/mg/ maths general: refereeing via blog comments edition

[Anonymous No.11539096]

talk maths, formerly >>11532134

https://www.math.columbia.edu/~woit/wordpress/?p=11709

[Anonymous No.11539134]

YOU WILL NEVER DEFEAT MOCHIZUKI

[Anonymous No.11539144]

>>11539096
Should i go into analytical number theory or information theory. The latter will give me lots of money but at the expense of being a 'real mathematician'.

[Anonymous No.11539557]

Threadly reminder to work with physicists.

[Anonymous No.11539633]

Question about partially ordered spaces.

Consider some locally convex topological space, with a pointed closed cone [math]K[/math] which defines the partial order as [math]x \leq y \iff y - x \in K[/math]. Consider some choice of points [math]x,y[/math]. When does there exist some [math]\lambda \in \mathbb{R}_+[/math] such that [math]x \leq \lambda y[/math]?

What I know is that if [math]y \in \operatorname{int}(K)[/math], then such a choice of [math]\lambda[/math] exists for any [math]x[/math]. But if [math]x[/math] is given, what are the necessary and sufficient conditions?

[Anonymous No.11539671]

>>11539096
Holy fucking shit, it was my time to create the thread, people aren't respecting the fucking order

[Anonymous No.11539689]

Should I study math olympiad problems books even though I'm already in college and learning analysis and abstract algebra? I just feel a little like a brainlet that I can't solve IMO problems, but I'm already on college, learning higher mathematics, idk if it would me useful to waste time on IMO problems, what do you think?

[Anonymous No.11539712]

>>11539689
I'm confused why she doesn't simply fly to get the apples.

[Anonymous No.11539954]

>>11539689
This is like asking "should I do Sudoku puzzles"?
If you enjoy the puzzles, then do them after you finish your real work. IMO problems are just a different kind of specialized puzzle, and studying them really only makes you better at solving more IMO problems.

> I just feel a little like a brainlet that I can't solve IMO problems
Again, this isn't really any different than feeling stupid because you can't solve XXXL big dick sudokus in 30 seconds like the world champions do.

[Anonymous No.11540057]

>>11539954
So it wouldn't be useful at all to me in college if I strenghten my IMO-problem-solving? If it isn't then I won't really bother.

[Anonymous No.11540074]

Has anybody published an in-depth checklist of every topic in elementary algebra, geometry, trigonometry and precalculus? I'm finishing linear algebra and I want to make sure I haven't missed anything before calculus.

[Anonymous No.11540094]

>>11540074
Did you make sure to learn symplectic linear algebra and the classification of quadrics?

[Anonymous No.11540113]

>>11540074
Also, did you make sure to learn about Pfaffians?

[Anonymous No.11540304]

>>11540057
>So it wouldn't be useful at all to me in college if I strenghten my IMO-problem-solving?
They're tangentially related skills. It might come in handy like once or twice a semester, and the soft skills you build from it like being able to persevere on a brutal problem for a few hours straight are always transferable.
It won't help that much. If your goal is to be better at college math, your time is substantially better spent just studying college math.

[Anonymous No.11540322]

>>11540094
I ONLY SEE FOUR DIFFERENCES WAT DO

[Anonymous No.11540358]

I'm trying to prove that lim x->0 (abs(x)^2)/x = 0. If I'm using sequences can I just simplify the sequences to x? Does it need additional justification? (analysis course)

[Anonymous No.11540360]

>>11540322
get trolled nerd

[Anonymous No.11540361]

>>11539144
>number theorist
>real mathematician
if these are your options go information

[Anonymous No.11540365]

>>11540304
I see, guess I'll just focus on college math then, and read something like 'Problem Solving Strategies' by Engels in my free time, it looks like a good book to start tackling IMO-level problems.

[Anonymous No.11540379]

>>11540322
On left image:
Shrimp. Egg. Mask. Middle lock of hair is parted. End of middle back slat of wood visible on right side of box.

[Anonymous No.11540382]

>>11540358
Yes, sure you can.

[Anonymous No.11540404]

>>11540358
>Does it need additional justification?
Normally I would say no, because it's pretty clear, but if this is going to be graded in an analysis class I would be safe and not skip the step. It's only one line.
One of the unspoken purposes of analysis courses is to be extremely, excessively anal about details. Profs will tend to dock you for things they will slide in a year or two, because this is where you're supposed to learn "rigor".

[Anonymous No.11540578]

>try to solve pde
>get unique surface, nice
>substitute back into equation
>doesn't work
>wtf. try to find error in derivation, fail
>check that your surface really isn't validated by the pde via cas
>check derivation again, compare with notes, examples
>nothing

any advice

[Anonymous No.11540579]

>>11540578
stop doing PDEs

[Anonymous No.11540666]

>>11540578
You guys think Frank survived the escape?

[Anonymous No.11540669]

>>11540578
Did you double check your boundary conditions?

[Anonymous No.11540670]

How do I prove this statement using the lemma? I feel like it should be obvious but I'm not sure how I can apply it

[Anonymous No.11540671]

>>11540670
subtract

[Anonymous No.11540672]

>>11540670
I forgot to say I want to prove a <= b

[Anonymous No.11540682]

>>11540670
[math]d_n = b_n - a_n \geq 0 [/math]

[Anonymous No.11540727]

Brainlet question. For formal limits, there's no problem if [math]|x_n|\leq\epsilon\Rightarrow x_n\rightarrow 0[/math], right?

[Anonymous No.11540750]

Hey guys, I recently found out Marx wrote a text on derivatives. I don't really expect anyone to do,you should probably take a look for mathematical curiosity, but can explain what his actual argument for his version of differentiation (he's arguing against Leibniz and Lagrange's derivations and saying they're a good start but not logically rigorous and presents his own method). I can't understand the proof, so I be interested to hear a mathematician's take on it.

https://www.researchgate.net/publication/238860293_Karl_Marx_and_the_foundations_of_differential_calculus

Its not a very long text, but the proof begins on part III.

[Anonymous No.11540870]

>>11540727
If you mean that for any ε > 0 we can find m for which |xn| =< ε for any n >= m, then yes.

[Anonymous No.11540975]

https://www.youtube.com/watch?v=MzB0-fA2604

[Anonymous No.11540986]

>>11540750
>Links to some obscure walled garden website
Fuck you for making me hunt down that paper.

But. Marx correctly rejects infinitesimals, although the reasons given seem to be ideological, which makes his arguments sound much closer to that of the typical 0.999... =\= 1 poster.

>the proof begins on part III.
There is no "proof" as I can see. Different techniques for calculating derivatives are presented and Marx rejects infinitesimals, although it seems he makes a total strawman argument or simply didn't understand it.
He then proceeds with an algebraic calculation to find the derivative, a result most definitely known at that time, which demonstrates that you can do derivatives without infinitesimals, which turned out to the better view.


In general nothing in there seems new or particularly relevant. Cauchy, without a doubt a much better mathematician (and more sane politically), made similar observations, and his methods to do analysis is what ultimately ended infinitesimals and brought forth modern analysis.
Compared to that, Marx's writing seems amateurish and out of touch with the actual mathematics at the time.

[Anonymous No.11541001]

>>11540986
>Compared to that, Marx's writing seems amateurish and out of touch with the actual mathematics at the time.
Well he was, this wasn't something widely known in Britain at the time.

[Anonymous No.11541216]

>>11540986
>cauchy
>more sane politically
lol. he was literally a monarchist. pretty much nobody who has read marx thinks he was a bad thinker, even if they don't agree with him. stop forming your politics on image boards my man.

[Anonymous No.11541223]

>>11541216
>lol. he was literally a monarchist
yes

[Anonymous No.11541341]

>>11539689
IMO problems are really hard if that's your first experience with math olympiad problems
the problems are usually divided into four groups:
1) geometry (2d or 3d, circles triangles etc)
2) combinatorics
3) number theory
4) algebra (equations, inequalities involving real numbers, polynomials)
the first one - useless for anything you will do in college
combinatorics, number theory - depends whether you are interested in those

I think calling olympiad problems "useless sudoku" is too harsh, they CAN teach you
- general problem solving skills: coming up with your own lemmas while solving a problem
- a lot of algebraic tricks, noticing that some expression factors, this kind of stuff
but generally I agree that you don't need to do them, just do them if you feel interested

[Anonymous No.11541393]

>>11539096
holy fuck why does Scholze have to suffer wasting time arguing with such mongoloids

[Anonymous No.11541448]

>>11541393
To be fair, Mochizuki has made brilliant contributions to mathematics in the past.

[Anonymous No.11541492]

>>11541448
I think the poster you replied to was referring to Dupuy

[Anonymous No.11541839]

>>11541341
What are some good books you can recommend then?

[Anonymous No.11542024]

>>11539096
>grown men arguing in the comments of a literal who blogger
the absolute state of PURE MATH

[Anonymous No.11542056]

>>11539633

go eat a dick stupid bitch

[Anonymous No.11542076]

>>11539096
Probably in the wrong place here but how do I make my TI-89 convert a decimal into a fraction without constantly changing the mode.

[Anonymous No.11542135]

>>11541839
I haven't learned this from books, so I can't recommend one. But I looked at the Engel's book you mentioned previously and it seems alright. The most important thing is that you don't pick something too hard.
You can try picking any national olympiad and starting from there, both the problems and solutions are available online. For example, here are USAMO problems with solutions:
https://artofproblemsolving.com/wiki/index.php/USAMO_Problems_and_Solutions
If a problem sheet has, say, 5 problems, then usually they will be sorted in the order of increasing difficulty. Small countries obviously have simpler olympiads. The competitions get harder and harder and more autistic every year, so if you're going to do them, start with the oldest ones. From 2000's onwards, USAMO are not much easier than IMO. For comparison, if you look at those problems from the first usamo edition ever, you will probably be able to solve at least one: https://artofproblemsolving.com/wiki/index.php/1972_USAMO_Problems
If you're studying at a university, you should be able to understand most solutions. Except for geometry (which I discourage you to do unless you enjoy it) where it is often assumed you know a shitton of lemmas and stuff.

