/mg/ math general

[Anonymous No.10062048]

/mg/ math general

For your stupid questions, go ask /sqt/.
For your homework, go ask /sqt/ or
>>>/wsr/

Previous thread >>10042555


p-adic edition

[Anonymous No.10062063]

Reposting my question about the appeal of alg. geometry because I want more (you)'s
>Why do people like algebraic geometry again? I am, admittedly, an undergraduate brainlet, but it seems like a field dedicated to fucking around with boring equations and trying to find integral solutions or whatever the fuck they do. Explain the appeal /mg/.

[Anonymous No.10062090]

>>10062048
I never understood what p-adic numbers are and why are they important.

[Anonymous No.10062113]

>>10062063
It pretty much is that. However, it is much more exciting than you think. You should play around with plane curves and see what comes off it. For example, can you parametrise the circle x^2+y^2=1 using only one parameter? How about In a continuous way? How about in a way that also gives you rational solutions? The answer to the latter is partially yes - you can do so, but only if you add a point at infinity - here it pretty much means that, that is, literally defining a function that admits infinity.

How to do this: Well, you can do it the boring way and just embed the circle on C and use e^ix. However, this doesnt give you any rational solutions. Instead draw (or imagine) the family of lines that pass through x = -1, which is on the circle. Notice every line through x = -1 will intersect the circle again, except one line, the tangent at x = -1 to the circle. You can calculate the other point of intersection using the gradient as a parameter, and every single real value in the parameter will provide you with a unique point on the circle. In a way, you have projected the circle onto the real line. There's a catch: the point x = -1 is not represented by this parametrization, but you can approach it as you go to infinity on either side of the real line. So if you add a single point "at infinity", then the circle is realized in a continuous way by a single parameter, while still making sense. But, there's more: this gradient representation in fact gives you every rational solution to the equation x^2 + y^2 = 1: if you put a rational input as your gradient, it will give you back a rational solution, and vice versa! So just using a pretty simple geometric picture, you have found all Pythagorean triples and parametrised them in a single stroke.

Then you can do similar things with other, more complicated plane curves, like y^2 = x^3 - x, and soon enough you arrive to the theory of elliptic curves

[Anonymous No.10062120]

>>10062090
p-adic numbers are a way of looking at solutions to polynomial equations mod p^n for every n natural at the same time. Then the Hasse principle says that an equation has a rational solution iff there is a real number solution and in the p-adics

[Anonymous No.10062138]

>>10062120
Give example pls, perhaps on some simple equation like x^2 -x +1=0

[Anonymous No.10062262]

Up to an affine transformation of coordinates in [math]\mathbb R^2[/math], there are only 3 quadratic plane curves, an ellipse: [math] x^2+y^2-1=0[/math] , a parabola: [math]y-x^2=0[/math], and a hyperbola: [math]xy-1=0[/math]. We can look at these equations in [math]\mathbb C^2[/math] and by taking their homogenization, we can look at them in [math]\mathbb P_{\mathbb C}^2[/math], so that we get the equations:
[math]C_1:X^2+Y^2-Z^2=0[/math]
[math]C_2:ZY-X^2=0[/math]
[math]C_3:XY-Z^2=0[/math]
Let [math]\sigma:\mathbb P_{\mathbb C}^2\to\mathbb P_{\mathbb C}^2[/math] be conjugation [math](A:B:C)\mapsto (\bar A:\bar B:\bar C)[/math], and consider the sets [math]\Sigma_{C_i}=C_i\cap \{Z=0\}[/math]. In particular:
[math]\Sigma_{C_1}: X^2+Y^2=0[/math]
[math]\Sigma_{C_2}: X^2=0[/math]
[math]\Sigma_{C_3}: XY=0[/math]

I am tasked to find some criterion that can recognize from the isomorphism type of [math]\Sigma_{C_i}[/math] and the [math]\mathbb Z/ 2[/math] action of conjugation, whether any given curve C in [math]\mathbb R^2[/math] is either of those three.

However, by some simply logic, [math]C_2=\{(0:1:0)\}[/math], [math]C_3=\{(0:1:0),(1:0:0)\}[/math], and neither of those satisfy the equation of [math]C_1[/math], so they cannot be isomorphic. So where does the conjugation action come in?

[Anonymous No.10062302]

I added this to the wikipedia page (because I had nothing better to do), but they later removed it, so now I'm fairly sad.
Let Df(x)=f(x). Then f=ce^x, for some c. (Old stuff, well aware)
By the median value theorem, if Df(x)=f(x) and Dg(x)=g(x), and for some a f(a)>g(a), then for all x f(x)>g(x). Then, by Dedekind's Axiom you show that some c exists that satisfies the equation.

[Anonymous No.10062339]

>>10062090
>why are they important
They aren't.

[Anonymous No.10062494]

>>10062063
Classical alg geo is like masturbating with your hands tied behind your back, while contemporary alg geo with sheaves and stacks is like getting your hole reamed out by your bf with your hand tied behind your back

[Anonymous No.10062528]

>>10062063
I was going to write my own answer when I remember reading one a while ago.
https://mathoverflow.net/questions/77195/how-has-modern-algebraic-geometry-affected-other-areas-of-math
https://math.stackexchange.com/questions/255063/why-study-algebraic-geometry
>>10062262
I mostly followed what you were saying until the last few lines. Do you mean any curve or any quadratic planar curve? I'm just wondering how you're supposed to get from a curve R^2 to [math]\mathbb P_{\mathbb C}^2[/math]. Are you using the same lifting procedure as before?

[Anonymous No.10062702]

as i progress with the degree i find it harder and harder to find PDFs of the books for my courses. Is there any trick to it? Is there some kind of maths PDFs source out there?

[Anonymous No.10062703]

>>10062120
Pretty sure that's only true of quadratic forms.

[Anonymous No.10062726]

>>10062494
Excellent post

[Anonymous No.10062735]

>>10062702
Even on libgen?

[Anonymous No.10062755]

>>10062735
Thanks a lot, that's exactly the kind of site i didn't know about.

[Anonymous No.10063047]

>>10062528
Any curve in the plane R^2 can be lifted to C^2. Then the homogenization makes it a curve in P^2_C

[Anonymous No.10063059]

>>10062048
I hate this general.

[Anonymous No.10063179]

>>10063059
>I hate this general.
Mathematicians use "we", not "I".

[Anonymous No.10063223]

>>10062494

[Anonymous No.10063324]

>>10062090
I understand what they are but I don't understand what the color pictures like the Op image are supposed to be

[Anonymous No.10063338]

>>10062494
https://youtu.be/jjnrLt3VuSM?t=26s

[Anonymous No.10063342]

1+2+3+4+... = -1/12
Prove me wrong

[Anonymous No.10063346]

>we write
cringe
>beauty
>elegance
haha
imagine either being french or emulating them

[Anonymous No.10063412]

Give me a short guide how to proof any limit using cauchy's definition. Can we say that function has limit only of |f(x)-A|=a*|x-x0|, where a is any possible factor?

[Anonymous No.10063737]

Help an anon out:
What book would you recommend a newbie on elementary graph theory? it should suffice (on its own) as a method of learning the subject, meaning it has clear and SIMPLE explanations, proofs, exercises and answers.
Thanks guys.

[Anonymous No.10063771]

Anybody know some good references for complex analysis? Currently going over integration in the complex plane. It isn't hard so far, but of the courses I'm taking so far, this one is the hardest. Textbook for the class is Brown and Churchill Complex Variables and Applications

[Anonymous No.10063795]

>>10063342

[Anonymous No.10063865]

>>10063737
Harary?

[Anonymous No.10063871]

>>10063865
Thanks I'll check it out. If anything else comes to mind please feel free to add to it.

[Anonymous No.10063888]

>>10063795
God exists!, No God exists!

Which one of these is the claim that needs proof?

[Anonymous No.10063910]

>>10063888
Dumb example, pick something not shrouded in metaphysical uncertainty

[Anonymous No.10063954]

>https://en.wikipedia.org/wiki/P-adic_number#Algebraic_approach

Am I retarded or is pic related wrong?
It says that every [math]\textbf{non-integer}[/math] p-adic number can be expressed as [math] p^{-n}u [/math], which makes sense but then equates [math]\textbf{all}[/math] of [math] \mathbb{Q}_p [/math] to [math] \left( p^{N} \right)^{-1} \mathbb{Z}_p^\times [/math].

[Anonymous No.10063956]

>>10063910
Why is it a dumb example? Its simply meant to shine light on the fact that it can be unclear whats the claim and whats the negation of the claim.

[Anonymous No.10063969]

>>10063956
Because it invites the person you're arguing with lots of opening to drag you into a metaphysics discussion. God is the worst example to use when talking about logic, everyone always gets sidetracked.

[Anonymous No.10064248]

Hi,Chilean fag here, we got an standardized test to uni,I'm 22 years and could score to the top two unis here, is 22 to late to begin to study maths?

[Anonymous No.10064250]

>>10064248
Nah man, go for it

[Anonymous No.10064258]

>>10063795
Bertrand declares that a teapot is not, at this very moment, in orbit around the Sun between the Earth and Mars, and that because no one can prove him wrong, his claim is therefore a valid one.

[Anonymous No.10064262]

>>10063059
based general hater

[Anonymous No.10064269]

>>10063954
Yeah, notice that p^N is not a number, but a set, and Z_p is not Z/pZ, but the p-adic integers

[Anonymous No.10064279]

need a linearly independent solution y2(x)

[Anonymous No.10064327]

>>10063956
That's why "burden of proof" isn't really a scientific principle. It's just another logical fallacy meme neckbeards use to act like their philosophers.

[Anonymous No.10064531]

Is there a way to generalize peano's axioms to posets with more initial objects? Like a bunch of zero elements 0_1, 0_2, etc instead of just one zero element. I remember that a professor of mine talked about it but i can't find anything.

[Anonymous No.10064660]

>>10063342
What was the maymay proof for that one again?
>>10064248
No such thing as too late.
>>10064531
>Peano axioms
>zero
You absolute shame to humanity.

[Anonymous No.10064707]

>>10063888
What? Both you retard. Saying some argument is fallacious doesn't mean the statement is false. Arguing that God doesn't exist because there is no proof of it is as retarded as the opposite. Arguing that only shit we have evidence for exists brings a lot of issues and it's a completely different discussion.

