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9401530 No.9401530 [Reply] [Original] [archived.moe]

Talk maths

The Work of Robert Langlands:


Previous thread >>9394546

>> No.9401535
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What's the worst mathematical Wikipedia page?


>> No.9401926

sup piggots

>> No.9401930
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why exactly is a linear transformation defined with these two properties in mind (over real vector spaces):

[math]T(\mathbf{u}+\mathbf{v}) = T(\mathbf{u})+T(\mathbf{v})[/math]
[math]T(c\mathbf{u}) = cT(\mathbf{u})[/math]

>> No.9401943

How else would you define it?

>> No.9401944

>why exactly is a linear transformation defined with these two properties in mind (over real vector spaces):
That's how it's defined for all vector spaces, not just real ones.

The only things you can do in a vector space are add vectors or scale vectors, so linear transformations are defined to "preserve" that structure.

This structure-preserving property is very useful when considering functions between two algebraic structures of the same type:

>> No.9401957
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>> No.9402008

like >>9401944 said, whenever you have some kind of structure, you should investigate maps which preserve this structure in some sense

in this case, it follows that every linear map can be represented by a matrix and composition of maps becomes matrix multiplication. this makes linear maps extremely easy to analyze. linear maps also have very clear geometric meaning: they are rotations, reflections, shears and scalings (i.e. the things you can do at photoshop).

>> No.9402021

>every linear map can be represented by a matrix

>> No.9402031

>thinks you can describe translations by a linear map.

>> No.9402044

Where was that implied?

>> No.9402050

>(i.e. the things you can do at photoshop).

>> No.9402075
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>> No.9402078

literally the easiest maths

>> No.9402079

Linear algebra is one of the simplest mathematical topics. Just because you can't handle taking an ordered basis doesn't mean you should insult La-chan.

>> No.9402080

i have no idea how to change coordinates.

>> No.9402082

Why are you even at college if you can't even understand linear algebra? Why waste your money like that?

>> No.9402083

To the anon here >>9401135 I answered your question here >>9402063.

>> No.9402107

if you have a basis [math]\{v_i\}_{i\leq n}[/math] and want to change it to a basis [math]\{w_k\}_{k\leq n}[/math], then in particular you can write any vector [math]w_k[/math] in terms of sums of the [math]v_i[/math]. But then you have a system [math]w_k = a_{1,k}v_1 + a_{2,k}v_2+...+a_{n,k}v_n[/math] for every[math]k[/math]. So you can write the matrix of the transformation in terms of the [math]a_{i,j}[/math], and this is your transformation matrix to change any coordinate in terms of the [math]v_i[/math] to coordinates in the basis [math]w_k[/math].

>> No.9402111
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linear algebra depends on the teacher

you can have either an autist that wants all proofs or someone that plugs and chugs into systems of linear equations for u engi majors

pic is a problem on the final that fucked pretty much everyone so he dropped it rofl

>> No.9402139

Why is/are there no linear transform(s) to get the transpose?

>> No.9402143

>Why is/are there no linear transform(s) to get the transpose?
What do you mean? The transpose is a linear transformation.

>> No.9402145 [DELETED] 

Aporogees for poor Engrish..

Why is there no way to find B such that

[math]A*B = A^T[math]

>> No.9402146

Aporogees for poor Engrish..

Why is there no way to find B such that

[math]A*B = A^T[/math]

>> No.9402148

Really? Did he not cover basic matrix factorizations in class, then?

(Though he should have put some quantifier on "n". As, "for any positive integer n".)

>> No.9402152

Is diagonalizing a matrix considered hard on Amerifatland?

>> No.9402158

Given square matrices [math]A,B[/math] and an inverse [math]M[/math] for [math]1-AB[/math], show that there is an inverse for [math]1-BA[/math] expressed in terms of [math]A,B,M[/math].

>> No.9402160
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I want to start learning more about discrete stuff / combinatorics. I have a strong background in differential geometry and functional analysis / PDE. Is it possible to use this knowledge to my advantage? Are there scenarios in combinatorics where methods / intuition from the aforementioned fields can be applied?

>> No.9402161

What have you tried?