[Anonymous No.11542326]

Should mathematicians be forced into celibacy just like priests?

[Anonymous No.11542340]

>>11539689
I'd say study them on the side when you have time. They can be fun to try, and their difficulty shouldn't get you too down if you can't solve them. Like other anons have said, they teach you lots of little techniques you don't often have a chance to learn elsewhere, and they make you into an overall stronger problem-solver for when you advance to your own field of study.

[Anonymous No.11542457]

>>11539096
the worst shitshow in mathematics

[Anonymous No.11542490]

>>11542457
but a very enjoyable shitshow

[Anonymous No.11542511]

>>11542326
Yes. It should be mandatory that every mathematician have a few gallons of their sperm frozen to make the next generation. After that they'll be put into compulsive celibacy and guarded house arrest, limited access to internet resources, no outside visitors besides their collaborators for short periods, classical music played for 30 minutes at noon. I guarantee you that every university's research output would skyrocket in a matter of weeks.

[Anonymous No.11542592]

How can changes of variables/Diffeomorphisms look when I require [math] |det(D \phi)| = 1[/math]? Linear transformations in [math]O(n)[/math] obviously work... but what about more complex stuff?

[Anonymous No.11542608]

>>11541839
>>11542135
I'd say Engels' book is probably fine for a uni student. Zeitz is my personal favourite.
Many of the resources intended for high schoolers, like the AoPS books, are probably "beneath" somebody in year 2+ of a math program.
Really as long as you stay away from Andreescu's autism fests I can't think of any competition-prep book that would be prohibitive for a decent uni student to work through.

Just grinding problems is not the worst idea, but books are important sources, especially if you aren't going to a prep class/group; most books are intended to clearly show you new techniques and give you enough practice to really be comfortable with them, whereas real contests don't care about the first and are structured exactly the opposite way of the second (a contest where 5 problems in a row used variations of the same trick would be a very bad contest).

[Anonymous No.11542650]

Brainlet here, best books on applied math?

[Anonymous No.11542680]

>>11542592
For example, start with any differentiable function f:R->R, then define g:R^2 -> R^2 by g(x, y) = (x, y + f(x)), then |det Dg|=1.

[Anonymous No.11542683]

>>11542650
>applied

[Anonymous No.11542694]

>>11542650
Any pure maths book related to the stuff. Anyone can swing a sword, but one can be a warrior only by understanding the weapon.

[Anonymous No.11542704]

>>11542694
How can one 'understand' a weapon? It's a non-sentient object devoided of any complex mechanisms or rules.

[Anonymous No.11542709]

>>11542704
Oh, how little you know...

[Anonymous No.11542727]

>>11542704
It is a tool the use of which one can master. It is possible to just blindly flail a sword and get a few things done. The same way one can use mathematical results to get some immediate applications. To go further, be it a longer period of using the sword or deeper applications, one is in trouble without mastering the art.

[Anonymous No.11542752]

dyscalculia fag here, I was given a task to practice mathematics, very easy for everyone else but I am struggling much with it, so the sum is 786 : 3, the answer is 262, but I need to get to the answer using the long division technique, so
3 /786\
3 fits 2 times into 7, but how do I get to the 6 now? It is really confusing for me.

[Anonymous No.11542762]

>>11542727
You must understand the techniques then, not the actual material weapon as if it's a living being

[Anonymous No.11542788]

>>11542608
Why exactly is Andreescu considered autistic? Give me a quick rundown.

[Anonymous No.11542797]

Is this a typo or something? Should it be linearly independent over F?

The [math]\tau_j(\alpha_i)\in K[/math] and the dimension of [math]K[/math] over itself is 1, so any 2 elements are linearly dependent.

[Anonymous No.11542857]

>>11542752
7 = 2*3 + 1. Then you drop the 8 down and get 18 = 6*3.

[Anonymous No.11542873]

>>11542797
The first K should be F.

[Anonymous No.11542902]

>>11542788
Andreescu is the real-life embodiment of the stereotype that competitive math is just stupid trick questions.
He really, really likes clever asspull solutions that give no mathematical understanding beyond
>because this is situation Y, we apply trick X. We guessed to try trick X because this is situation Y and that's where it works.

[Anonymous No.11542942]

>>11542902
that's a really deep approach though when you think about it, there doesn't really have to be a reason other than experience why a particular tool out of one's toolbox is more or less suited to a particular problem

[Anonymous No.11542955]

>>11542902
But that's math in a nutshell my nigga

[Anonymous No.11542979]

>>11542592
Recall that on a Riemannian/Kahler manifold [math]M[/math], diffeomorphisms [math]\phi\in \operatorname{Diff}(M)[/math] induce an isomorphism [math]U_\phi \in \operatorname{Aut}(TM)[/math] on the tangent bundle, which acts on the metric tensor [math]g \rightarrow U^*_\phi gU_\phi[/math] by the adjoint representation. The unimodularity requirement [math]|\operatorname{det}d\phi|=1[/math] is then equivalent to unitarity/orthogonality [math]U_\phi \in \operatorname{Isom}(TM)[/math].
Suppose, to first order, we take [math]\phi(x) \sim 1 + f(x)[/math] and [math]U_\phi \sim 1 - \epsilon H_\phi[/math] for some self-adjoint [math]H_\phi[/math], we obtain the constraint [math](d f)_S = \epsilon[H_\phi,g][/math] where [math]S[/math] denotes the symmetric part. This is the stationarity equation for the energy functional [eqn]E[f] = \frac{1}{2}\int_M d(\operatorname{vol}M) |df -\epsilon [H,g]|^2,\quad \text{or} \quad S[\phi] = \frac{1}{2}\int_M d(\operatorname{vol}M) |d\phi - \operatorname{Ad}_U(g)|^2,[/eqn] hence we conclude that [math]\phi[/math] induces a unimodular [math]U_\phi \in \operatorname{Isom}(TM)[/math] iff it is generated by [math]f[/math] such that [math]\delta E[f] = 0[/math], or [math]\delta S[\phi]=0[/math].

[Anonymous No.11542996]

>>11539144
I think information theory would be cool. You still would get to do a lot of fun mathematics but also have some ability to apply what you know in Industry. I mean, you can kind of do that with any math but it would probably a little harder with analytic number theory off the bat.

[Anonymous No.11543063]

yukariposter, do you think [math]0.999... = 1[/math] is true?

[Anonymous No.11543082]

>>11543063
>>11543063
What we can say for certain is that he think to address the question you first ought to learn the rational homotopy theory of p-adic orbifolds

[Anonymous No.11543085]

>>11543063
What we can say for certain is that he thinks that to address the question you first ought to learn the rational homotopy theory of p-adic orbifolds.

[Anonymous No.11543092]

Dupuy responded plus memes
https://www.math.columbia.edu/~woit/wordpress/?p=11709#comment-235955

[Anonymous No.11543093]

if you're not yukariposter, don't respond to me

[Anonymous No.11543100]

>>11543085
>he

[Anonymous No.11543112]

>>11543093
Watch me.

[Anonymous No.11543122]

>>11543093
Dude, that's exactly what he wants, idiots masturbating his ego everytime in this thread, you're being used like an idiot by him and you don't even realize it

[Anonymous No.11543123]

>>11543092
literally who

[Anonymous No.11543125]

>>11543123
some jabroni number theorist trying to make a name for himself by mooching off all the Mochizuki publicity

[Anonymous No.11543130]

>>11543122
shut up, Iove him

[Anonymous No.11543147]

>>11543122
AAAAAAAAAAAAA AN ANON JUST REPLIED THAT HE CAN'T UNDERSTAND MY POST BECAUSE HE'S NOT AS SMART AS ME I'M GONNA COOOOOOOOM

[Anonymous No.11543151]

>>11543147
s-source?

[Anonymous No.11543162]

>>11539557

>>11541393
If you're referring to Dupuy, I agree.

[Anonymous No.11543166]

>>11543151
https://asmhentai.com/gallery/220535/14/
I don't remember whether you could get banned for this sort of link.
Ah, such is life.

[Anonymous No.11543192]

>>11539096

This is a really great book to read if you've never touched type theory before because it challenges your understanding of math.

Something that I think few mathematicians realize is that ZF set theory is obsolete as a foundation for mathematics. Although ZF set theory can not be replaced by Martin-Lof type theory because it's not as useful for many practical purposes, TT is much more theoretically appealing.

The main advantage is that proofs and theorems are actually objects within martin-lof type theory. You were taught "In set theory everything is a set," but if you ask "what is a proof?" the answer is "not a set." Type theory solves this problem. Proofs and theorems are objects in type theory.

[Anonymous No.11543212]

>>11543192
>foundations

[Anonymous No.11543232]

>>11539633
No bruh, there's no such λ.

[Anonymous No.11543276]

>>11543122
>he
>his
>him
I'm not a "he".

[Anonymous No.11543350]

yukariposter please love me

[Anonymous No.11543359]

>>11543166
Thank you. Any other recommendation while you’re here?

[Anonymous No.11543375]

>>11543276
ok gender non queer binary trans alphabetsexual

[Anonymous No.11543379]

>>11543192
Thanks, I've been trying to get into that stuff recently, got any more recommendations?