[Anonymous No.10064717]

>>10062262
Wait, I thought in the definition of projective spaces you can't set any of the coordinates to be zero?

[Anonymous No.10064740]

>>10062262
nevermind, im retarded, there are only 2 solutions to C_1 which vary under conjugation so you do in fact get the result

>>10064717
no point can be (0:0:0), but you can have zeroes in your point

[Anonymous No.10064744]

>>10064717
No. You can't set all coordinates of a point to 0.
(0,0,1) yes
(0,0,0) no

[Anonymous No.10064751]

>>10064660
Please post relevant content only

[Anonymous No.10065119]

>>10064269
I understand that but how would you express, say, the p-adic integer p in that form? p is a non-unit because it corresponds to the sequence (0, p, p, p, ...) = p(1, 1, ...), but the exponent of p here is positive.

[Anonymous No.10065159]

>>10064751
Kill yourself.

[Anonymous No.10065170]

Can anyone recommend some good resources on formal verification and/or program synthesis?

[Anonymous No.10065174]

>>10065170
http://nobrain.dk/

[Anonymous No.10065222]

>>10065174
I forgot you aren't allowed to talk about math that's actually used to do things around here

[Anonymous No.10065241]

>>10065222
>I forgot you aren't allowed to talk about math
What you're discussing is completely nontrivial undergrad bullshit that I would rather see shitposting than have your nonsensical questioning

[Anonymous No.10065243]

I'm going to learn about groups.

[Anonymous No.10065271]

>>10063954
The two things mean are the same.

[math]\mathbb{Z}_p^ \times [/math] is the units and thus [math]{\left( {{p^\mathbb{N}}} \right)^{ - 1}}\mathbb{Z}_p^ \times [/math] is all elements of the form u/p^n for u a unit and n a natural number.

[Anonymous No.10065274]

>>10065243

[Anonymous No.10065279]

>>10065241
I really appreciate your insight about an active area of research being "undergrad bullshit" thanks

[Anonymous No.10065281]

>>10065274
Ultramarine orb elixir?

[Anonymous No.10065326]

>>10065279
Off to the >>>/trash/bin
what i am saying is that YOU'LL NEVER UNDERSTAND ANY OF IT ANYWAY YOU FUCKING WORTHLESS, CONTRIVED, BOURGEOISIE, SELF-CONGRATULATING DECADENT BULLSHIT THE INTERNET EVER HAD THE MISFORTUNE OF KNOWING.
However, I do not believe that you are beyond redemption! All you have to do is DRINK BLEACH AND DIE YOU EMO, SELF-INSISTING, SELF-DEPRECATING, SELF-INDULGENT EMPTY HUSK OF A HUMAN BEING.
REPEAT AFTER ME: I WISH I WAS PROFOUND, BUT I'M NOT! I WISH I WAS ORIGINAL, BUT I'M NOT! I WISH MY IMPENDING DEATH WAS OF ANY CONSEQUENCE, BUT IT IS MOST CERTAINLY NOT!

[Anonymous No.10065343]

What's finite math? I have to take it for a CS minor and it's a level 100 class. Physics advisor gave me a weird look having that and DE in the same quarter. They told me I could substitute finite math for discrete math, but why take the more difficult course. Neither math or CS is my major, only Physics.

[Anonymous No.10065457]

>>10062048
sup sci
how does one go around reading euclid? im rebuilding all my fundamentals in math, and also wanting to tour some older texts... im curious though as how one should really study some of these things? with the principia mathematica, my main plan of attack is just verifying the algebra, asking myself what a proposition is telling me, and also giving myself an example of how all of this is playing out... idk what to do with euclid - should i draw out all of his results, then interpret them algebraically? maybe also prove some of them if they aren't too trivial? any other suggestions?

[Anonymous No.10065485]

>>10065457
Well, the correct way to _study_ Euclid is to not study Euclid. Everything, from style to writing to content, is so hopelessly outdated that it's not really accessible or useful to modern mathematicians.
By all means read it for personal interest if you find old books interesting, but you should always keep in mind that you're just a historical tourist looking for interesting tidbits and not a student trying to absorb the text. Just browse around, look for cool things that nobody remembers anymore, enjoy the autistic prose, etc.

[Anonymous No.10065521]

>>10065457
I won't say reading older texts is necessarily a bad thing, but understanding, notation, and picking the "right" definitions only increase in number with time, so I wouldn't disregard modern texts. If you want a really autistic (albeit rigorous and kind of fun) way to start from the ground up and build to geometry try
http://www.geometry.org/tex/conc/dgchaps.html

[Anonymous No.10065551]

Why do many functions and polynomial exhibit interesting properties at integers even when there's no Gamma function involved?

[Anonymous No.10065723]

>>10065551
Elaborate

[Anonymous No.10065749]

>>10062090
https://youtu.be/XFDM1ip5HdU?t=10m

[Anonymous No.10065783]

Does the proof of FLT depends on axiom of choice? I don't trust axiom of choice because of Banach-Tarski paradox.

[Anonymous No.10065788]

>>10065783
No choice causes issues too though, for example you can have vector spaces that don't have a basis

[Anonymous No.10065789]

How do I marry a mathfu?

[Anonymous No.10065793]

>>10065789
I really want to get a girl who likes math and is good at it pregnant.
Like, I want to fuck her brains out, until she believes 1+1=9 because we have 9 children.

WHERE ARE YOU MY MATHEMATICIAN WAIFU!?

[Anonymous No.10065797]

No mathematician waifu = no laifu.

[Anonymous No.10065829]

>>10063737
>>10063865
>>10063871
Hanary won't cut it. it lacks the "exercises and answers."

Help an anon out:
What book would you recommend a newbie on elementary graph theory? it should suffice (on its own) as a method of learning the subject, meaning it has clear and SIMPLE explanations, proofs, exercises and answers.
Thanks guys.

[Anonymous No.10065837]

>>10065829
I feel like graph theory isn't a subject that admits a "good" introductory text. You're just gonna be getting familiar with what a graph is, how graph theory proofs usually go, etc. It's not like analysis where you get a totally different perspective on the subject between different texts. Just get a book and start learning anon.

[Anonymous No.10065845]

>>10065837
Thanks for the insight. You see, I already did an introductory course in it. I passed. barely. I feel like I know very little and need to somehow build the skills for it. Staring at proofs without context didn't do me much good... I need a "coaching" text of some sort, not unlike what you mention in analysis. I have done quite a bit of analysis, and abstract algebra, its only graph theory that feels so unstructured...

[Anonymous No.10065850]

>>10065845
Ah alright. Sorry then anon I can't help much, I've only done an intro course as well and I've only ever used graphs as a convenient representation for algebra things.

[Anonymous No.10065857]

>>10065850
Thanks for trying

[Anonymous No.10065873]

>>10065159
No you!spookyanimegirl.jpg

[Anonymous No.10065896]

What's the difference between Cauchy sequences and convergent sequences?

[Anonymous No.10065912]

>>10065896
A Cauchy sequence has all terms arbitrarily close to each other, a convergent sequence as all terms arbitrarily close to a limit.

In the reals these are equivalent, but it's not the case in all spaces. For example you can have a Cauchy sequence in the rationals that converges to a point outside of it (and thus does not converge). Take a rational approximation of root 2 for example.

[Anonymous No.10065914]

>>10065912
ps Cauchy sequences of rationals are one way of constructing the reals

[Anonymous No.10065960]

>>10065912
Cheers.

[Anonymous No.10065962]

How did Mochizuki get into the position he's in? Shouldn't the fact that he developed an incredibly complicated theory that seems to do nothing but prove a single conjecture have concerned him?

[Anonymous No.10066323]

>>10065783
>I don't trust axiom of choice because of Banach-Tarski paradox.
Banach-Tarski isn't really a paradox or anything. All it says is that given the axiom of choice you can construct non-measurable sets, and using those sets you can get bunch of wacky shit. This is really easy to remedy however, just don't use non-measurable sets and there are no paradoxes. That's it. That's why these measure-theoretic proofs always only consider measurable sets. Also this >>10065788.
>>10065829
I think this
https://www.cambridge.org/us/academic/subjects/mathematics/discrete-mathematics-information-theory-and-coding/course-combinatorics-2nd-edition?format=PB&isbn=9780521006019
would be good for you, at least I enjoyed the parts I needed to read.
>>10065962
He was a full prof by 2002 and already had publications beforehand. Prior to working on IUT in solitude he had a solid enough track record to be a well established mathematician. It was after that that he started working hard on IUT and building on his previous work for several years.

[Anonymous No.10066388]

Is it wrong to think of any space with the discrete metric as a circle (any two distinct points are 2r apart?) and the British Railways metric as a disc? (any two distinct points are [math]\lVert \mathbf[x+y] \rVert[/math] apart)

[Anonymous No.10066413]

What field of math should I study if interested in algorithm design

[Anonymous No.10066432]

>>10066413
Hindi

[Anonymous No.10066443]

>>10062048
Post things man was not meant to know

[Anonymous No.10066450]

>>10066323
Thanks ill check your book tonight and get back to you

[Anonymous No.10066594]

>>10066443
>man was not meant to know mathematical constructs made up by men

[Anonymous No.10066635]

why hasn't shinichi committed sudoku yet

[Anonymous No.10066646]

>>10066450
>>10066323
That book fits the bill, Thank you very much!

[Anonymous No.10066785]

>>10065485
>>10065521

huh i see, ty anons... i might skim euclid a bit, but i guess there is no use trying to a absorb everything

>http://www.geometry.org/tex/conc/dgchaps.html

ive seen this b4... i guess im gunna have to try it out myself sometime... should you already be familiar with all of these topics before you start though?

[Anonymous No.10066805]

Potentially stupid: Is this right?
[eqn]\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\mathrm{d}\psi \left(z(x)\right)}{\mathrm{d}z}\right)\equiv \frac{\mathrm{d}}{\mathrm{d}z}\left(\frac{\mathrm{d}\psi \left(z(x)\right)}{\mathrm{d}x}\right)[/eqn]

I'd argument using Schwarz's theorem.