>> No.9402162

I know how to solve it. This is a nice exercise for you guys. You can try convincing yourself that 1-AB is invertible if and only if 1-BA is invertible; that can be done abstractly without expressing the inverse for one in terms of A,B and the inverse for the other.

>> No.9402163

I'm not a "guy".

>> No.9402164


its more of wtf is the question asking

>> No.9402167

That's a very simple problem, once you know that every self-adjoint matrix is diagonalizable.

>> No.9402169

faggot mentally ill nigger bitch ass

>> No.9402170

>Given square matrices A,B and an inverse M for 1−AB, show that there is an inverse for 1−BA expressed in terms of A,B,M.
Please no homework in this thread

>> No.9402171

>pic is a problem on the final that fucked pretty much everyone so he dropped it rofl
Which school for brainlets do you go to?

>> No.9402173

1-BA trivially has inverse BA-1, no need for M.

>> No.9402174

This is apparently an interview question from Microsoft.

>> No.9402178

Multiplicative inverse is what's asked for, not additive.

>> No.9402179

>faggot mentally ill nigger bitch ass
Are you okay?

>> No.9402182

saint louis university in stl

>> No.9402209

The closest interactions with combinatorics those fields have is ergodic theory. Mainly the representation theory of discrete amenable groups has some nice connections with geometric group theory and ergodic theory both of which are connected to combinatorics. There is some more abstract work that is connected with combinatorics under the guise of operads, specifically through Stasheff polytopes.

>> No.9402210

How old are you?

>> No.9402310
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>> No.9402322

If * is matrix product then it definitely is possible sometimes, like when A is invertible. Can you be more specific? Maybe you mean *when* is there no way?

>> No.9402365

This is why brainlets should stay away from universities.

>> No.9402396
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Thanks for the laugh

>> No.9402397
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I've seen the light. After journeying to hell and back, and mustering every last IQ point I have, I now understand where I went wrong. Forgive me, linear algebra. Take me back into your embrace.

>> No.9402421


If you're that brainlet who was struggling with LA, check this playlist out for a nice intuition based introduction to LA.It's pretty good and not too long either.


>> No.9402433
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Are these the most challenging maths textbooks of all time?

>> No.9402449

Are those TAOCPs even worth the time?

>> No.9402576


>> No.9402672

Why are physishits so retarded?

>> No.9402674

>solving other people's homework for free

>> No.9402681

lel gb2 >>>/g/

>> No.9402686

Why the homophobia?

>> No.9402855

No, IUT is

>> No.9402858

Good lord it's actually real

>> No.9402904


>> No.9402918

For small real numbers a and b
Be inspired therefrom.

>> No.9402919

I'm not your guy, buddy.

>> No.9402920

Everyone else who answered this is educated stupid; I'll tell you the real reason because this is a good question not left to undergrads who just memorized this 2 years ago or whatever. Linear transformations are defined this way because it is just the precise way of saying "Knowing how the basis vectors transform tells you how ALL vectors in your space transform."

>> No.9402925

>>thinks you can describe translations by a linear map.
you can tho

>> No.9402928

[eqn]\begin{bmatrix}a & b & r \\ c & d & s \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix}x \\ y \\ 1\end{bmatrix} = \begin{bmatrix} ax+by+r \\ cx+dy+s \\ 1\end{bmatrix} [/eqn]

not that other guy but you can represent translations with matrix multiplication, this is called homogeneous coordinates where you look at translations as being rotations restricted to a surface.

>> No.9402934 [DELETED] 

No, you need an affine map. Brainlet.

>> No.9402959

[math](1-BA)B = B(1-AB)[/math], hence [math](1-BA)B(1-AB)^{-1} = B[/math] and [math](1-BA)B(1-AB)^{-1}A = BA[/math].
Finally, [math]1-BA + (1-BA)B(1-AB)^{-1}A = (1-AB)(1+B(1-AB)^{-1}A) = 1[/math].
It's easy to check that [math]1+B(1-AB)^{-1}A[/math] is also a left inverse.