[Anonymous No.11543403]

>>11543359
https://asmhentai.com/g/243190/

[Anonymous No.11543413]

>>11543403
Tohou-poster, I'm honestly disappointed that you read hentai, liking tohou and waifus is ok, but hentai? Really? That's below you...

[Anonymous No.11543419]

>>11543413
Nothing is below the human spirit

[Anonymous No.11543441]

Holy shit bros, I love maths so much, but there is simply too many fucking things to learn and besides that there's other fields of knowledge like philosophy, physics, medicine, psychology, computation, music, literature, holy shit, how can I learn all that in my lifetime?

[Anonymous No.11543476]

>>11541341
>the first one - useless for anything you will do in college
What about IMO-level trigonometry?

[Anonymous No.11543521]

yukariposter please be my husband
I'll do whatever you ask

[Anonymous No.11543537]

>>11543441
if you wanna learn to the point of research, it is impossible.
just learning a subfield textbooks can be done. 4 years of undergrad plus 2 years of master and you know a lot about physics or music in a subfield. add 2 more years for another master in another subfield. etc.
you can probably cut all that time in half if you cut some crap. assuming you are a neet with the whole day available.

[Anonymous No.11543543]

If someone gives a counterexample to the Riemann hypothesis and then follows it up with strong regularity results on the zeta function's zeroes, i.e. "it's the ones Riemann thought plus these other three", will number theorists literally spend years looking through old papers that assumed RH and trying to deform the proofs into new results?

[Anonymous No.11543557]

>>11543537
I just wanna do research in mathematics, maybe physics, the others are more like hobbies, but I still wanted to learn about them and all, of course It doesn't have to be everything, for instance, in philosophy my main interests are philosophy of mathematics/science and epistemology, in medicine it would be mostly just reading books on neurology, in music I just wanted to learn the violin well, but even then it's pretty tough finding time to do all that, guess I'll just focus mostly on math and physics

[Anonymous No.11543568]

>>11543441
You won't — dedicate your time to study one discipline you really like, and learn about the rest through outreach/pop-sci type articles, videos and books. There's tons of people applying mathematics and statistics to medicine, the social and natural sciences and computing, so you could look into doing something of that sort.
>>11543537
See also: http://matt.might.net/articles/phd-school-in-pictures/

[Anonymous No.11543576]

>>11543543
Most likely, yes. The most chad proof technique has got to be showing that both RH and (not RH) imply your result, because fuck intuitionists: https://en.wikipedia.org/wiki/Riemann_hypothesis#Excluded_middle

[Anonymous No.11543579]

http://superlust.bandcamp.com/album/inter-universal-teichm-ller-theory

[Anonymous No.11543586]

>>11543476
When do you ever use complicated trigonometry in college? I think I could count on one hand the number of trig identities I ever had to remember in my entire undergrad, and they're all the really basic ones.

[Anonymous No.11543595]

>>11543586
>really basic ones.
Like what?

[Anonymous No.11543623]

>>11543595
Pythagoras and the addition formulas.
I thought about it for a bit and I seriously don't remember ever using anything beyond those in uni.

[Anonymous No.11543689]

>>11543375
You're new here.

[Anonymous No.11543691]

>>11543595
>Like what?
Like that sin = opposite over hypotenuse one.
Or like A^2 + B^2 = C^2.

[Anonymous No.11543716]

>>11543192
>You were taught "In set theory everything is a set," but if you ask "what is a proof?" the answer is "not a set."
Isn't a proof just an ordered set of implications?

[Anonymous No.11543849]

>>11542979
ok thx yukarifag. I understand half the words but I think I get it a bit. Sounds like physics. Can you give a physical example of mimimizing this energy functional?

[Anonymous No.11543861]

in b4 Mochizuki comments on Woit's blog

[Anonymous No.11543927]

>>11543543
>prove that RH implies the result
>prove that NOT RH implies the result
>therefore result
The ultimate Chad proof technique

[Anonymous No.11544062]

Redpill me on the Koszul complex, boys and girls.

[Anonymous No.11544113]

>>11542650
>applied math
Any more specific?
When in doubt, some ODE or ODE stuff.

[Anonymous No.11544195]

Does anyone have an updated version of Aluffi, alas 2016+ printing? There aren't any on libgen.

[Anonymous No.11544201]

>>11544195
(depsite there being one that claims otherwise)

[Anonymous No.11544387]

>>11544195
there's an errata sheet for the first edition, just use it. there's no content changes other than typo corrections between printings.

[Anonymous No.11544409]

>>11539096
If I want to learn undergraduate maths on my own after highschool, what texts should I go and read?

[Anonymous No.11544420]

>>11544409

[Anonymous No.11544449]

>>11544420
Ok I know this is a meme, can I get a serious answer?

[Anonymous No.11544451]

>>11544449
>Ok I know this is a meme, can I get a serious answer?
That's our official curriculum.

[Anonymous No.11544476]

>>11544195
https://libgen.is/book/index.php?md5=FB1FD224719A0F0AC4B63E9D75F8328B

There you go, 'reprinted with corrections'

[Anonymous No.11544482]

>>11543379

No, I like that book more than anything else I have read.

>>11543716

something like that.

[Anonymous No.11544514]

Can I post a math question in this thread? Would you help?

[Anonymous No.11544533]

>>11544514
Yes, I would help you. Just shoot.

[Anonymous No.11544537]

>>11544514
Depends. If it's "I need help in my calculus class", then it's better off in /sqt/. If it's a legitimate math question then either thread is fine.

[Anonymous No.11544538]

>>11544514
Depends. If it concerns relatively deep mathematics and I have got a clue about it, yes.
If it's basic stuff /sqt/ is the thread for you.

[Anonymous No.11544542]

>>11544537
>>11544538
Trigonometry. They don't like me at /sqt/, I stole their GET.

[Anonymous No.11544544]

>>11544542
>Trigonometry
fuck off then m8

[Anonymous No.11544625]

>>11544409

[Anonymous No.11544996]

>>11544625
Numerical analysis can be skipped.

[Anonymous No.11545036]

>>11539096
I found an old topology textbook my grandpa had for some reason.
Some of the exercises seem pretty darn hard. Imma post some.

Let [math]\Delta[/math] be the family of all relatively compact subsets of a metric space [math](E,d)[/math].
Define for [math]S,T \in \Delta[/math] :
[eqn]S \sim T \iff \bar{S} = \bar{T}[/eqn]
Prove that [math]\sim[/math] is an equivalence relationship over [math]\Delta[/math].
Name [math]X[/math] the quotient set [math]\Delta / \sim[/math] and define:
[eqn]\rho :X \times X \rightarrow \mathbb{R}[/eqn]
Such that if [math]\xi , \eta \in X[/math] we take [math]S \in \xi , T \in \eta[/math] and so:
[eqn]\rho (\xi , \eta) = d(S,T)[/eqn]
Proof that [math]\rho[/math] is well defined and that [math](X, \rho)[/math] is a metric space.

Can post more.

[Anonymous No.11545049]

>>11545036

> prove ~ is an equivalence relation
> prove p is well defined
> prove (X, p) is a metric space

these do not actually require any creativity at all. Just go through the lists of properties and verify them.

[Anonymous No.11545058]

>>11545049
Hey dude, i just took a random exercise from the book.
Can post a different one if ya like

[Anonymous No.11545107]

>>11545058

why don't you just read the book

[Anonymous No.11545185]

>>11544625
any specific texts?

[Anonymous No.11545196]

>>11545185
>Calculus
Apostol
>Proofs
Chartrand
>Set Theory
Jech
>Logic
Enderton
>Linear Algebra
Axler
>Topology
Munkres
>ODEs
Tannenbaum
>Analysis
Pugh
>Algebra
Dummit and Foote
>Differential Geometry
Do Carmo
>PDEs
Evans
>Complex Analysis
Kodaira
>Smooth Manifolds
Lee

The rest I can't recommend anything on.

[Anonymous No.11545205]

>>11545196
thanks a lot man

[Anonymous No.11545227]

>>11545205
Don't thank me, thank the authors who put a lot of hardwork into those books that you'll probably just steal online.

[Anonymous No.11545252]

>>11545227
kek ye

[Anonymous No.11545326]

>>11544409
>maths on my own after highschool
>>11545196
>>Calculus
>Apostol
into the trash it goes

[Anonymous No.11545337]

>>11545326
what's wrong with that?

i'm high IQ

[Anonymous No.11545349]

>>11545337
apostol is great when you already know some calculus, not good to learn from, specially not on one's own

[Anonymous No.11545433]

>>11545349
>not good to learn from
Wow, that must be one of the biggest brainlets lies ever, Apostol's theory is extremely clear and easily understandable, it's not like Rudin or anything, it's not even an extremely hard book, if you're looking for hard books Sternberg is much harder. Apostol is great for anyone starting calculus who is a little above average in intelligence.

[Anonymous No.11545487]

>>11545227
If the attainment of knowledge is “stealing” then I am gladly a thief.

These people (rightfully) get a cushy salary, often from the taxpayers, or if that isn’t the case, from their students. But for that they shouldn’t be allowed to hide their research from the general public.

[Anonymous No.11545501]

>>11545487
writing textbooks isn't part of their job description though

[Anonymous No.11545516]

>>11545107
I am, I just wanted to share some cool problems with my /mg/ bros.

[Anonymous No.11545610]

>>11545433
>tfw bellow average in intelligence

[Anonymous No.11545651]

I thought the category theory tranny stuff was just a meme, but I've been feeling girly lately.

[Anonymous No.11545670]

>>11545610
By how much? If you have an IQ of 100 you can still accomplish somethings with a lot of hard work.