[Anonymous No.10066806]

>>10066785
I disagree with those guys. Reading the Elements is actually a useful exercise, since it will teach you a certain style of thinking. Math is never "outdated". Math is math.
There are certain problems with Euclid's books (mainly, incomplete proofs in parts), but being "outdated" certainly isn't it.

Try reading this edition: https://archive.org/details/ByrneOFirstSixBooksOfTheElementsOfEuclid1847/page/n5
>Byrne O First Six Books Of The Elements Of Euclid 1847
It's fully illustrated.

[Anonymous No.10066815]

>>10066805
ye, provided psi has the right properties.

[Anonymous No.10066830]

>>10066594
You can't even tell me whether pi and e are in the same coset.

[Anonymous No.10066868]

>>10066815
Thanks, much appreciated.

Actually [math]\psi[/math] is meant to be the Wave Equation, so it is required to be (totally) differentiable at least two times. So it is partially differentiable in both "variables" [math]\psi(x, z(x))[/math], thus making the Hesse Matrix symmetric, which -- in return, if looking at [math]\psi(z(x))[/math] again -- makes the two (total) Differentials commutable.
Feels kinda wierd, just like Lagrange's Formalism where everybody's understanding of Calculus collided...

[Anonymous No.10066877]

sup bros, writing a 20 page assignment in applied mathematics in LaTeX
I've been typing for over 4 hours and I have barely gotten to page 10
Who else fucking hate LaTeX? The syntax is ironically much more complicated than what you'd actually expect from this. I thought it was supposed to be simple but no, you need to write a bunch of \ and {}. Can't write a fraction as simply x/y, nope, gotta write \frac{x, y}, which fucking sucks if you have 15 fractions in one equation
Don't even get me started on writing down matrices. Gees....

[Anonymous No.10066879]

>>10066877
whoops meant \frac{x}{y} which is somehow even worse

[Anonymous No.10066888]

>>10066877
shut up fag

[Anonymous No.10066894]

>>10065783
I don't know about choice but I do know there is a inaccessible cardinal assumption in the proof.

[Anonymous No.10066900]

>>10066877
Yeah, it kinda can get annoying, but it looks sooo good. It's really frightening me that Donald Knuth was de-facto the only developer of Latex in it's early stages.

But it's more annoying to me that it isn't implemented in an easy Scripting Language like Python, instead it is "programmable" in this wierd Macro-Language. Because -- let's face it -- could you imagine an easier language for math. Expressions?
Because you can always use tabs or define your own macros to make life easier. But if you, let's say, want to load-in Data and iterate over it, oh boy, that's where LaTeX get's onto your private parts.

[Anonymous No.10066902]

>>10066888
ok grandpa

[Anonymous No.10066907]

>>10066877
\frac{x}{y} is just because everything else would f-up the macro language [even more].
There actually is Infix notation for fractions, but it is not recommended to use it.

[Anonymous No.10066912]

>>10066877
This. LaTeX successor when?
>inb4 two centuries

[Anonymous No.10066924]

>>10066912
Excuse me?
Does this Man look to you like he wanted you any evil when using his typesetting system based upon programming standards of the 70s?

[Anonymous No.10066925]

failed my midterm

[Anonymous No.10066928]

>>10066925
Which field?

[Anonymous No.10066930]

>>10066925
Study harder, keep trying

[Anonymous No.10066935]

>>10066928
Number theory, or as I have taken to calling it Gay theory.

[Anonymous No.10066946]

>>10066935
Pff, pure Mathematics then?

I'd never even nearly consider taking anything related to that subject. Even sounds autisticly boring...

[Anonymous No.10066956]

>>10066450
>>10066646
No prob anon.
>>10066785
I mean, it's not that it's the best way to do geometry, but it is "complete" and doesn't really expect you to know too much before hand.
>>10066806
>but being "outdated" certainly isn't it.
It's not that I think it's "outdated" per se, more that the type of thinking that it uses isn't as relevant to geometry. Most geometric proofs I've seen don't use the same kind of techniques or constructions as Euclid, but it is certainly a fun read. It's just that the landscape of geometry has changed quite a bit and we're working in far more general spaces.
>>10066877
It takes a while but once you get used to it it's not so bad. The real key to getting good is setting up macros. You can usually define things beforehand in such a way that makes using LaTeX a lot easier.

[Anonymous No.10066966]

>>10066956
Are macro languages Turing-complete?

[Anonymous No.10066977]

How seriously does the average graduate program take the math gre, provided that the rest of the application is strong?
More or less everything else on my resume is good to very good (GPA, research+publications, letters, grad courses, etc.) but I massively understudied for the subject GRE and landed at 65th percentile, and it's too late to retake it.
I'm unsure how high in school rankings I can go before I start getting auto-binned because of that score.

[Anonymous No.10066997]

>>10066966
Apparently LaTeX actually is: https://stackoverflow.com/a/2968527

Wouldn't have thought that, because I couldn't imagine that looping/recursion was possible in LaTeX. Mhh.

[Anonymous No.10067024]

>>10066946
Y-y-your on this board and not into autistically boring things?
Normie GTFO. >>>/sp/

[Anonymous No.10067038]

>>10066877
t. Mathematica brainlet who can't do his own typesetting

[Anonymous No.10067077]

>>10066977
Many schools use the math gre as an initial cutoff to weed out potentially weaker candidates, not perfect but effective. The GPA can be a bit difficult when comparing people from very different universities and with very different coursework. A 4.0 from an average school with mediocre coursework isn't better than a 3.5 from MIT taking grad courses since their sophomore year. The math gre seeks to fill that gap and provide a even playing field to compare students against each other. Schools like ucla want people typically in the top 80 percentile and don't consider anyone below that score as much, usually disregarding them.
https://www.math.ucla.edu/grad/frequently-asked-questions
So you being barely better than average isn't a good sign. Frankly, and this is not to knock you, but even if you didn't study you shouldn't do poorly on the math gre. In my opinion over 75% of the test is not only standard material, but fairly basic stuff you should know without having to review, the rest can be a bit of a grab bag. When I took it there were some approximation theory and graph theory problems that screwed me over. Still got into the 78 percentile but not as good as I would have wanted. And yes, that was with no prep time. I don't want you to think I'm trying to brag (cause my score isn't impressive by any means) or enforce my opinion over you, but I just say this as someone who's talked to other grad students and faculty members to get a grasp of what people think of the math gre and how it reflects your performance. I don't want to discourage you though, having great letters and recommendations will still get you into a lot of places, and frankly math gre itself doesn't mean a while lot, but it will be used by people as a cutoff.

[Anonymous No.10067133]

>>10067077
While you probably shouldn't need to relearn any of the actual math on the subject GRE, the reason the test is hard is not because of the mathematical content. It's because you're given less than 3 minutes a question and need to keep that pace for 3 hours.
If you did not grind practice problems first and you even made it to the end of the test at all you already should be considering yourself fairly talented.

60-70 percentile is dead-on where everyone in my cohort who didn't spend a month or two practicing scored.

[Anonymous No.10067948]

bump

[Anonymous No.10068033]

>>10066877
pleb

[math] \displaystyle
x = \bigg( - \frac{1}{3a} \bigg) \sqrt[3]{ \frac{1}{2} \bigg( 27a^2 d - 9abc + 2b^3 + \sqrt{ 729a^4 d^2 - 486a^3 bcd + 108a^3 c^3 + 108a^2 b^3 d - 27a^2 b^2 c^2 } \bigg) } + \bigg( - \frac{1}{3a} \bigg) \sqrt[3]{ \frac{1}{2} \bigg( 27a^2 d - 9abc + 2b^3 - \sqrt{ 729a^4 d^2 - 486a^3 bcd + 108a^3 c^3 + 108a^2 b^3 d - 27a^2 b^2 c^2 } \bigg) } - \frac{b}{3a}

\\

\displaystyle x = \bigg( \frac{ 1 + \sqrt{3} i }{6a} \bigg) \sqrt[3]{ \frac{1}{2} \bigg( 27a^2 d - 9abc + 2b^3 + \sqrt{ 729a^4 d^2 - 486a^3 bcd + 108a^3 c^3 + 108a^2 b^3 d - 27a^2 b^2 c^2 } \bigg) } + \bigg( \frac{ 1 - \sqrt{3} i }{6a} \bigg) \sqrt[3]{ \frac{1}{2} \bigg( 27a^2 d - 9abc + 2b^3 - \sqrt{ 729a^4 d^2 - 486a^3 bcd + 108a^3 c^3 + 108a^2 b^3 d - 27a^2 b^2 c^2 } \bigg) } - \frac{b}{3a}

\\

\displaystyle x = \bigg( \frac{ 1 - \sqrt{3} i }{6a} \bigg) \sqrt[3]{ \frac{1}{2} \bigg( 27a^2 d - 9abc + 2b^3 + \sqrt{ 729a^4 d^2 - 486a^3 bcd + 108a^3 c^3 + 108a^2 b^3 d - 27a^2 b^2 c^2 } \bigg) } + \bigg( \frac{ 1 + \sqrt{3} i }{6a} \bigg) \sqrt[3]{ \frac{1}{2} \bigg( 27a^2 d - 9abc + 2b^3 - \sqrt{ 729a^4 d^2 - 486a^3 bcd + 108a^3 c^3 + 108a^2 b^3 d - 27a^2 b^2 c^2 } \bigg) } - \frac{b}{3a}

[/math]

[Anonymous No.10068181]

>>10068033
Nice redundant parentheses
*GOTTA LOOK MORE COMPLICATED THAN IT IS*

[Anonymous No.10068188]

>>10068033
>doesn't know about the [math] \big(big \bigg) \Big( \Bigg) [/math] brackets

[Anonymous No.10068327]

[math] {\mathrm d} f(t) = \left(-\log\left(g(t)\right)\right) f(t) \implies {\mathrm d} \left( g(t) f(t) \right) [/math]

[Anonymous No.10068329]

[math] {\mathrm d}f(t) = \left(-{\mathrm d}\log\left(g(t)\right)\right) f(t) \implies {\mathrm d}\left( g(t) f(t) \right)=0 [/math]

[Anonymous No.10068333]

[math] {\mathrm d}f(t) = f(t)\left(-{\mathrm d}\log g(t)\right) f(t) \implies {\mathrm d}\left( g(t) f(t) \right)=0 [/math]

[Anonymous No.10068378]

>>10068333
Please walk me through this.