>> No.9402978

So that it is a homomorphism for the operations of vector spaces.
Generally a homorphism is a map between algebraic structures A and B (f:A-->B) with which:
computing in A and then sending in B
is the same as
sending in B and computing in B

Why study homomorphisms?
Because with them you can study part of B from the point of view of A, and vice versa.
Partition A in cells where in each cell you have elements that are sent to the same element of B. i.e. a1,a2 are in the same cell whenever f(a1)=f(a2).
It is possible define "new" operations on these cells: (Cell where a1 is) * (Cell where a2 is) = (Cell where a1*a2 is) and this operation is independent of which elements of the cell you picked.
The cells along with those operations form a structure which is the same (except in names) as the image of A under f ( f(A) ).
This is called "First Isomorphism theorem".

>> No.9402990

Does anyone have the wolframalpha android app?
Is there a point buying it or is it the same as using the browser version?

>> No.9402998

Still not technically a linear map, since you always need the last component to be 1. They're elements of the projective general linear group on R^n, which is the quotient group of GL(R^n+1) by scalar multiplication (isomorphic to R*).

>> No.9403012

Of course it is a linear map, but it is not a translation in all of R^3. Still, it restricts to a translation on the plane {z=1}

>> No.9403015

In the same vein, but easier: Let A and B be square matrices such that [math]A+B=AB[/math]. Prove that A and B commute

>> No.9403035

Oh...you mean a B that works for all A.

If you set A = I then you get B = I so obviously that won't work.

>> No.9403087

it's asking you to prove that n-th root of a matrix exists if the matrix is hermitian and positive semidefinite.

>> No.9403123

This is a REALLY simple problem. It's there to test if you know the real spectral theorem.
Your class is retarded / you professor did a terrible job at teaching.

>> No.9403159
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no point, he has enough shekels already

>> No.9403203

Linear maps are defined on vector spaces, which {z=1} is not. It's the restriction of a linear map on an affine subset.

>> No.9403209

>No, IUT is
IUT is a series of papers, not a textbook.

>> No.9403217

>Linear transformations are defined this way because it is just the precise way of saying "Knowing how the basis vectors transform tells you how ALL vectors in your space transform."
But that's not true at all, linear transformations are still defined that way even for vector spaces that don't have a basis.

>> No.9403321

meant to reply to >>9403012

>> No.9403323
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>vector spaces that don't have a basis
I sense a rain of pro-AC posts incoming.

>> No.9403347
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>vector spaces that don't have a basis
In non-retarded circles "vector space" means "free module over a field". Perhaps you meant to say that they were defined the same way for all homomorphisms of modules?

>> No.9403350

>vector spaces that don't have a basis
autism or ignorance

>> No.9403354

he means that defining linear transformations in infinite dimension is independent of AC, clearly

>> No.9403356

I'm not a "he".

>> No.9403359

>autism or ignorance
Speak for yourself.

>> No.9403364

shut the fuck up, faggot

>> No.9403367

>In non-retarded circles "vector space" means "free module over a field".
You meant "module over a field".

>> No.9403369

>independent of AC
Yes, "every free module over a field is free" is independent of AC. Your point?
No, I meant "free module over a field".

>> No.9403381

>No, I meant "free module over a field".
Then your statement is not true.

Find one (1) source that defines a vector spaces as such.

>> No.9403386

Why the homophobia?

>> No.9403401

>Find one (1) source that defines a vector spaces as such.
The book I'm currently writing.

>> No.9403439

I'm not a "homophobe".

>> No.9403456

Because faggots are not human.

>> No.9403481
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But with AC comes a basis, and so that wording is there to tell xį does not require choice to be axiomatically true. Your interpretation of žůr post is incorrect.

>> No.9403490

What are your preferred axioms?

>> No.9403501
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Axiom of Equality: every human is to be treated the same way.
Axiom of Infinity: there is an infinite amount of genders.
Axiom of Racial Purity: only white people (at least 57%) are to be considered human.

>> No.9403519
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Ps. I'm not a "you". Please refer to me as "thou" from now on.

>> No.9403529

The second two can be derived from the axiom of faggotry

>> No.9403543
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Why the homophobia? You should try homotopia instead. Just imagine you were sitting on the lap of some nice guy explaining him Quillen's model categories work, and he would then reward you with an intense kiss. So much more fun!

