YOU YOU FUCKING BITCH!! READ IT! FUUUUCKING REEEEAAAD ITTTT!!!!!
>>10788054

>>10794495
how do you fucking tell someone that they should learn algebraic geometry
>ohhh look at all those theorems you can prove with this EPIC topological invariant!!!!
you have to be autistic to think this way
when i tell someone what algebraic topology is, i start with "hey you know point set topology but can you look at a topological space and tell me how many holes are in it?"
and the answer will be "uhhh no"
and i can go "well here you fucking go"

How do I email a professor to ask about research opportunities without sounding like a Sheldor?

I am debating whether or not to namedrop topics which I have studied in the past

>>10794677
>namedrop topics which I have studied in the past
Cringe tbqhwyf. If you insist on showing off, just read his research and comment on it.

>>10794677
you will sound like a sheldor no matter what. go fucking talk to your professor in real life you autist.

>>10794645
>It isn't "for"
Refer to >>>/lit/.
>>10794495
>is for
Refer to the above message.