[Anonymous No.10068382]

>>10068378
I added one f(t) too much. It's a triviality written down using a log

[Anonymous No.10068385]

>>10068382
>I added one f(t) too much.
Makes sense now.

[Anonymous No.10068416]

>>10062048
You have one hour (1h) to explain HOW TO PRODUCE SOMETHING with the knowledge of ADIC INTEGERS.

[Anonymous No.10068440]

I'm very interested in cryptography. Is algebraic number theory a good thing to pursue given this interest? Is there something better?

[Anonymous No.10068520]

>>10068440
number theory, abstract algebra, complexity theory and obviously anything directly related to crypto itself

[Anonymous No.10068574]

>>10068440
number theory, algebraic geometry, computer science, information theory

[Anonymous No.10068636]

WHAT DO i major in if I like set theory and want to be a logician

[Anonymous No.10068748]

legit brainlet question that I can only ask here to get a quick answer
What did you call that operation in complex numbers where you change the sign of the imaginary number, eg. (1+2i) become (1-2i)
Quick hurry tell me

[Anonymous No.10068755]

>>10068748
It's for my assignment

[Anonymous No.10068758]

>>10068748
>>10068755
It's not a "hurr durr help me I am too retarded to do this task". It's a question of vocalbulary

[Anonymous No.10068765]

>>10068748
Generator of Gal(C/R)

[Anonymous No.10068791]

>>10068765
nononooo it's not that is this bait

[Anonymous No.10068805]

>>10068636
Philosophy, not even kidding.

>>10068791
That anon isn’t wrong but yeah you’d be better off calling it conjugation.

[Anonymous No.10068809]

>>10068748
Complex conjugation.

[Anonymous No.10068910]

>>10068805
>>10068809
ahah sorry guys sorry for being retarded. That's it, yes

[Anonymous No.10068931]

what's your go-to (geometric or otherwise) intuition for topics in measure theory?

[Anonymous No.10068988]

What are some fun math exams I can take for an undergraduate? I know calc 2. Something like the american math competition except for undergrads.

[Anonymous No.10069005]

>>10068988
>What are some fun math exams I can take for an undergraduate?
https://en.wikipedia.org/wiki/William_Lowell_Putnam_Mathematical_Competition

[Anonymous No.10069191]

Does anyone know this formula for pi? I want to know if there is a form of it with some name so I can see if it is correct.

[Anonymous No.10069196]

>>10069005
wtf why is this so hard

[Anonymous No.10069219]

>>10069191
holy shit I checked it it actually works

[Anonymous No.10069223]

>>10069219
kek

[Anonymous No.10069275]

Does anyone know how to compute the minkowski sum of a square and circle where you only know the square's side length and radius of the circle?

[Anonymous No.10069293]

Can anyone recommend some really good books about applications of quadratic forms in physics?

[Anonymous No.10069294]

>>10069191
https://en.wikipedia.org/wiki/Vi%C3%A8te%27s_formula

[Anonymous No.10069358]

>>10069294
>>10069294
trying to get it into this form

[Anonymous No.10069840]

Anyone participate in Simon Marais?
I got the first 2 questions of paper 1 and half of 3 when I ran out of time.
Paper 2 I got question 1 and a partial proof for question 3.

[Anonymous No.10069850]

>>10062702
b-ok.org also excellent

[Anonymous No.10069867]

>>10069005
>>10069196
I was always a bit disappointed by the lack of an accessible math contest for undergrads.
There's a huge opportunity for undergraduate contest circles to actually have value because there's basically an infinite pool of cool stuff outside the standard curriculum you can teach undergrads that you can't really teach high school kids, but the audience for the Putnam is basically IMO medalists + people who are just writing it because profs give out grade bonuses to boost their numbers.
A contest where the average math student could actually answer a few questions correctly would attract a lot more people.

[Anonymous No.10069900]

>>10069196
because you have to learn all the special tips and tricks that no one ever could figure out by themselves unless they were a hyper-autist genious.

[Anonymous No.10069931]

>>10063342
>Prove me wrong
its divergent by a p-series test, so it cant equal a finite number
https://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/SandS/SeriesTests/p-series.html

[Anonymous No.10070086]

>>10069840
What did you do with B3?
I tried a triple circuit cover of the polyhedral graph but didn't figure out what speed to set

[Anonymous No.10070102]

>>10070086
It's kinda hard to explain without pictures and I doubt I'll get that many points for it. I found it by trying first to do it on a tetrahedron and cube. I only worked out how to get the fast spider within distance 1 of the bug though. Not sure what to do from there.
The basic idea is you can show that as the fast spider moves towards the bug if the bug wishes to maintain distance then at each vertex only one of the three paths works (at the other two the fast spider can reactively choose a path to get closer). This path is then a fixed cycle and so you can move a slow spider into it, forcing the bug to divert and thus allowing the fast spider to get closer. There's a bit of trickery because the bug and fast spider are not necessarily at integer distance from each other but you can fix this with a slow spider too.
This argument gets you within distance 1 and then you need something else which must involve both slow spiders.

Did you get B2 by any chance? Seemed like it shouldn't be that hard but I couldn't get anything to work.

[Anonymous No.10070379]

So when you construct the rationals as equivalence classes of ZxZ that’s some kind of quotient, right? But in what category?

[Anonymous No.10070390]

>>10063342
It's easy to confuse yourself with this shit but it's quite simple.

All that the ramanujan summation stuff, cutoff and zeta regularization does, is look at the smoothed curve at x = 0.
What sums usually do is look at the value as x->inf.

It's just a unique value you can assign to a sum, really they have many such values.

https://en.wikipedia.org/wiki/File:Sum1234Summary.svg

[math] \displaystyle
\zeta \neq \Sigma
[/math]

https://youtu.be/sD0NjbwqlYw?t=10m

[Anonymous No.10070651]

>>10070379
Set

[Anonymous No.10071028]

>>10070651
I feel like there it should be in a more structurally rich one though.

[Anonymous No.10071048]

>>10066443
It's basically still R right? Like yeah we don't know where our normal irrationals get collapsed to but it's still more or less R. If you ignore topology that is.

[Anonymous No.10071057]

>>10071028
What do your feelings have to do with anything?

[Anonymous No.10071062]

>>10071057
Your intuition is a valuable tool you dummy. Sometimes it's wrong but you haven't given any indication that's the case.

[Anonymous No.10071201]

>>10069972
any takers?

[Anonymous No.10071208]

>>10071201
>any takers?
What have you tried?

[Anonymous No.10071308]

>>10063771
stein and shakarchi's text is pretty well written. if you want a very nice book for intuition i strongly recommend Visual Complex Analysis by Tristan Needham

[Anonymous No.10071316]

>>10071048
That's the quotient [math]\mathbb{R}/\mathbb{Q}[/math] not set difference [math]\mathbb{R}\setminus \mathbb{Q}[/math] you fuckface. Also the irrationals aren't closed under addition so that wouldn't even be an Abelian monoid you mongoloid

[Anonymous No.10071352]

>>10066877
Imagine getting this far in your math degree and not once considering that you should make a couple macros

[Anonymous No.10071393]

>>10071048
The only thing it has in common with R is that it's an uncountable abelian group.

[Anonymous No.10071407]

>>10071316
Chill the fuck out anon jeez

[Anonymous No.10071425]

>>10063771
The book you're using is good. Ahlfors' book is fantastic

[Anonymous No.10071427]

>>10071048
not at all, R is torsion-free and R/Q has a lot of torsion

[Anonymous No.10071786]

>>10069191
>left handed Ramanujan

[Anonymous No.10071822]

>>10071427
That's wrong. The additive group R/Q has no torsion. Suppose r is irrational but r+r+...+r = nr is rational, say nr = p/q. Then r = p/nq is rational, a contradiction. This means that every nonidentity element [r] in R/Q has infinite order, i.e., no torsion.

[Anonymous No.10071827]

>>10062048
Optimized.

[Anonymous No.10071860]

>>10071827
Based picture optimisation poster.

[Anonymous No.10071904]

>>10071201
The question is unreadable.
Do you mean something like, "Is the limit of a sequence of bounded functions bounded?"

[Anonymous No.10071924]

>>10071822
oh absolutely, my bad ! I must have been thinking of R/Z

[Anonymous No.10071931]

>>10071822
oh absolutely, my bad ! I must have been thinking of R/Z.
Then actually, since it is still divisible and torsion-free, it is a vector space over Q and has the same dimension as R so it is abstractly isomorphic to R as a group no ?

[Anonymous No.10072101]

>>10065119
Can someone answer this guy? In particular, 0 is a p-adic integer. How would one produce 0 as p^-n times a p-adic unit? Same for p.

[Anonymous No.10072345]

>>10071904
He's asking to find a pointwise bounded sequence of measurable functions on that interval such that each is bounded by the limit suprmum of the sequence of functions is not.

[Anonymous No.10073222]

>>10071827
What the FUCK does that picture mean?

[Anonymous No.10074137]

>>10071931
>it is a vector space over Q and has the same dimension as R so it is abstractly isomorphic to R as a group no ?

[Anonymous No.10074204]

>>10073222
Don't know, but it's real neat and I plan to go over it one of these days.

[Anonymous No.10074804]

I posted this in another thread but this might be better. Can someone give me a purely algebraic/topological definition of homotopy. I don’t want R in there at all. Even ncatlab’s definition uses R.

[Anonymous No.10074823]

>>10074804
>I don’t want R in there at all.
Then it wouldn't be homotopy.

[Anonymous No.10074833]

>>10074823
Are you sure we can’t have a ‘general’ topological definition of path-connectedness that can let us define homotopy without parameterizing the transformation over the unit closed interval in R? It seems strange we have homotopy as a weaker notion of topological equivalence than homeomorphism but only one is defined ‘purely’ topologically.

[Anonymous No.10074860]

>>10074833
>It seems strange we have homotopy as a weaker notion of topological equivalence than homeomorphism but only one is defined ‘purely’ topologically.
Well, when ever you define curves on manifolds people usually parameterize them in terms of the local coordinates, so it's not a new thing. I guess you could think of a homotopy as something acting on the space of continuous functions, basically a line in the space of continuous functions with the endpoints being the two homotopic functions, though this would have to be a very specific path in the space. And yes, this would imply there being a thing like "homotopies between homotopies" which is indeed a thing.