[Anonymous No.11545671]

>>11545651

[Anonymous No.11545757]

>>11543849
Sure. Consider the flat Euclidean space [math]M=\mathbb{R}^n[/math] with [math]g =\delta[/math], then the tangent bundle is trivial and the minimization condition [math]d\phi = \operatorname{Ad}_U(g) = U^T U = 1[/math] must be satisfied globally. This leads to [math]\phi(x) = Ux + B[/math] for [math]U\in O(n) [/math] and [math]B\in \mathbb{R}^n[/math]; the group of such transformations is the Euclidean group [math]{\bf E}(n)[/math]. This, in other words, shows that the group [math]\operatorname{Isom}\mathbb{R}^n[/math] of unimodular diffeomorphisms of flat Euclidean space is isomorphic to [math]{\bf E}(n)[/math].
My previous post can be generalized to construct energy functionals that characterize other forms of diffeomorphisms. One such class is the conformal class of transforms, which sends [math]g\rightarrow g' = \Lambda g[/math] and the energy functional for this reads [eqn]E[\phi] = \frac{1}{2}\int_M d\operatorname{vol} \operatorname{tr}|df - \lambda g|^2,[/eqn]
where [math]\lambda[/math] generates the conformal factor [math]\Lambda[/math].

[Anonymous No.11545847]

I often hear people say that classical combinatorics is an insanely difficult field.
Could someone point me to contemporary research so I can see what it is about?

[Anonymous No.11545861]

>>11545847
There's a common saying in the IMO training schools:

'if there's a question you can't solve, it will be in combinatorics, don't waste too much time on it because you'll probably not get the right answer'

[Anonymous No.11545898]

We should completely replace ZFC and our foundation of mathematics with a para-consistent-quantum logic based on a fundamental unit (corresponding to the Planck length) and based on the logic of quantum mechanics (which are an inherently para-consistent logic that do not adhere to non-contradiction or law of excluded middle) and from there build up the corresponding logic lattice, and then from there build up a new form of geometry, calculus, etc.

[Anonymous No.11545905]

>>11545898
hun, you don't know what math, foundations or physics are.

[Anonymous No.11545908]

>>11545898
No. Go away.

[Anonymous No.11545929]

>>11545905
how do you figure?
>>11545908
why not?

[Anonymous No.11545931]

>>11545905
Care to explain what's so wrong with what he said then?

[Anonymous No.11545940]

>>11545931
>Planck length
>Pure math

[guess who No.11545966]

https://www.reddit.com/user/NonlinearHamiltonian

[Anonymous No.11545974]

>>11545861
ok, that makes sense.
But it's not research level stuff, if you were to publish a paper, you'd have enough time to double check and not release anything incorrect just because it's hard to really get right..
From what I looked up so far, it seems like topics in graph theory like drawing, coloring and Ramsey-type problems are out there, but it seems like a considerable amount of the research is driven by CS people, which seems unlikely that /sci/ would find based.

[Anonymous No.11545979]

>>11545898
How would this Planck length change anything? No idea what its official notation is, so I'll just call it P. If we replaced 1 with P, we would then have essentially nP instead of n, and so on, where nP means the sum of n copies of P. Or did I misunderstand what you meant? Note that we could then simply fix the system so that nP would become merely n, essentially the same way you can set c=1 in physics.

[Anonymous No.11545980]

>>11545974
>not release anything incorrect just because it's hard to really get right
that's a bold statement, I guess mistakes may happen, but I think you get what I mean.

[Anonymous No.11545990]

>>11545898
>We should completely replace ZFC and our foundation of mathematics with a para-consistent-quantum logic based on a fundamental unit (corresponding to the Planck length) and based on the logic of quantum mechanics (which are an inherently para-consistent logic that do not adhere to non-contradiction or law of excluded middle) and from there build up the corresponding logic lattice, and then from there build up a new form of geometry, calculus, etc.
The universe is modeled (in a graph thereotic fashion) by a metric graph isomorphic to [math] \mathbb{Z}^3 [/math], where every edge has a Planck length

[Anonymous No.11546006]

Comment section of Woit's blogpost is very comfy to follow along

[Anonymous No.11546026]

How's this book fellas?

[Anonymous No.11546030]

>>11546006
has mochizuki arrived yet?

[Anonymous No.11546035]

>>11546026
>How's this book fellas?
Why don't you read it and find out?

[Anonymous No.11546037]

>>11546030
>has mochizuki arrived yet?
Soon...

[Anonymous No.11546038]

>>11546035
Assume it's actually a bad book, then why would I waste my time going through it when someone who read it already could warn me in advance?

[Anonymous No.11546079]

Kek, and here my friends, is what happens when a physishits tries to tackle a math-oriented book. Hilarious.

[Anonymous No.11546106]

>>11546079
Thanks for telling the name of this a books.

[Anonymous No.11546110]

>>11546106
https://www.amazon.com/gp/aw/d/3319309382/ref=tmm_hrd_title_0?ie=UTF8&qid=&sr=

[Anonymous No.11546158]

>>11545196
God, this is such a great list except you put a fucking proofs book
Why?

[Anonymous No.11546162]

>>11546038
outing yourself as new is kind of pathetic

[Anonymous No.11546169]

>>11546110
>12 5 star reviews
>one seething brainlet who thinks it's bad
Seeing this always brings me a smile

[Anonymous No.11546177]

>>11546162
I didn't do it deliberately though

[Anonymous No.11546188]

>>11546177
that's why it's pathetic
in case it's not obvious "why don't you read it and find out?" or "why don't you read it and tell us?" is a long running /mg/ meme

[Anonymous No.11546195]

>>11545990
yee boy that's what I'm talking about

[Anonymous No.11546204]

>>11546026
Other Analysis friendly book
The Real Numbers and Real Analysis - Ethan bloch, as modern.
But Apostol analysis is good.

[Anonymous No.11546208]

>>11546037
ALL HAIL KONIG VON SCHOLZE

[Anonymous No.11546210]

>>11542902
>every problem in math should be solved with set theory

have fun with that

>>11542650

Linear Algebra, Statistics, Calculus, Graph Theory.

>>11539689

Bad place to ask friend- this board is infested with theorists who vomit at the idea of numbers. you

>>11540986
>Marx thought infinitesimals were a "slight of hand"and enjoyed replacing h with 0

wtf I love communism now

[Anonymous No.11546215]

https://youtu.be/jckyLXSWgZM

Proving fun things today, such as

[math] J^3=j^2 J \implies \exp(i\,z\,J/j) = \left(E-(J/j)^2\right) + \cos(z)\,(J/j)^2 + i\,\sin(z)\,J/j [/math]

>>11546006
summary?

>>11545898
I know some have considered it because it allows for then adopting also unresetricted comprehension, but - if you go down that particular road - I don't know what to make of theorems such as
>the class of ordinals doesn't contain itself, but also it's a actually a set and it contains itself

[Anonymous No.11546220]

>>11545898
>probabilities are contradictions

Reading what you just wrote gave me brain AIDS.

[Anonymous No.11546536]

>>11543276
You are. In fact, you are far and wide the most fucking annoying he in this general.

[Anonymous No.11546543]

>>11546536
Terrible post. Stop wasting my time.

[Anonymous No.11546554]

>>11543441
Simple. Are you von Neumann-tier intelligent? No? Then you fucking can't, no matter how hard you try.

[Anonymous No.11546619]

>>11546554
>Then you fucking can't, no matter how hard you try
Then I'll gladly prove you wrong, punk...

[Anonymous No.11546644]

how come 3b1b hasnt caused a revolution in stem courses?
he makes every class by every professor look so damn inadequate.

[Anonymous No.11546657]

>>11544996
I highly recommend against this. Knowing the basics is a fantastic backup plan in case you end up disenchanted with the world of academia (as has happened to me).

>>11545185
- For Numerical Analysis, check Faires' and Burden's Numerical Analysis. There's also Bulirsch, Stoer.
- For Calculus, instead of Apostol you might also want to check out Spivak.
- For Complex Analysis, Ahlfors is the usual recommendation.
- For Commutative Algebra (Abstract Algebra in the image posted previously), Atiyah-MacDonald is pretty good and exercise heavy.
- For Functional Analysis, check out Alt's Linear Functional Analysis (Brezis's book is also good).
- For Multivariable and Vector Calculus I used Marsden-Tromba, it was fine enough.
- For Algebraic Topology is difficult to make good recommendations, because it depends very heavily on what you want to study more specifically; just read Hatcher, and once you start noticing it's shit, go check out Fomenko-Fuchs.
- For Analysis on Manifolds, there's Jost (this book is tough as fuck).
- For Measure Theory, I remember that Folland's Real Analysis was quite fine; I've also heard good things of Cohn's Measure Theory.
- It's not in the image, but I also want to add that Lee's Riemannian Geometry book is quite a nice extension of his Smooth Manifolds book.

Overall, don't worry too much about which book to follow. Pick one, stick to it, and only once you encounter problems do you start checking others. Also, don't pay much attention to the flowchart's precise order. I agree with the choices made in >>11545196.

[Anonymous No.11546672]

>>11546657
>as has happened to me
Why? Tell me more about your story, traveller.

[Anonymous No.11546677]

>>11546543
Not only are you a faggot, you're also a pig. It's why I'm going to call you a piggot.
>>11546619
heh, it's nothin personnel, kid

[Anonymous No.11546680]

>>11546644
That guy is immensely annoying with his whole schtick of "Wow guys, isn't maths le beautiful!". He never mentions how fucking excruciating it is, because that doesn't give clicks from brainlet know-it-alls. He's also irrelevant in the grand scheme of things. Good videos otherwise though.