[Anonymous No.10074872]

>>10074833
>>10074860
You know what, I just remembered something. Operads. That's what you're looking for
https://arxiv.org/pdf/1202.3245.pdf

[Anonymous No.10074877]

>>10074872
Thanks anon I'll check it out.

[Anonymous No.10074882]

>>10074877
Here are some better links
https://www.ams.org/notices/200406/what-is.pdf
https://math.mit.edu/~ebelmont/operads-talk.pdf

[Anonymous No.10074954]

If I want to prove bijection can I just claim pigeonhole principle, prove that and be done? I feel like this is naive. But I also wanna be lazy.

[Anonymous No.10075038]

>>10074954
That's not naive. It's stupid. Think about it.

[Anonymous No.10075221]

>>10062048
Hello /mg/
In the Einstein–Infeld–Hoffmann equations:
[math]{\begin{aligned}{\vec {a}}_{A}&=\sum _{{B\not =A}}{\frac {Gm_{B}{\vec {n}}_{{BA}}}{r_{{AB}}^{2}}}\\&{}\quad {}+{\frac {1}{c^{2}}}\sum _{{B\not =A}}{\frac {Gm_{B}{\vec {n}}_{{BA}}}{r_{{AB}}^{2}}}\left[v_{A}^{2}+2v_{B}^{2}-4({\vec {v}}_{A}\cdot {\vec {v}}_{B})-{\frac {3}{2}}({\vec {n}}_{{AB}}\cdot {\vec {v}}_{B})^{2}\right.\\&{}\qquad {}\left.{}-4\sum _{{C\not =A}}{\frac {Gm_{C}}{r_{{AC}}}}-\sum _{{C\not =B}}{\frac {Gm_{C}}{r_{{BC}}}}+{\frac {1}{2}}(({\vec {x}}_{B}-{\vec {x}}_{A})\cdot {\vec {a}}_{B})\right]\\&{}\quad {}+{\frac {1}{c^{2}}}\sum _{{B\not =A}}{\frac {Gm_{B}}{r_{{AB}}^{2}}}\left[{\vec {n}}_{{AB}}\cdot (4{\vec {v}}_{A}-3{\vec {v}}_{B})\right]({\vec {v}}_{A}-{\vec {v}}_{B})\\&{}\quad {}+{\frac {7}{2c^{2}}}\sum _{{B\not =A}}{{\frac {Gm_{B}{\vec {a}}_{B}}{r_{{AB}}}}}+O(c^{{-4}})\end{aligned}}[/math]
What does [math]\sum_{B\neq{A}}[/math] mean?
How is this sum calculated? What does an inequality mean when put in an index?

I'm not posting in /sqt/ (yet) because I don't think I'll get an answer there.

[Anonymous No.10075250]

>>10075221
If the index is not a simple iterator condition like i=0, it is generally an abbreviated set condition. A=/=B means sum over all combinations (of two bodies in this case) where they are different. Something like i<j means sum the terms that satisfy this condition, but of course this wouldn't make sense in your problem since there is no notion of ordering the bodies.

[Anonymous No.10075260]

>>10075250
Thank you. Now I understand the math/notation part, I think.
But I am more confused when it comes to solve it numerically.
Is that equation describing the acceleration of A interacting with 2 other bodies? (there's a third element "C") And when iterated, it can be used to describe N bodies? With the usual precision loss associated with those, of course. (matter of choosing an integrator, verlet or runge kuttas)

[Anonymous No.10075425]

First year CS student here, I know most of you already think we are brainlets when it comes to math, but hear me out.

We started with discrete maths - proofs. Like a lot, there's so much of proofs and professor goes so fast through lectures I'm left with notes, but no logic behind the steps to answers, since you must understand and gradually know how to extract shit and re-arrange it.

While I find it hard (we never did this in high school), I also sat down, learned and got a feeling for it and see things and how they happen, but I still can't put stuff into words a lot of the times.

Is this just because I'm facing a lot of new stuff or should I record lectures and go over them later. As said, we go through an insane amount of stuff in 3 hours and you gradually forget stuff as you go along, since there is so much new shit and new proofs that I just can't keep up and 90% of class also.

I did good in hs math, but feel complete brainlet, but I'm not sure if it's because it's new math to me or if the professor is just going too fast and let's out the steps.

[Anonymous No.10075432]

>>10075425
CS math is notoriously bad and CS professors are terrible at communicating. Can you try to just sit in math lectures from the math department? if not find out when they have open office hours and visit some of those professors and ask for their help? or find a drop in centre or something. math is something you need a person sitting next to you slowly showing you each step until you understand it.

alternatively you could try to do 100s of problems and just drill it like a machine until it comes second nature to you.

[Anonymous No.10075445]

>>10075432
I know that it's either going to be grind on my own, but I'm not sure how much value I can get if I get some tutor for few hours that will go through with me slowly and make sure I know my shit before I proceed.

At this point I really don't know, if it's the professors fault or why it is the way it is. Few guys that are year higher said that professor just goes through lessons too fast and it's nothing new, then other guys from our class that had discrete math in HS said that they did 2 weeks of certain topics in HS, that this professor went over in 1 hour.

So, I mean.. it looks pretty fucking bad, but it's incredibly intense math course and you are left only writting shit down for 3 hours that you sit in there, without learning anything, unless you have perfect memory and cocaine focus.

[Anonymous No.10075561]

>>10074954
You can do that if the set is finite and you've proven injection or surjection

[Anonymous No.10075622]

>>10064660
>>Peano axioms
>>zero
>You absolute shame to humanity.
Maybe originally, but the superior definition uses zero.

[Anonymous No.10075662]

>>10075445
supplement lectures with youtube videos. it sounds retarded but especially crammed cs math lectures are awful sometimes and theres some people on youtube who are pretty good at teaching

[Anonymous No.10075813]

>classmates insist on using fullpage package for group projects
How can convince them to see the error of their ways? Having equations almost running off the side of the page is getting to me, and their reasoning is basically "well it looks better to me" which is only true because that's the way you did it in high school. Literally all lecture notes and handouts use larger margins, but they refuse to adapt.

t. autist

[Anonymous No.10075843]

>>10066443
what structure is this? it seems like R quotient Q but that isnt closed under addition?

[Anonymous No.10075862]

>>10075843
Q is clearly normal in R so the quotient is well defined.

[Anonymous No.10076078]

>>10075260
You are confusing yourself with A, B and C, they are simply placeholders for a body in your system. The equation assumes there are N bodies, not 3 as you suggest. C is used because the sum is inside another sum and is presumably there to include some modified gravitational effect on A and B within the sum. All of the values in the equation are assumed to be known and so you can calculate the acceleration on each body when they are all fixed in place. There is no 'iteration', the equation already handles N bodies. The 'integration' or whatever you call it refers to what happens when time continues and all the positions and velocities update.

[Anonymous No.10076083]

>>10066925
Keep working, anon. I have faith in you.

I almost always fail the first exam in a course, but I've never ended lower than a B+ in any course, and I'm sure you can do better than I.

[Anonymous No.10076090]

>>10076078
Poor choice of words - replace 'when they are all fixed in place' with 'at a fixed point in time'.

[Anonymous No.10076695]

Any of /mg/ using sage for symbolic calculus?
I have a polynomial
>sage: (2*s*w*z + s^2 + w^2)^2*(p + s)
and I want to expand it a polynomial in "s". By using the expand function I get
>4*p*s^2*w^2*z^2 + 4*s^3*w^2*z^2 + 4*p*s^3*w*z + 4*s^4*w*z + 4*p*s*w^3*z + 4*s^2*w^3*z + p*s^4 + s^5 + 2*p*s^2*w^2 + 2*s^3*w^2 + p*w^4 + s*w^4
which is of course a correct expression, but too expanded for me.
I finished this by hand, but is there a way to automatize this?
I'm too lazy and carefree to manipulate expressions

[Anonymous No.10076774]

>>10076695
Isn't there collectLikeTerms() or something?
t. someone who watched a professor use sage once

[Anonymous No.10077197]

>Starting at a community college with intent to transfer to a 4 year
>highest level math course offered is Differential equations 1
>cost effective uni that I want to transfer too only has only topology, graph theory, and tensor calc as the highest level undergraduate courses.

am I fucked?

[Anonymous No.10077310]

Was doing some discrete logarithms in finite groups and I got

[math]0^0 = 1[/math] mod n

Does 0 raised to 0 in finite groups create singularities? I havn't thought about it too much

[Anonymous No.10077399]

>>10077197
I'm in a somewhat similar situation though I didn't go to CC first. The most advanced courses I've taken are the groups, rings, and fields sequence, analysis, and baby's first complex analysis (skipped a lot of proofs and general rigor because we don't even have a topology course). The flip side is that the professors love to have actually motivated students and will work in depth with you, I've learnt a fair share of representation theory, galois theory, Fourier analysis, algebraic topology, etc this way. Some of this has extended to actual research which is good.

When I've asked how fucked I am it ranges from "completely" to "not at all". The mean opinion seems to be that I probably won't get into a top 10 or whatever but should manage to land in a comfy program.

I'm in my final year now, though I'm only taking a few meme classes I need to graduate and spending most of my time with research and independent study (munkres and rudin atm since my analysis is pretty fucking lacking). Probably won't end up applying until next year because I have some publications in the works that I imagine I'll be relying heavily on.

So I don't think you're fucked per se anon, just be aware that you'll need to really engage with your department and supplement your education manually.

[Anonymous No.10077408]

>>10077310
Which group (family) are you working in? Modular integers under addition?

[Anonymous No.10077452]

>>10077408
multiplication

[Anonymous No.10077454]

>>10077452
Then zero isn't in the group to begin with

[Anonymous No.10077589]

What's a good book on Ordinary Differential Equations?

I don't want brainlet-tier books, but good/hard ones.

[Anonymous No.10077649]

>>10077589
anyone with the words "boundry value" in the title

[Anonymous No.10077762]

Yo fellas, can someone help me out here? I can't be a brainlet that gets stuck on chapter 1 section 1
https://math.stackexchange.com/questions/2959793/hartshorne-exercise-1-9-are-these-answers-correct

[Anonymous No.10077809]

>>10077762
fellas, i indeed am a brainlet, figured out the solution on my own

[Anonymous No.10077846]

>>10077762
Stop bellitling yourself bro, you're not a brainlet if you post on /sci/ and is majoring in math, you're a true warrior who'll overcome extreme hardships and contribute in the future to the advance of mankind.

Remember, Stephen Smale, Field's Medal winner, was terrible at College.

You can do it! We all can!

https://en.wikipedia.org/wiki/Stephen_Smale

[Anonymous No.10077896]

I've started working my way through Spivak's calculus, and I'm getting my butt kicked by the summation problems in chapter 2.

Any recommended resources on summation?

Thanks.

[Anonymous No.10077903]

>>10077896
>summation
Please call them series.
Anyway, read and reread the solutions on the chapter, try to imitate their methods and approaches, and feel free to post some for us to walk you through.

[Anonymous No.10077969]

>>10077903

This is the one that has me stumped: 2.21.c.

I'm not sure how you'd arrive at the generalization
(x1y2 - x2y1)^2 => sigma( (xiyj - xjyi)^2) for i<j
to start with.

I understand the two 'note that' series.

The calculation of the difference, no clue again.
From i different from j, to 2 times some sum running over i<j.
Is the factor of two there because i<j excludes half the "space"
of running over i different from j? (The portion where j<i?).

Where did the xj^2*yi^2 term suddenly pop up from?

[Anonymous No.10078019]

>>10077969
If i=/=j then its clear that there are i terms less than j and i terms greater than j, right? Well then I can split the sum into two parts, one where i<j and one where i>j. But notice that I can swap the roles of i and j on the RHS of the difference equation and the terms remain the same, so what does this tell us? If you split the RHS into two equal sums, then interchange the roles of i and j for one of the sums, and then re-sum, then you get the LHS equation.

[Anonymous No.10078118]

>>10077589
Arnold's book (conveniently titled ODE)

[Anonymous No.10078155]

>>10077969
First, join the sqrts. So
sigma xiyi <= sqrt(sigma xi^2 sigma yi^2).
Pass the sqrt to the other side.
Show that the statement is true by the Schwarz inequality.

[Anonymous No.10079225]

I studied engineering (I know) but like math. Going through Understanding Analysis by Abbot and the Francis Su lectures on youtube.

Is this stuff even worth learning as a hobbyist?

[Anonymous No.10079326]

Who is the strongest mathematician? I bet I can beat Atiyah in a one-on-one fist fight.

[Anonymous No.10079332]

How does the law of cosines fall out of this proof by picture?

[Anonymous No.10079878]

>>10079225
Yes

[Anonymous No.10080007]

Does anyone know how can I get access to the 'The n-th prime is greater than n log n' from Barkley Rosser?

[Anonymous No.10080119]

>>10080007
>Does anyone know how can I get access to the 'The n-th prime is greater than n log n' from Barkley Rosser?
1. google 'The n-th prime is greater than n log n'
2. find a journal's website that hosts the article
3. put that link into https://www dot sci-hub dot tw/

[Anonymous No.10080127]

>>10080007
Step one: the prime counting function behaves like ln(x)
Step two: use the limit definition to show the statement is true for higher than some n.
Step three: Prove manually for the smaller ones lmao.

[Anonymous No.10081568]

>>10079225
Do you enjoy it? If so, what more reason could you need? If not, why would you even bother asking?

[Anonymous No.10081590]

>>10079225
>hobbyist
An amateur, from French amateur "lover of"

[Anonymous No.10081835]

Should I just off myself now?

[Anonymous No.10081871]

tell me something cool about HOMOmorphisms.

[Anonymous No.10081935]

>>10081871
Any ring homomorphism from a field is injective.

[Anonymous No.10081956]

>>10081871
Fields don't have any non-trivial homomorphisms.
A fairly obvious fact but one with far-reaching implications for how field theory is studied.

[Anonymous No.10082263]

can someone explain to me dummy style how you ACTUALLY do analytic continuation? Its been a while since I did complex variables so be gentle UwU.

[Anonymous No.10082478]

Is there an area of math that studies sets of functions related by derivatives and anti derivatives, like 0, 2, 2x, x^2, x^3/3, etc.?

[Anonymous No.10082668]

>>10081835
What's this from?

[Anonymous No.10082962]

>>10082478
What kind of things do you think they would study? I mean you could probably quotient under that equivalence but I don't know if you'd get anything interesting.

[Anonymous No.10083015]

Roughly speaking, how many mathematicians, including college student who took math-related major, are there? Are we fewer than say, physicists in our manpower?

[Anonymous No.10083037]

how does he make such aesthetically pleasing websites, lads?

[Anonymous No.10083057]

>>10082263
There's no algorithmic way to do it. Usually it is along the lines of
>you start with a function that is holomorphic in some region R
>in this region, you can write this function in terms of a function that is holomorphic in a larger region R'
>this can be for example by writing a functional equation of it, or perhaps there is a nice form of its Taylor expansion
>check that this latter expression is equal everywhere on the region R
>identity theorem implies that the original function must be the same function on R
>define the new function to be the analytic continuation of the old function

For example, the series [math]\sum_{n=1}^\infty z^n[/math] converges for [math]|z|<1[/math]. However, we know that this is precisely the geometric series that, in this region that "makes sense", has the formula [math]\frac{1}{1-z}[/math]. Note that the latter formula is holomorphic everywhere but [math]z=1[/math], and anywhere inside the region [math]|z|<1[/math], both functions are holomorphic and equal everywhere. By the identity theorem, they must be the same. Hence you can analytically extend the original to a holomorphic function on [math]\mathbb C- \{1\}[/math]. You can do it in a somewhat more complicated way for the Riemann zeta function (using some functional equation with the Gamma function that takes it 1 step back everytime forever).

[Anonymous No.10083078]

>>10083057
[math] \frac {1} {2+\epsilon } = \frac {1} {1-(-1)\cdot (1+\epsilon) } = \sum_{n=0}^\infty (-1)^n \cdot (1+\epsilon)^n [/math]

[Anonymous No.10083080]

>>10083078
[math] \displaystyle
\frac {1} {2+\epsilon } = \frac {1} {1-(-1) \cdot (1+ \epsilon) } = \sum_{n=0}^ \infty (-1)^n \cdot (1+ \epsilon)^n

[/math]

Optimized

[Anonymous No.10083082]

>>10083078
what's that supposed to mean? Standard convergence tests fail to give a radius of convergence greater than 1 precisely because of the pole at z=1

[Anonymous No.10083440]

>>10083082
then

[math] \displaystyle \frac {1} {2-\epsilon } = \frac {1} {1-(-1) \cdot (1- \epsilon) } = \sum_{n=0}^ \infty (-1)^n \cdot (1- \epsilon)^n [/math]

[Anonymous No.10083709]

Why does the fourier transform of the equation on the top end up as this? I understand that the unit step funtion is where the limits come from, but I don't get why cosine(pi*t) ends up as 2cos(9*pi*t)

[Anonymous No.10083951]

*blocks your path*

[Anonymous No.10084224]

>combining intuitionistic linear logic and constructive S4 logic in natural deduction style yields judgements with 4 contexts
Sheeeeeeeeeeeit

[Anonymous No.10084248]

>>10084224
[math]\Gamma[/math] valid, intuitionistic
[math]\Delta[/math] true, intuitionistic
[math]\Theta[/math] valid, linear
[math]\Xi[/math] true, linear

[Anonymous No.10084395]

>>10083951
>assume your lemma is true when it's not
>be able to prove anything because you reduced ZFC to T
He's a genius

[Anonymous No.10084407]

>>10084395
>assume axioms are true
What’s the problem with this? Is this a new sci meme?

[1 No.10084413]

>>10065343
Finite math is retard level math. Like matrix elimination, basic set theory, little programming in matlab. Your Physics professor prob does not think you're a brainlet and is wondering why you would take Finite over Discrete. Take Discrete.

[Anonymous No.10084471]

>>10083951
Damn, I would do anything to be a part of that brotherhood. It’s so fucking cool.

[Anonymous No.10084487]

>>10084407
There's a difference between assuming an axiom and making up a proof for your lemma so that you can prove abc

[Anonymous No.10084500]

>>10084395
>>assume your lemma is true when it's not
Which lemma are you referring to?

[Anonymous No.10084756]

*calls you a brainlet for doubting the validity of IUT*

[Anonymous No.10084775]

>>10067038
Mathematica actually can export/clipboard to LaTeX, but getting it to look right in context may require tweaking. You can use it though to generate expressions in a more free-form manner and then just copy-paste them into your document.

[Anonymous No.10084812]

>>10075425
get the book how to prove it by velleman

[Anonymous No.10084929]

I can only choose four out of these

Which one do I remove?

[Anonymous No.10084935]

>>10084929
Combinatorics. Discrete math is for faggpts.

[Anonymous No.10084937]

>>10084929
I would say rings/fields, the spaces course, and combinatorics are things you should definitely take.
I cannot really make any suggestion on the last one because "geometry" is a completely fucking meaningless course title. That said, PDEs are super gay so choose carefully.

[Anonymous No.10084943]

>>10084812
>get the book how to prove it by velleman
proof books are a literal meme

[Anonymous No.10085207]

>>10084943
Agreed. Your uni should offer "intro to proofs" or similar anyway.

[Anonymous No.10085341]

>>10085207
>Your uni should offer "intro to proofs" or similar anyway.
No it shouldn't.

[Anonymous No.10085342]

>>10085341
Why not? Or are we larping that we were studying first order logic in middle school?

[Anonymous No.10085372]

>>10085342
>Why not?
Because there's not enough content for an entire course. If someone had to take such a course before taking an actual math course, they would be pointlessly delaying learning more important material. And if someone took such a course during/after an actual math course it would all be redundant.

[Anonymous No.10085383]

>>10085372
Mine was a half-semester course, dunno if that's weird or not

[Anonymous No.10085411]

I know context and judgements and theories like S4 in principle.
Explain why what you point out is interesting

[Anonymous No.10085413]

>>10085383
>Mine was a half-semester course, dunno if that's weird or not
What did you take for the other half of the semester?