[Anonymous No.11546683]

>>11546680
>Hey let me make a math video that calls math ugly and shows that it is actually really hard work that surely will give me views and present mathematics to new audiences in a positive light
I mean, really?

[Anonymous No.11546704]

>>11546683
It's immensely dishonest. You can't just show the nice stuff with cute little pi people and live it at that, you're doing a disservice to whoever might be interested in pursuing mathematics as a career. I can only imagine many poor fuckers got completely obliterated in uni.
>>11546672
I wasn't made for this world, simple as.

[Anonymous No.11546724]

>>11546704
>I wasn't made for this world, simple as.
No one was made for anything in particular, we were just put in this world without asking and now we all gotta play with the cards that we've been dealt with, you could have become an academic if you wanted to, simple as that

[Anonymous No.11546730]

>>11546680
It's not excrutiating. The process is fun. He does a better job of presenting the thinking process than numberphile or other channels.
>>11546704
It's not dishonest at all. Do you really have trouble with the computations? Doing analysis and geometry feels a lot like going through the motions of a 3b1b video, I'm drawing all kinds of pictures, moving my hands around a lot, and then backing it up with a proof later. The ideas and creativity is where the work is. All the computations and mess is just relaxing and gives time to get thinking about the next problem.

[Anonymous No.11546763]

>>11546724
I don't want to play, but I feel compelled to by others. I don't want to try to become an academic, either. I doubt I could manage.
>>11546730
>It's not excrutiating. The process is fun. He does a better job of presenting the thinking process than numberphile or other channels.
I'd be surprised if you always found the process of mathematics "fun". Good for you if that's the case, but it's not for either me or a lot of my friends.
>All the computations and mess is just relaxing and gives time to get thinking about the next problem.
Again, not the case for me. I'm willing to accept it's just me the one who has a problem that I'm taking out on the guy, but even then he paints too nice a picture of things.

[Anonymous No.11546786]

>>11543716
>>11544482

Set and type theory are both wrong and parasitic. turing machines are the foundation of math. how do you "make" a set without a turing machine? likewise how do you "make" a type without a turing machine? you can't. computation is the foundation of math. no computation no math. we've been so spoiled by the results of computation we have forgotten how we arrived at these results. computation.

[Anonymous No.11546788]

>>11546763
First, it doesn't matter if you want to play or not, you're already playing.
Second, you do want to play, otherwise you'd just quit, and there's only one way to quit this game, you already know that

Some people were dealt with better cards, others with worse, but we all keep playing regardless and only those who give their true effort into this game, in every single play, no matter how bad their hand is, only they will reach victory.

[Anonymous No.11546792]

Fucckk I want to go to math grad school but I don't want to be homeless when I graduate
Why couldn't I have been Jeff Bezos' son

[Anonymous No.11546800]

>>11546792
You won't, there are lots of universities out there, even if you can't become a professor at harvard, you can at least manage to find a job in a nameless place

[Anonymous No.11546802]

>>11546788
I don't care about winning anymore, and the only reason I "want" to keep playing is for the sake of not making other people's lives worse. Lastly, it does make a big difference whether you want to play or not, I don't understand why you say it doesn't.

[Anonymous No.11546824]

>>11546800
man i hope so

[Anonymous No.11547039]

>>11544409
The minimum to get into grad texts is Rudin and and abstract algebra book. I found Judson online is free, good, and at appropriate difficulty.

If those are too hard, you need some intros to get you prepared like spivak calc. But your goal is to get through those books above.

[Anonymous No.11547042]

>>11544625
functional analysis is a prereq for PDEs.

[Anonymous No.11547071]

>>11546215
>summary
>>Wow. I thought that Fesenko’s criticisms of people who don’t understand Mochizuki were harsh, but now I see that also when it comes to hurling insults, he is just an apprentice of the great IUT master.

[Anonymous No.11547100]

>>11546644
3b1b's videos aren't meant for a course; they're essentially an outreach meant to show people, mathematicians or not, some cool things that appear in math — that's roughly what Grant says about the channel in his public appearances (such as a recent TEDx talk he did). A good course in mathematics helps with some intuition but also goes through the necessary formalities, which often aren't fun as some anons pointed out.

[Anonymous No.11547103]

>>11546786
False. There are problems that are undecidable computationally and yet decidable in some axiomatic systems, e.g. ZFC. Sets and types may be regarded as platonic sort of structures which have nothing to do with computation. To start, see:
https://en.wikipedia.org/wiki/Undecidable_problem
https://en.wikipedia.org/wiki/List_of_undecidable_problems
I could define the set of solutions to some of those particular problems and it would be perfectly sound mathematically, as it won't lead to contradictions.

[Anonymous No.11547124]

>>11545501
>writing textbooks isn't part of their job description
Then change their job description.
I have no issue with them getting paid for their work, but paywalled information always makes me mad.

[Anonymous No.11547128]

>>11546215
>math.uni-Hamburg
WHAT THE FUCK

Are there seriously people I have seen in real life on this board?

[Anonymous No.11547133]

>>11546644
Why would he?
His videos aren’t complete courses, although I am sure they help some students to understand some concepts, they do not really replace anything.

[Anonymous No.11547197]

>>11547042
Same for real analysis and measure theory, although that goes mostly for the non-classical theory, you can do a lot of classical theory which can not use functional analysis. Oddly enough you need very little from ODEs.

[Anonymous No.11547204]

>>11545847
>>11545974
bump

[Anonymous No.11547207]

>>11547128
Are you really surprised? Every department has its token chad, but every department has at least that one extremely awkward mega nerd. I don't think you are either of those, because you are here, but you don't think every single one of the other members is. There is therefore at least one more hamburger posting in these threads.

[Anonymous No.11547217]

>>11547207
>Are you really surprised
Yes.

> Every department has its token chad, but every department has at least that one extremely awkward mega nerd
Well, I know the chad already. And the autistic mega nerd is female. I doubt they are here...

> There is therefore at least one more hamburger posting in these threads.
I would guess there are no more than 200 regulars in these threads, thus it seems kinda unlikely that somebody I knew, or had seen at least, would be posting here.

[Anonymous No.11547297]

I may be trapped in the body of a guy, but I can still do my CT.

[Anonymous No.11547325]

>>11547128
>>11547217
I merely screencapped this short document here
https://www.math.uni-hamburg.de/home/khomskii/ALST/hrafn.pdf
I'm not a student or in Hamburg. Allerdings spreche ich Deutsch. And I always have the feeling there's a lot of Germans here, I'd not be surprised.

Some years ago I was in Stuttgart, posting on /lit,/ and completely randomly added a pic out of the Starbucks window I was sitting there. A Stirner scholar saw it and we ended up meeting for some literary discussions.
That's the best case scenario of what you can get out of these sort of interest focused boards/forums.

>>11546644
They won't replace paperwork and student interaction, so any "revolution" may be of limited scope.
In any case, while I can see the color and position dynamics of equations be helpful, the shit-ton amount of work it would take to produce enough content also for less 101 things.
I played around with his tool for a day some years ago, it's available here
https://github.com/3b1b/manim
Not sure I like it, it would need a "more agile" approach to make something like that a common open source gadgets replacing overhead projectors.

It's work, but it might take a few such people to get enough animations to cover a reasonable amount of topics.
If you know the links to click, you can find that animation sort of guy who made a lot of the gifs on Wikipedia. Here:
https://en.wikipedia.org/wiki/User:LucasVB/Gallery

[Anonymous No.11547437]

>>11546657
What about the Princeton analysis books?
Are they any good?

[Anonymous No.11547454]

>>11546188
haha man youre so cool

[Anonymous No.11547457]

>>11547454
Ow the butthurt.

[Anonymous No.11547458]

>>11547457

[Anonymous No.11547719]

>>11547217
>And the autistic mega nerd is female.
[X] Doubt

[Anonymous No.11547723]

>>11547437
Stein Shakarchi are a classic, hard to go wrong with them.

[Anonymous No.11547859]

>>11545847
please

[Anonymous No.11547916]

>>11545847
Read Keevash's paper on the existence of designs.
I did a REU in combinatorics and it just felt as if the problem was fighting against me. I can't describe it, but whenever I felt as if I had a big idea or found a workaround for some obstacle I felt as if I had advanced 1 mm before facing yet another concrete wall. At the end I straight up told my advisor that I don't expect to see the problem solved in my lifetime.
>>11545974
dunno where you get that idea from, but Ramsey theory is one of the most based areas of maths

[Anonymous No.11547984]

>>11545847
it's because combinatorics makes you think and gives you nontrivial problems right from the start
there are no harder and easier fields at the "top quality research level", it's just how brainlets feel about combinatorics
compare this two experiences:
1) an undergrad is learning algebraic topology. He learns what is a module, a long exact sequence, cycles and boundaries, homology groups, Mayer-Vietoris sequence. In the end of semester, he is asked to prove that the unit balls in R^m and R^n are not homeomorphic if n and m are different. He happily computes the homology groups and the problem is done. He feels smart because he learned many new definitions.
2) an undergrad is learning combinatorics. There are no new objects introduced, just graphs and numbers and vertices and edges, but the proofs are long enough to fuck him up because he can't keep all of them in his memory while reading the proof. In the end of semester, give him a problem which requires a nontrivial application of pigeonhole principle and he will still fail to do it.

[Anonymous No.11548043]

>>11539096
Question about linear combinations.

I want to know if it's possible to use a linear combination of 30, 50 and 80 to create 982880.
I don't know how to approach this though. Since it's an equation with three unknowns, I imagine you'd need three equations in order to pinpoint the constants needed but, based off my brief stint with Linear Algebra in the past, if there's an insufficient number of equations, all that happens is you get a range of answers rather than something specific. So, with that in mind, how do I approach this?