[Anonymous No.10085420]

>>10085413
Nothing, it was in addition to my normal course load

[Anonymous No.10085443]

>>10063338
That pastor did nothing wrong. Unironically gas the homos poopoo war now.

[Anonymous No.10085711]

>>10062048
Brainlet here, is it possible for a genus 0 curve to have zeros as in pic attached (hyperbola with line in each quadrant of x,y axes)? Thx me brehs.

[Anonymous No.10085719]

>>10084929
Depends. Geometry is gonna be tough, probably, but quite rewarding, and extremely useful for intuition if you take more advanced geometric courses (like algebraic/differential geometry, possibly topology). Rings and Fields will also probably be a bit tough, depending on the lecturer, however, for anything algebraic that is interesting, this will be an extremely necessary course. Metric spaces is very useful too, especially if you like analysis and/or topology. The rest are interest-based, but PDEs is extremely important if you wanna do just that.

[Anonymous No.10085727]

Does anyone know a good in depth book on Symplectic Geometry? Our lecture is loosely based on the one from Ana Cannas da Silva and I am completely lost after a few lectures already(yes I am a brainlet I know).

[Anonymous No.10085868]

>>10085711
Well, (yx-1)(xy+1) ain't no gen=0 curve. In fact bro, if you look at gen=0 curves in the plane you already know them by gen(deg) formula, at least if you are talking about algebraic curves. You can glue two spheres to get a sphere, but you should normalize. Ok bye.

[Anonymous No.10085879]

>>10085868

Thanks! Yep I understand that the conic sections are derived from literally a conic, but I was unsure if there is some weird transformations that are possible. Kthxsir

[Anonymous No.10085886]

>tropical and arctic semirings
Based algebraicists

[Anonymous No.10085980]

What area of math will the next big field come off of? Will there even be another big field? I get depressed looking at current research and seeing how trifling it is. I want to push things forward desu.

Maybe I should just give up and dedicate myself to category theory autism.

[Anonymous No.10086087]

Suppose that the set of sentences T and P are such that every model is either a model of T or P. Show that a finite D subset of T exists that has exactly the same models as T.

Seems like compactness is the solution. Any insights?

[Anonymous No.10086088]

>>10064531
check out well-orders

or, nonstandard models of N.

[Anonymous No.10086097]

>>10065783
>>10066894

If you believe Brian Conrad it can all be done in ZFC. Not sure about ZF though.

[Anonymous No.10086111]

>>10086097
ah, I forgot, McLarty actually addressed this question: https://math.stackexchange.com/questions/2888541/is-fermats-last-theorem-provable-in-peano-arithmetic

[Anonymous No.10086126]

>>10068636
math

[Anonymous No.10086132]

>>10074804
look up the interval characterized as the universal set equipped with two distinct points.

[Anonymous No.10086159]

Newman's proof of Prime Number Theorem page 707 consider re(z)>0 and re(z)<0. What about re(z)=0?

https://www.maa.org/sites/default/files/pdf/upload_library/22/Chauvenet/Zagier.pdf

[Anonymous No.10086162]

>>10074804
But R is topological? It is the completion of Q at the infinite place.

[Anonymous No.10086267]

>>10077454
>you can't multiply things by 0

It's still a monoid

[Anonymous No.10086339]

>>10084929
Remove PDE and Combinatorics

[Anonymous No.10086401]

>>10086267
Then it's not a group

[Anonymous No.10086808]

>>10085727
Bumping for this

[Anonymous No.10087041]

>>10084929
Get rid of combinatorics and geometry

[Anonymous No.10087091]

>>10084929
Looks like I'm taking Metric, Normed and Hilbert Spaces.

[Anonymous No.10087100]

>>10083951
*steps over the yellow midgets*

[Anonymous No.10087372]

>>10086159
You men in the "proof of analytic theorem section"? They are doing some kind of integral, and the intersection of the curve they're integrating along and Re(z) = 0 is just two points, so the integral in that region will be zero.

[Anonymous No.10087863]

How do I justify simplifying equation (1) here

Can't use wolframalpha as answer and I absolutely do not want to calculate the full stuff

[Anonymous No.10087873]

>>10087863
Also I'd preferred to show that the function of k has a maximum at 0 on the interval [0,r] using a differential argument, but the derivative is nonzero at k = 0 and second derivative is positive

[Anonymous No.10087880]

>>10087863
Is this homework? You generally don't have to justify tedious algebraic manipulations unless you do something weird.

Also you should use \pmod{} not write the word "mod"

[Anonymous No.10087942]

>>10087863
First of all, you don't have an equation (1), learn to TeX. Second of all, stop having a fear of simplistic math and do your own homework.

[Anonymous No.10088017]

>>10087942
it's literally right here

[Anonymous No.10088021]

>>10088017
His point is that if you use the equation environment you'll get numbers done for you and they'll look nicer

[Anonymous No.10088070]

Isn't it possible to guess the right lottery numbers through statistical analysis and it's common frequency + intervals

[Anonymous No.10088082]

>>10088070
wut

[Anonymous No.10088121]

>>10088070
You ever saw a statistician repeatedly won lottery?

[Anonymous No.10088124]

>>10088070
No
You need to use Esoteric Numerology for that

[Anonymous No.10088135]

>>10088121
https://www.dailymail.co.uk/news/article-2023514/Joan-R-Ginther-won-lottery-4-times-Stanford-University-statistics-PhD.html

>Joan R. Ginther, 63, from Texas, won multiple million dollar payouts each time.
>First, she won $5.4 million, then a decade later, she won $2million, then two years later $3million and in the summer of 2010, she hit a $10million jackpot.
>[...]Ms Ginther is a former math professor with a PhD from Stanford University specialising in statistics.

[Anonymous No.10088182]

>>10088135
Then the draw was implemented by brainlets, a random draw is impossible to predict
Also
>unironically citing the dailymail

[Anonymous No.10088422]

Brainlet undergrad here. I'm majoring in a science but i don't know shit about math. Do you guys have a reading list or anything else you recommend I could get? I do really want to be able to do math at a decent level.

[Anonymous No.10088435]

>>10088422

[Anonymous No.10088481]

>>10088435
Thanks

[Anonymous No.10088590]

>>10088435
That anime girl on bottom left is so unnecessary

[Anonymous No.10088789]

>>10088422
>Do you guys have a reading list or anything else you recommend I could get?
High School:
• Euclidean geometry, complex numbers, scalar multiplication, Cauchy-Bunyakovskii inequality. Introduction to quantum mechanics (Kostrikin-Manin). Groups of transformations of a plane and space. Derivation of trigonometric identities. Geometry on the upper half-plane (Lobachevsky). Properties of inversion. The action of fractional-linear transformations.
• Rings, fields. Linear algebra, finite groups, Galois theory. Proof of Abel's theorem. Basis, rank, determinants, classical Lie groups. Dedekind cuts. Construction of real and complex numbers. Definition of the tensor product of vector spaces.
• Set theory. Zorn's lemma. Completely ordered sets. Cauchy-Hamel basis. Cantor-Bernstein theorem.
• Metric spaces. Set-theoretic topology (definition of continuous mappings, compactness, proper mappings). Definition of compactness in terms of convergent sequences for spaces with a countable base. Homotopy, fundamental group, homotopy equivalence.
• p-adic numbers, Ostrovsky's theorem, multiplication and division of p-adic numbers by hand.
• Differentiation, integration, Newton-Leibniz formula. Delta-epsilon formalism.

[Anonymous No.10088791]

>>10088789
Freshman:
• Analysis in R^n. Differential of a mapping. Contraction mapping lemma. Implicit function theorem. The Riemann-Lebesgue integral. ("Analysis" by Laurent Schwartz, "Analysis" by Zorich, "Theorems and Problems in Functional Analysis" by Kirillov-Gvishiani)
• Hilbert spaces, Banach spaces (definition). The existence of a basis in a Hilbert space. Continuous and discontinuous linear operators. Continuity criteria. Examples of compact operators. ("Analysis" by Laurent Schwartz, "Analysis" by Zorich, "Theorems and Problems in Functional Analysis" by Kirillov-Gvishiani)
• Smooth manifolds, submersions, immersions, Sard's theorem. The partition of unity. Differential topology (Milnor-Wallace). Transversality. Degree of mapping as a topological invariant.
• Differential forms, the de Rham operator, the Stokes theorem, the Maxwell equation of the electromagnetic field. The Gauss-Ostrogradsky theorem as a particular example.
• Complex analysis of one variable (according to the book of Henri Cartan or the first volume of Shabat). Contour integrals, Cauchy's formula, Riemann's theorem on mappings from any simply-connected subset C to a circle, the extension theorem, Little Picard Theorem. Multivalued functions (for example, the logarithm).
• The theory of categories, definition, functors, equivalences, adjoint functors (Mac Lane, Categories for the working mathematician, Gelfand-Manin, first chapter).
• Groups and Lie algebras. Lie groups. Lie algebras as their linearizations. Universal enveloping algebra, Poincaré-Birkhoff-Witt theorem. Free Lie algebras. The Campbell-Hausdorff series and the construction of a Lie group by its algebra (yellow Serre, first half).

[Anonymous No.10088806]

>>10088791
Sophomore:
• Algebraic topology (Fuchs-Fomenko). Cohomology (simplicial, singular, de Rham), their equivalence, Poincaré duality, homotopy groups. Dimension. Fibrations (in the sense of Serre), spectral sequences (Mishchenko, "Vector bundles ...").
• Computation of the cohomology of classical Lie groups and projective spaces.
• Vector bundles, connectivity, Gauss-Bonnet formula, Euler, Chern, Pontryagin, Stiefel-Whitney classes. Multiplicativity of Chern characteristic. Classifying spaces ("Characteristic Classes", Milnor and Stasheff).
• Differential geometry. The Levi-Civita connection, curvature, algebraic and differential identities of Bianchi. Killing fields. Gaussian curvature of a two-dimensional Riemannian manifold. Cellular decomposition of loop space in terms of geodesics. The Morse theory on loop space (Milnor's Morse Theory and Arthur Besse's Einstein Manifolds). Principal bundles and connections on them.
• Commutative algebra (Atiyah-MacDonald). Noetherian rings, Krull dimension, Nakayama lemma, adic completion, integrally closed, discrete valuation rings. Flat modules, local criterion of flatness.
• The Beginning of Algebraic Geometry. (The first chapter of Hartshorne or Shafarevich or green Mumford). Affine varieties, projective varieties, projective morphisms, the image of a projective variety is projective (via resultants). Sheaves. Zariski topology. Algebraic manifold as a ringed space. Hilbert's Nullstellensatz. Spectrum of a ring.
• Introduction to homological algebra. Ext, Tor groups for modules over a ring, resolvents, projective and injective modules (Atiyah-MacDonald). Construction of injective modules. Grothendieck Duality (from the book Springer Lecture Notes in Math, Grothendieck Duality, numbers 21 and 40).
• Number theory; Local and global fields, discriminant, norm, group of ideal classes (blue book of Cassels and Frohlich).