Pic related. I want to find values for x, y and z. Help?

[Anonymous No.11548057]

>>11548043
x=0,y=0, z=12286. Two variables are free, pick two numbers and solve for the last one. Keep trying until you get a triple that works.

[Anonymous No.11548064]

>>11548043
I mean it's clearly possible. With the "50" you can make multiple of 100, and 982880 = 982800 + 80.
This is more of a number theory problem in general, look up Bezout's identity.

[Anonymous No.11548073]

>>11548043
How about taking [math] 30\cdot 3 + 80\cdot (-1) = 10[/math] and multiplying both sides by 98288? Or is there some addition constraint saying that x,y,z have to be non-negative, which you did not tell us about anon?

[Anonymous No.11548082]

>>11548043
x=0, y=0, z=982880/80

[Anonymous No.11548092]

>>11548057
>>11548064
Thanks guys.

>>11548073
Oh yes, x, y and z are absolutely natural numbers (I notice Mathematicians seem to prefer saying "non-negative integers" over "natural numbers". Why would that be?). I didn't think that needed to be said.

[Anonymous No.11548118]

>>11548092
>(I notice Mathematicians seem to prefer saying "non-negative integers" over "natural numbers". Why would that be?
Autism over whether 0 is a natural number

[Anonymous No.11548126]

>>11547916
I did an REU in knot theory. The problem quickly became combinatorial and seemly intractable. Just so many cases.
>>11547984
This is very true.

[Anonymous No.11548151]

>>11548043
Suppose a solution exists, (a,b,c). Given this solution, can you find other values?

If you fix two values, say, a and b, then there is only one possible value for c. ie, no.

Hence, suppose now you fix only one value, a. We can rewrite the equation as 5y+8z=98288-30a. Also note that 5x5-8x3=1. Therefore (*) 5x5x(98288-30a) - 8x3x(98288-30a) = 98288-30a, meaning that (a, 5x(98288-30a), 3x(98288-30a)) is a solution (I multiplied by 10 since i removed that factor earlier). Now notice: we can "add 0" to the left side of equation (*), since we know (obviously), that 5x8 - 8x5 = 0. This way we can find all solutions given a possible "a".

Repeat this for the other solutions, find one solution, then youre done.

[Anonymous No.11548153]

>>11548151
ignore that part about multiplying by 10

[Anonymous No.11548169]

>>11547916
>>11547984
It seems to me like combinatorics is a field of problems that seem easy after they are done with.
Stuff that stays unproven for decades and then someone comes up with an obscure double counting probabilistic proof.
(this obviously doesn't apply to all of combi, ie computer-aided proofs etc)

[Anonymous No.11548185]

How can I differentiate between meme and non meme book recommendations? These books should be good, no?
>1. Lang Basic Mathematics or Axler Precalculus
>2. Apostol or Spivac calculus
>3. Fischer Lineare Algebra and Forster Analysis (standard first semester books in germany afaik)
>1. Chartrand Mathematical Proofs

I know there are some German Anons here. I want to study math this fall. I have been out of highschool for over one year and I would not be able to solve Leistungskurs Abitur Mathematik right now, because I forgot so much stuff.
Then again it seems to me that you learn everything from the ground up in university anyway so should I bother with re-learning all the high school stuff?
Or in other words: What are the necessary prerequisites for a pure math degree in Germany?
Pic related is the stuff you do in Analysis and Linear Algebra 1.

[Anonymous No.11548194]

>>11547454
It's not about being "cool." It's board culture. Lurk for a few months or fuck off.

[Anonymous No.11548195]

>>11548194
muh club

[Anonymous No.11548197]

>>11548185
Yeah sure they're all fine
Listen, most math books aren't all that different.

[Anonymous No.11548201]

>>11548195
>muh
Mathematicians use "our", not "my".

[Anonymous No.11548218]

>>11548185
>Fischer Lineare Algebra and Forster Analysis
This. Don't bother with the English stuff, it will inevitably get confusing.
Get the books online (and by that I do not mean you should buy them) and work through the first few chapters.

>What are the necessary prerequisites for a pure math degree in Germany?
There legitimately are none. You will start from first principles anyway and be very rapidly dragged through them and many will drop out because they can't stand it.
Being prepared means having already understood where it all is going.

>so should I bother with re-learning all the high school stuff?
The issue isn't that you will not have the prerequisites, but that you will get lost in the middle of it. Just *understanding* where all of that "abstract bullshit" is going is invaluable.

Also, which Uni and how is corona treating you? Here all classes will be online, which I imagine will be a hurdle to first semester students.

[Anonymous No.11548235]

>>11548185
A meme book will tend to just be a list of theorems, lemmas, corollaries, propositions, etc, without much explanation for the why's and how's. They aren't necessarily bad, they are meant for being a reference within a lectured course. However, if your prof sucks or you are self studying, it's best to grab a book with more meat to it. The only way to know if a book is good is to start reading it. Skip to the first chapter of new stuff, and read the first couple pages, that should be enough for you to get a feeling of the authors purpose and style.

[Anonymous No.11548267]

>>11548235
>>11548197
Thanks.

>>11548218
Universität Münster. This Sommersemester will be online.
Can you expand on the "*understanding* where it all is going is invaluable" part?

[Anonymous No.11548314]

>>11548169
>It seems to me like combinatorics is a field of problems that seem easy after they are done with. Stuff that stays unproven for decades and then someone comes up with an obscure double counting probabilistic proof.
I don't think I would agree with this. First, it's really not that common that the proof to an old problem is just some galaxy brain elementary argument that everybody else has missed for 30 years. It's very surprising when this happens. It makes news. Just like in "the rest" of math, when something hard is resolved there's usually some serious conceptual content needed, some new way of thinking about the problem that softens it up enough to be doable.
And clever technical proofs aren't usually the kind of things that make a problem seem easy in hindsight anyway. 20 pages of furious manipulations of sums and inequality-bashing rarely gives you the "oh yeah, of course" impression when you're done.

>>11545847
Gil Kalai keeps a fairly active blog mostly talking about news in combinatorics, pretty heavily focused on the old-school kind (by old-school or "classical" I think of Hungarian-style extremal shit, inequalities and probabilities, rather than the more formal algebraic stuff that Americans tend to do). Scrolling through some of the posts should give you a decent snapshot of some stuff that's immediately been happening in the field.

[Anonymous No.11548355]

>>11548267
>Can you expand on the "*understanding* where it all is going is invaluable" part?
The first few lectures will be mostly random collection of things you never heard of previously, sets, relations, equivalence classes.
Nothing will seem like it has anything to do with what you did in Gymnasium all the while you have to figure out how to do proofs and deal with all the abstractions thrown at you.
At least that was basically my experience.

I think what helps in that situation is an understanding of where the things connect. If you already have read a bit into these books, it will make it easier to understand what the point of all that nonsense might be and it will you get a bit more comfortable with the basic mathematical techniques.

[Anonymous No.11548367]

>>11541216
dumb little teenager.

[Anonymous No.11548466]

this is how the rest of 4chan does maths

[Anonymous No.11548475]

>>11548466
Okay?
Where is the mistake?

[Anonymous No.11548484]

>>11548466
That ain't mathematics, that's basic arithmetic, I hate when people talk like math is just mindless counts, no you fucking retards, math is about proving statements, not playing with numbers

[Anonymous No.11548602]

Can someone help me? I really like the location of this uni, but the math major seems weak as fuck. Check it out: https://www.pdx.edu/math/bsba-in-mathematics (also check out the electives requirement link within, where most the real math courses are hiding).

Anyway, they seem to teach abstract algebra in a weird way, breaking it down into group theory and ring and field theory. Can you guys help me come up with the best class list to take? A complete list of classes can be found here: http://pdx.smartcatalogiq.com/2019-2020/Bulletin/Courses/Mth-Mathematical-Sciences, since the vanilla BS is so dull I’m going to get in contact with an advisor this week about adjusting it and having /mg/ guide my choices would be great. My minor is CS, so I’m hoping I can take some extra math classes like graph theory and have count as CS elective.

Is this degree weak as fuck or what? The school does kinda suck I think, but it’s hard to get objective rankings on that. But if I go to school here, I get to live an extremely comfortable life with excess money and leisure, if I go somewhere else I’d have to work a part time McJob and that sounds horrible.

[Anonymous No.11548608]

>>11548602
Also, I’m coming in as a transfer student so all my lower division courses are already done. I’ve self studied a little bit of real analysis and abstract algebra, I just don’t want to exit without having formal grounding in analysis, algebra, topology, combinatorics, probably and statistics, and it looks like this degree will have trouble getting me there.

[Anonymous No.11548616]

>>11548602
OOF

[Anonymous No.11548625]

>>11548475
They only have 4 times as many cases per capita but 9 times as many people.
If they only had 4 times as many cases and 9 times as many people, he would have had a point.

[Anonymous No.11548639]

http://www2.math.technion.ac.il/~room/ps_files/EP_genpos.pdf
>To observe that this bound is tight let P be the set of vertices of a regular n-gon and let R be the set of n points on the line at infinity that correspond to the directions of the edges (and diagonals) of P.
what did he mean by this?
(third paragraph of introduction)

[Anonymous No.11548663]

>>11548616
elaborate pls :( I don't wanna waste my money

[Anonymous No.11548778]

>>11548663
Doesn't matter where you study at, but how much effort you put into your studies. Actually, it might be a good thing that you're going to such a weak college, it will be easier to maintain high grades and you'll still have free time to study the harder subjects on your on.

I say go for it.