[Anonymous No.10088811]

>>10088806
Sophomore (cont):
• Reductive groups, root systems, representations of semisimple groups, weights, Killing form. Groups generated by reflections, their classification. Cohomology of Lie algebras. Computing cohomology in terms of invariant forms. Singular cohomology of a compact Lie group and the cohomology of its algebra. Invariants of classical Lie groups. (Yellow Serre, the second half, Hermann Weyl, "The Classical Groups: Their Invariants and Representations"). Constructions of special Lie groups. Hopf algebras. Quantum groups (definition).

Junior:
• K-theory as a cohomology functor, Bott periodicity, Clifford algebras. Spinors (Atiyah's book "K-Theory" or AS Mishchenko "Vector bundles and their applications"). Spectra. Eilenberg-MacLane Spaces. Infinite loop spaces (according to the book of Switzer or the yellow book of Adams or Adams "Lectures on generalized cohomology", 1972).
• Differential operators, pseudodifferential operators, symbol, elliptic operators. Properties of the Laplace operator. Self-adjoint operators with discrete spectrum. The Green's operator and applications to the Hodge theory on Riemannian manifolds. Quantum mechanics. (R. Wells's book on analysis or Mishchenko "Vector bundles and their application").
• The index formula (Atiyah-Bott-Patodi, Mishchenko), the Riemann-Roch formula. The zeta function of an operator with a discrete spectrum and its asymptotics.
• Homological algebra (Gel'fand-Manin, all chapters except the last chapter). Cohomology of sheaves, derived categories, triangulated categories, derived functor, spectral sequence of a double complex. The composition of triangulated functors and the corresponding spectral sequence. Verdier's duality. The formalism of the six functors and the perverse sheaves.

[Anonymous No.10088832]

>>10088811
Junior (cont):
• Algebraic geometry of schemes, schemes over a ring, projective spectra, derivatives of a function, Serre duality, coherent sheaves, base change. Proper and separable schemes, a valuation criterion for properness and separability (Hartshorne). Functors, representability, moduli spaces. Direct and inverse images of sheaves, higher direct images. With proper mapping, higher direct images are coherent.
• Cohomological methods in algebraic geometry, semicontinuity of cohomology, Zariski's connectedness theorem, Stein factorization.
• Kähler manifolds, Lefschetz's theorem, Hodge theory, Kodaira's relations, properties of the Laplace operator (chapter zero of Griffiths-Harris, is clearly presented in the book by André Weil, "Kähler manifolds"). Hermitian bundles. Line bundles and their curvature. Line bundles with positive curvature. Kodaira-Nakano's theorem on the vanishing of cohomology (Griffiths-Harris).
• Holonomy, the Ambrose-Singer theorem, special holonomies, the classification of holonomies, Calabi-Yau manifolds, Hyperkähler manifolds, the Calabi-Yau theorem.
• Spinors on manifolds, Dirac operator, Ricci curvature, Weizenbeck-Lichnerovich formula, Bochner's theorem. Bogomolov's theorem on the decomposition of manifolds with zero canonical class (Arthur Besse, "Einstein varieties").
• Tate cohomology and class field theory (Cassels-Fröhlich, blue book). Calculation of the quotient group of a Galois group of a number field by the commutator. The Brauer Group and its applications.
• Ergodic theory. Ergodicity of billiards.
• Complex curves, pseudoconformal mappings, Teichmüller spaces, Ahlfors-Bers theory (according to Ahlfors's thin book).

[Anonymous No.10088836]

>>10088832
Senior:
• Rational and profinite homotopy type. The nerve of the etale covering of the cellular space is homotopically equivalent to its profinite type. Topological definition of etale cohomology. Action of the Galois group on the profinite homotopy type (Sullivan, "Geometric topology").
• Etale cohomology in algebraic geometry, comparison functor, Henselian rings, geometric points. Base change. Any smooth manifold over a field locally in the etale topology is isomorphic to A^n. The etale fundamental group (Milne, Danilov's review from VINITI and SGA 4 1/2, Deligne's first article).
• Elliptic curves, j-invariant, automorphic forms, Taniyama-Weil conjecture and its applications to number theory (Fermat's theorem).
• Rational homotopies (according to the last chapter of Gel'fand-Manin's book or Griffiths-Morgan-Long-Sullivan's article). Massey operations and rational homotopy type. Vanishing Massey operations on a Kahler manifold.
• Chevalley groups, their generators and relations (according to Steinberg's book). Calculation of the group K_2 from the field (Milnor, Algebraic K-Theory).
• Quillen's algebraic K-theory, BGL^+ and Q-construction (Suslin's review in the 25th volume of VINITI, Quillen's lectures - Lecture Notes in Math. 341).
• Complex analytic manifolds, coherent sheaves, Oka's coherence theorem, Hilbert's nullstellensatz for ideals in a sheaf of holomorphic functions. Noetherian ring of germs of holomorphic functions, Weierstrass's theorem on division, Weierstrass's preparation theorem. The Branched Cover Theorem. The Grauert-Remmert theorem (the image of a compact analytic space under a holomorphic morphism is analytic). Hartogs' theorem on the extension of an analytic function. The multidimensional Cauchy formula and its applications (the uniform limit of holomorphic functions is holomorphic).

[Anonymous No.10088843]

>>10088836
Specialist: (Fifth year of College):
• The Kodaira-Spencer theory. Deformations of the manifold and solutions of the Maurer-Cartan equation. Maurer-Cartan solvability and Massey operations on the DG-Lie algebra of the cohomology of vector fields. The moduli spaces and their finite dimensionality (see Kontsevich's lectures, or Kodaira's collected works). Bogomolov-Tian-Todorov theorem on deformations of Calabi-Yau.
• Symplectic reduction. The momentum map. The Kempf-Ness theorem.
• Deformations of coherent sheaves and fiber bundles in algebraic geometry. Geometric theory of invariants. The moduli space of bundles on a curve. Stability. The compactifications of Uhlenbeck, Gieseker and Maruyama. The geometric theory of invariants is symplectic reduction (the third edition of Mumford's Geometric Invariant Theory, applications of Francis Kirwan).
• Instantons in four-dimensional geometry. Donaldson's theory. Donaldson's Invariants. Instantons on Kähler surfaces.
• Geometry of complex surfaces. Classification of Kodaira, Kähler and non-Kähler surfaces, Hilbert scheme of points on a surface. The criterion of Castelnuovo-Enriques, the Riemann-Roch formula, the Bogomolov-Miyaoka-Yau inequality. Relations between the numerical invariants of the surface. Elliptic surfaces, Kummer surface, surfaces of type K3 and Enriques.
• Elements of the Mori program: the Kawamata-Viehweg vanishing theorem, theorems on base point freeness, Mori's Cone Theorem (Clemens-Kollar-Mori, "Higher dimensional complex geometry" plus the not translated Kollar-Mori and Kawamata-Matsuki-Masuda).
• Stable bundles as instantons. Yang-Mills equation on a Kahler manifold. The Donaldson-Uhlenbeck-Yau theorem on Yang-Mills metrics on a stable bundle. Its interpretation in terms of symplectic reduction. Stable bundles and instantons on hyper-Kähler manifolds; An explicit solution of the Maurer-Cartan equation in terms of the Green operator.

[Anonymous No.10088845]

>>10088843
Specialist (cont):
• Pseudoholomorphic curves on a symplectic manifold. Gromov-Witten invariants. Quantum cohomology. Mirror hypothesis and its interpretation. The structure of the symplectomorphism group (according to the article of Kontsevich-Manin, Polterovich's book "Symplectic geometry", the green book on pseudoholomorphic curves and lecture notes by McDuff and Salamon)
• Complex spinors, the Seiberg-Witten equation, Seiberg-Witten invariants. Why the Seiberg-Witten invariants are equal to the Gromov-Witten invariants.
• Hyperkähler reduction. Flat bundles and the Yang-Mills equation. Hyperkähler structure on the moduli space of flat bundles (Hitchin-Simpson).
• Mixed Hodge structures. Mixed Hodge structures on the cohomology of an algebraic variety. Mixed Hodge structures on the Maltsev completion of the fundamental group. Variations of mixed Hodge structures. The nilpotent orbit theorem. The SL(2)-orbit theorem. Closed and vanishing cycles. The exact sequence of Clemens-Schmid (Griffiths red book "Transcendental methods in algebraic geometry").
• Non-Abelian Hodge theory. Variations of Hodge structures as fixed points of C^*-actions on the moduli space of Higgs bundles (Simpson's thesis).
• Weil conjectures and their proof. l-adic sheaves, perverse sheaves, Frobenius automorphism, weights, the purity theorem (Beilinson, Bernstein, Deligne, plus Deligne, Weil conjectures II)
• The quantitative algebraic topology of Gromov, (Gromov "Metric structures for Riemannian and non-Riemannian spaces"). Gromov-Hausdorff metric, the precompactness of a set of metric spaces, hyperbolic manifolds and hyperbolic groups, harmonic mappings into hyperbolic spaces, the proof of Mostow's rigidity theorem (two compact Kählerian manifolds covered by the same symmetric space X of negative curvature are isometric if their fundamental groups are isomorphic, and dim X> 1).
• Varieties of general type, Kobayashi and Bergman metrics, analytic rigidity (Siu)

[Anonymous No.10089621]

>>10074804
Most compact definition is in homotopy type theory, there it's a path between paths, and path is a primitive notion.