[Anonymous No.11548860]

How important is the study of Representation Theory if I want to specialize in complex geometry and QFT?

[Anonymous No.11548897]

>>11548860
its important for qft

[Anonymous No.11548902]

>>11548860
Representation theory is a subfield of physics, anon, aside from geometry.

[Anonymous No.11548904]

>>11548897
What about lie groups?

[Anonymous No.11548909]

>>11548902
Is it a hard subject to learn?

[Anonymous No.11548918]

>>11548235
a meme book is a book for intellectuals yet idiots, all of /mg/ except me.

[Anonymous No.11548928]

>>11548904
Important too

[Anonymous No.11548938]

>>11548902
and QM is a subfield of representation theory

[Anonymous No.11548945]

>>11548909
Not from a jedi.

[Anonymous No.11548984]

>>11548185
People place too much emphasis on book choice. I think it's just because window shopping books is so much goddamn fun. At least that's what makes me spend so much time picking out what to read.
But basically any book is "good enough"; it's very rare that a book is so fucking atrocious that it's just impossible to productively learn from (though they do exist). Two or three generations of mathematicians now learned their analysis from Blue Rudin and most of them suffered no permanent damage.
Even bad books can be good teachers in a perverse way because if the book explanations are useless or simply don't exist you have to figure it all out yourself, and you'll build a solid understanding that way.

[Anonymous No.11548998]

>>11548602
Can you not go to Oregon State at least? I don't think it's that far from Portland and it's a substantially less crappy state school.

[Anonymous No.11549011]

>>11548938
>and QM is a subfield of representation theory
Subcategory*

[Anonymous No.11549015]

>>11548185
>>1. Lang Basic Mathematics or Axler Precalculus
Lang is a meme.

[Anonymous No.11549017]

>>11548860
https://arxiv.org/abs/math-ph/0005032

[Anonymous No.11549018]

>>11548945
Did you ever hear the tragedy of Darth Mochizuki, the wise? It's not a tale the Jedi would tell.

Ironic, isn't it? He could prove others' theories, but not his own.

[Anonymous No.11549021]

>>11548602
not recommending anything in particular but abstract algebra is always split into groups, rings, and fields. that's normal.

[Anonymous No.11549027]

>>11549017
Holy shit, thanks a lot, I love his books

[Anonymous No.11549029]

>>11548355
I see, thanks for the answers.

>>11549015
>>11548984
Alright, ty. Ill just go with the books and get into studying.

[Anonymous No.11549033]

>>11548602
Also, the 400 level classes look perfectly advanced enough.

[Anonymous No.11549043]

>>11549017
Isn't this just a really early version of his representation theory book?
Seems kind of silly to read a preliminary outline when the book is in its 2nd edition.

[Anonymous No.11549048]

>>11549043
Oh damn, is it really? Is that good a book introduction?

[Anonymous No.11549057]

>>11549018
From my point of view Scholze is evil.

[Anonymous No.11549064]

>>11549048
I really like it. I used Part II on Lie algebras to replace Humphreys' book (which is fucking garbage, legit worst math textbook I have ever read) when I took a course that used it.

[Anonymous No.11549075]

>>11549064
I see, thanks, what would be the prerequisites?
>legit worst math textbook
Care to expand more on this?

[Anonymous No.11549085]

>>11547719
I've seen it happen as well. Stop learning everything you know about women from /r9k/, maybe.
>>11548984
>People place too much emphasis on book choice. I think it's just because window shopping books is so much goddamn fun.
I think it's more of a leftover from the old times when you actually had to buy a physical book and stick with it until you had the money to buy another. Nowadays that's irrelevant if you don't mind piracy (and if you own a printer maybe). Then again, Ramanujan supposedly learnt from an old-ass compendium of results that were presented without proof, so it all boils down to how much effort you're willing to invest. That's a hard pill to swallow for most people.

[Anonymous No.11549109]

>>11548860
Very

[Anonymous No.11549125]

Fesenko's text on mochizuki and iut (dated 3 April 2020)
https://www.maths.nottingham.ac.uk/plp/pmzibf/rpp.pdf

[Anonymous No.11549143]

>>11549109
Care to expand?

[Anonymous No.11549147]

thoughts on my proof of 0.99.. = 1?

[Anonymous No.11549167]

>>11549043
Yes, but preliminary outline had linear algebra and group theory introduction.

[Anonymous No.11549187]

>>11549143
Depends on what your background/level is.
One obvious way to get at it is "you want to know what the objects in a theory you cook up could be, whos expression doesn't change w.r.t. some popular transformations."

Maybe this question points you to something?
https://physics.stackexchange.com/questions/19323/groups-acting-on-physics-a-clarification-on-electrons-and-spin/19325

[Anonymous No.11549219]

>>11548118
The only autistic ones are those who insist it isn't.

[Anonymous No.11549242]

>>11548267
Man, I should have gone there to do my Master's and do cool applied shit. Instead I'm in fucking pure maths paradise REEEEEEEEE.

[Anonymous No.11549291]

>>11549147
With people like you no wonder there are idiots who still be .999...!=1.

[Anonymous No.11549293]

>>11549147
It is idiotic to an extent which has never been reached elsewhere.

[Anonymous No.11549294]

>>11549057
That's because you haven't met him. He's a pretty chill dude, especially when he's drunk.

[Anonymous No.11549302]

>>11549064
Worst book I read was Berline-Getzler-Vergne, which is somehow what everyone recommends to learn about Atiyah-Singer index theorem. It was fucking painful.

[Anonymous No.11549311]

>>11549085
>Stop learning everything you know about women from /r9k/, maybe.
No, they're the only ones saying the truth. What's happening here is that women have the opportunity to be way more autistic than men while still being somewhat social and successful, but men don't, so you will never meet all the many men who are worse than her unless you get out of your way to do so. Selection bias, my friend.

[Anonymous No.11549314]

>>11549147
Wait, so the result is correct until someone proves it's not? Based.

[Anonymous No.11549318]

>>11549294
I would imagine not too many of Scholze's drinking buddies are on /mg/. Are you Hungarian by any chance?

[Anonymous No.11549323]

>>11548778
Do you think it still doesn't really matter if I want to go to grad school? Neither school 'good', one is just better than the other and feels real enough.

[Anonymous No.11549325]

>>11549318
I'm not a drinking buddy, I just got to see him wasted. And no, I'm not Hungarian. Who might you be thinking of? And who are you?!?!

[Anonymous No.11549344]

>>11549325
Lol, as if I would reveal my identity on the chan. I'm just an undergrad who happens to know a Hungarian guy who sometimes drinks with Scholze and who's a bit edgy irl, so I thought he might post here.

[Anonymous No.11549352]

>>11549147
How about this? Given any [math]r>0[/math], there is an [math]n\ge1[/math] such that [math]1- 9 \sum\limits_{j=1}^n 10^{-j} < r[/math]. Suppose [math]x=0.999\dots \neq 1[/math], and consider the intervals [math][0, x], [1, 2]\subset [0, 2][/math]. Since [math][0, 2][/math] is compact and Hausdorff, it is also T4, and so there are [math] x < y < z <1[/math] as the intervals [math][0, x], [1, 2][/math] are contained in disjoint open sets [math]U\ni y, V \ni z[/math], respectively. Now [math]0 < z-y < 1- x < 1 - 9\sum\limits_{j=1}^n 10^{-j} < r[/math]. Now, choose [math]r \le z-y[/math] to get a contradiction.

[Anonymous No.11549356]

>>11544476
That's the one I referenced in the first reply; thank you for the try, though.
>>11544387
I know and thatś fantastic, but it's kind of a bother to have to use it alongside the book. Oh well, the second printing already has 50+ reported errors, so I guess I would have to anyway.

[Anonymous No.11549362]

>>11548602
A couple of things, some of the 400 level courses don't seem bad, as long as they cover material at a suitable depth, you should be fine. Also worth mentioning that you are in control of the pace with which you take the courses. If you want, you can take a bunch of those courses simultaneously and then move on to the unis graduate courses. Being at a lower level school also has its advantages, since it means you'll stand out more and have more opportunities to work with profs and pursue your own interests. That being said, it really depends on what you want from the degree.
>>11549302
>Berline-Getzler-Vergne
Really terse and dry, which is weird though, I thought people's go to was Roe.

[Anonymous No.11549368]

What's the future of lecturing? It's undeniable that classes will move online more, and it feels kind of retarded to listen to my mid-grade professor when I'm on an internet connected computer with world class talent ready to lecture me for free. Will schools move toward using pre-made lecture series, and basing courses around that? And maybe the instructor then hosting some Q&A / explatory seminars retouching on the themes? that would be so baste and save professors tons of time.

[Anonymous No.11549394]

Is this right? This class has been extremely hard so far. I'm not sure if the solution is this easy.

[Anonymous No.11549414]

>>11549356
>already has 50+ reported errors
How do you know all that? Also, are those errors big?

[Anonymous No.11549433]

>>11549323
>those remedial math courses
The absolute fucking state.

Well, because I'm feeling nice today I'm going to go through the whole catalogue of math courses and give you my feedback.
First of all, I don't fucking understand why there are 5 Calculus courses, all of that stuff is super basic, but if you're smart and can work on your own (if you aren't, DO NOT GO to this uni) just take the necessary ones and avoid the rest. Still though, keep in mind that, for reference, all of them are mandatory in any European uni, so be careful. Mth 311,312, 313 (probably 324) would also be included, I don't know why it's all separate. Moreover, they have Mth 444, 445 in like a different hundreds code or some shit. This is EXTREMELY fucking basic and important.
Overall, anything below and including the 500s is the most basic shit, which you MUST know to get anywhere so just beeline for 600s courses (even here there are courses which are standard in Europe). The most important ones are the Research courses and such (Mth 301, 401, 405, 407, 501, 601). If this is some sort of research project you do with a professor, then definitely do as many as you can manage and then some more. Considering the shitty uni you're going to (because it is really crappy), this kind of stuff is really important because you'll need recommendation letters to go from there to anywhere nice for grad school.

If you want to know more, tell me, because I'm very willing to help you. This will only take a good couple minutes from me, but many years from you.

[Anonymous No.11549437]

>>11549344
Not that many undergrads in Uni Bonn anon, don't make me try and guess who you might be.

[Anonymous No.11549441]

>>11549352
Mate, you assumed what you want to prove.

[Anonymous No.11549444]

>>11549362
>Really terse and dry, which is weird though, I thought people's go to was Roe.
When I was looking for recommendations, I'm pretty sure BGV was the most recommended. Roe was there too, but honestly my favourite one for the moment are Nicolaescu's notes. They're very comprehensive and clear.

[Anonymous No.11549448]

>>11549394
>he didn't prove that the maximum exists
Never gonna make it.

[Anonymous No.11549451]

>>11549448
>>he didn't prove that the maximum exists
I'm not a "he".

[Anonymous No.11549455]

>>11549451
Prove it UwU.

[Anonymous No.11549469]

>>11549437
Sorry m8, I'm not at Bonn. The Hungarian guy did his PhD and first postdoc in Germany and met Scholze at Bonn, and he's now a prof at my uni and mentioned Scholze over some drinks once.

[Anonymous No.11549495]

am I a faggot if I choose what book to read based on the number of problems?? if I'm doing every problem in the book I'm not going to waste my time with textbooks that give you 30 problems every 5 pages and the first 15 are "prove this by looking at the definition and moving things around"

[Anonymous No.11549549]

>>11549167
Should somebody really be reading a representation theory book in the first place if they need a linear algebra introduction?

[Anonymous No.11549554]

>>11549495
>and the first 15 are "prove this by looking at the definition and moving things around"
Those problems are how you get comfortable actually using the definitions to do things. They only take a few minutes at most to do.

[Anonymous No.11549663]

>>11549549
Yes if they are a physicist.
>>11549495
Who cares, just don't do them. Don't be so autistic.

[Anonymous No.11549701]

>>11548998
Oregon State is the “other” school. I’ve always planned on going there, but I’ve built up a customer base since I’ve lived in Portland, and I’d have to give a lot of that up in the move. I just wish they’d offer the classes online.

I’m still unsure

[Anonymous No.11549719]

>>11548998
Also, I get free rent in Portland, in Corvallis (OSU) I’d have to pay. So there tons of financial incentive for me to just stay in PDX. I know OSU’s reputation is way better, but I’m not sure how I’d really objectively compare the two departments.

[Anonymous No.11549720]

>>11549701
>I’ve built up a customer base since I’ve lived in Portland
??

[Anonymous No.11549744]

>>11549021
I guess Ijust hadn’t seen it titled like that before, thanks though. You don’t seem to put off by the school, so that’s nice

>>11549033
But I only get to take so many for the degree, if you read the elective requirements it just seems sort of narrow AND shallow, but maybe I’m wrong.

>>11549362
Good reply, thanks. I’d like the degree to open up grad school possibilities. So based on what you said it might be good for research opportunities, with the downside being lower perceived quality

>>11549433
>remedial
Yeah, yeah. Whatever. Almost all USA unis start out with calculus unless your an Ivy League student. I think it sucks too, but I’m past those courses anyway. Thank you for what you’ve said though, you captured all my thoughts about it.

What I don’t understand is how to “set myself up for 600s” - this school seems to require that I spend my entire undergraduate slogging through intro level classes. Let me get off mobile, I want to hear as much as you’ll share and I really appreciate you looking so in depth. You seem to see my own criticisms of the school but know how to get around them.

[Anonymous No.11549795]

>>11549433
also, thanks for the research course recommendation - i overlooked those big time initially

[Anonymous No.11549898]

>>11549744
>What I don’t understand is how to “set myself up for 600s” - this school seems to require that I spend my entire undergraduate slogging through intro level classes.
I'm not sure how the American system works in this regard, so maybe I'm not the best to tell you, you should definitely mention it to some professor, but NEVER to a beaurocrat. Although I've always read that burger unis allow a lot of flexibility, so there's probably a way to only take the minimum required of the basic courses.
Also, really important detail I missed, if I'm not mistaken sometimes you can take credit for courses of other unis (I have a friend who took gradute course credits from Berkeley doing this), but maybe it's just a Commiefornia thing. Still, you should look into it.
Try to find out if there's any big or at least medium name in the department, and take his stuff, even if you're not that into it. If you excel, if only in effort, he might take notice and give a nice recommendation letter that could have some more weight than with others. Going to such a uni means you need to really game the system as much as you can.
As to how to set yourself for the 600s, keep in mind that the work you do on your own is, in the end, more important than whatever you do in the courses. You could also consider just blazing through the intro courses and go to more advanced classes without taking credit, I'm sure the profs won't mind if you just take the time to explain your plan briefly, and again, they might take some notice.

[Anonymous No.11550020]

>>11549898
>you should definitely mention it to some professor, but NEVER to a beaurocrat. Although I've always read that burger unis allow a lot of flexibility, so there's probably a way to only take the minimum required of the basic courses
Ok, I'll try asking some professor's for sure. I'm asking all this stuff right now because I'm roughly preparing to ask some beaurocrats for help, but I'll be sure not to rely only on them.

>sometimes you can take credit for courses of other unis
Thanks for bringing this up, this was one of the things I was going to talk to them about. I'd also like to take some classes like graph theory and have that count towards my CS minor, at other schools graph theory is sometimes a CS class.

>Try to find out if there's any big or at least medium name in the department, and take his stuff, even if you're not that into it. If you excel, if only in effort, he might take notice and give a nice recommendation letter that could have some more weight than with others. Going to such a uni means you need to really game the system as much as you can.
Good idea, I'll be sure to search all professor's prior to taking classes and base my schedule off this idea as much as possible.

>You could also consider just blazing through the intro courses and go to more advanced classes without taking credit
I guess that's possible, I'd have no objections to that as long as I'm not charged a premium, I'll look into that when the time comes

[Anonymous No.11550025]

>>11550020
>I guess that's possible, I'd have no objections to that as long as I'm not charged a premium, I'll look into that when the time comes
I mean, I imagine that if you just go to class, participate in a constructive manner, and are not a bother otherwise, then the profs might not give a fuck. It's not like the administration needs to know you're there.
>Good idea, I'll be sure to search all professor's prior to taking classes and base my schedule off this idea as much as possible.
Be sure to ask /mg/ if you're not quite sure who's a big guy for you and who isn't.

[Anonymous No.11550768]

>>11548639
>regular n-gon and let R be the set of n points on the line at infinity that correspond to the directions of the edges (and diagonals)
bump
what does 'n points on the line at infinity' mean?

[Anonymous No.11550834]

Did the Finnish category theorist weeb (?) really kill himself?

[Anonymous No.11550846]

>>11550834
who?

[Anonymous No.11550851]

>>11550846
An old regular.

[Anonymous No.11550878]

>>11550834
Now that I think about it, what happened to that category theorist with no academic background who namefagged?

[Anonymous No.11550891]

>>11550878
OHP? I liked his posts.

[Anonymous No.11550898]

>>11550891
Yeah that's the one.

[Anonymous No.11550939]

>>11545049
>>11545107

Lmfao.

>>11545516

It's okay kid- it's cool that you're interested in math and especially older topology texts. What year was it published? I have in interest in the history of topology personally since I find it interesting to see the progression from point-set topology to algebraic topology. Also why does your grandfather have a topology textbook? Is/Was he a math professor? My great-grandfather taught Calculus at a public university.

[Anonymous No.11550976]

>>11544409

Depends on what kind of mathematics you like anon? There are a few different paths you can take and they lead to different kinds of math. You can take "concrete/applied/practical" math, or you can take the "abstract/pure/theoretical" math path, or some combination of them. Now, unfortunately those dichotomies "concrete/abstract, applied/pure, practical/theoretical" are mostly false but common enough to help you find the textbooks you need. Comptational topologists for example deal with abstraction and "hard" computations. The proper distinction is really between the order of the concepts. Some concepts are made out of other concepts and those would be higher-order and their instances lower order. Practical/applied/concrete math tends to be lower order, and you are expected to have a facility with the concepts at the level of manipulating the concepts themselves- i.e. decomposing numbers and recomposing them quickly, solving integrals, etc. With higher-order math, the process is the same, of taking things apart and putting them back together, but the objects are not numbers anymore- they are concepts that may be a few levels of abstraction above a number- i.e. a space of equations.

So if you like numbers and solving problems that have numerical or at the most lower-order algebraic answers, then be clear about that in your goal. If you don't have any preference for whether you work with numbers or concepts about numbers, then you are going to want to have a strong foundation in logic, starting with propositional (0th-order), then predicate (1st-order) and possibly even higher orders although that won't be necessary until graduate school. Logic helps with numerical math too, but is more critical to higher-order because you need to be able to treat higher and lower order math with the same mindset without getting vertigo. Logic helps you see no difference between the orders- arguments are valid or not.

>>11544996

Theorists already on defense.

[Anonymous No.11551260]

>>11550898

He's still on Quora. He did a few too many psychedelics and now he struggles with keeping a focused stream of thought. He can get tangential quickly and superficial in his reasoning, kind of like a schizophrenic but without the paranoid delusions. Shame because he seemed like he had some talent.