>> No.9403573 [DELETED] 
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i know i'm retarded but i can't for the life of me get this same answer for the lcm

>> No.9403583

>i know i'm retarded but i can't for the life of me get this same answer for the lcm
What do you get?

>> No.9403586 [DELETED] 


>> No.9403590

oh fuck me i'm so stupid i realized what i've done.
i blame the calculator interface it confused me. nevermind i'll delete my posts now

>> No.9403591

I don't understand. What's the point of studying LCD's in the real numbers? The real numbers have no non-trivial divisibility structure. Everything divides everything. What the fuck?

>> No.9403597

it's a trick question. The answer is [math]\forall\epsilon>0:\epsilon[/math]

>> No.9403600

Is it really? Because it looked like a high school problem. I just thought it was shitty math education teaching retarded bullshit as usual.

>> No.9403601

>log in
>no new yous
>log out

>> No.9403657
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I'm not sure what this question wants exactly. Is it something like, e.g. for (1, 1)-tensors [eqn]
\mathbf{T} = \left\{\begin{align}
& \mathbf{x} \in V_n \mapsto (\mathbf{\omega} \in V_n^* \mapsto T_j^i\ \mathbf{e}_i \otimes \mathbf{\theta}^j (\mathbf{\omega}, \mathbf{x})
\in \text{Hom}_\text{Vect}(V^*_n, \mathbb{R}) = V^{**}_n = V_n) \in \text{Hom}_\text{Vect}(V_n, V_n) \\
& \mathbf{\omega} \in V^*_n \mapsto (\mathbf{x} \in V_n \mapsto T_j^i\ \mathbf{e}_i \otimes \mathbf{\theta}^j (\mathbf{\omega}, \mathbf{x})
\in \text{Hom}_\text{Vect}(V_n, \mathbb{R}) = V^{*}_n) \in \text{Hom}_\text{Vect}(V^*_n, V^*_n) \\
\right. \\
\therefore \mathbf{T} \in \text{Hom}_\text{Vect}(V_n, V_n) \cup \text{Hom}_\text{Vect}(V^*_n, V^*_n)
[/eqn] and then I do something like that for all the other (r, s)?

>> No.9403763


>> No.9403776

>Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, once the concept is translated into projective geometry's terms.

What the fuck. How is this possible?

>> No.9403787

>What the fuck.
Do you need to swear?

>> No.9403789

yes. how can two lines that are parallel "meet", especially "at" """"""infinity"""""""""? that makes no sense.

>> No.9403801

is there an extension of linear algebra to "non" linear transformations?

>> No.9403805

>is there an extension of linear algebra to "non" linear transformations?

>> No.9403829

Pretty much algebraic geometry.

>> No.9403838

What do I need to approach Langlands program?

>> No.9403846

algebraic number theory, algebraic geometry, functional analysis, representation theory

>> No.9403852

What is calculus and why do we need it?

>> No.9403857
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see pic

>> No.9403941

>functional analysis

>> No.9403946

Infinite dimensional representations of Lie groups come into play

>> No.9403984

I'd just like to interject for a moment. What you're referring to as calculus, is in fact, real analysis, or as I've recently taken to calling it,
[math]\Bigg(\mathbf{R},+,\times, \leq, |\cdot|,\tau = \{ A\subset \mathbf{R}\hspace{0.1cm} | \hspace{0.1cm}\forall x \in A, \exists \epsilon > 0 ,\hspace{0.1cm} ]x-\epsilon,x+\epsilon[\hspace{0.1cm} \subset A \},\hspace{0.1cm} \displaystyle \bigcap_{\substack{\text{A} \hspace{0.1cm}\sigma-\text{algebra of}\hspace{0.1cm}\mathbf{R}\\
\tau \subset A}}A , \hspace{0.1cm}\mathscr{L}\Bigg) [/math] -analysis. Calculus is not a branch of mathematics unto itself, but rather another application of a fully functioning analysis made useful by topology, measure theory and vital [math]\mathbf{R}[/math]-related properties comprising a full number field as defined by pure mathematics.

Many mathematics students and professors use applications of real analysis every day, without realizing it. Through a peculiar turn of events, the application of real analysis which is widely used today is often called "Calculus", and many of its users are not aware that it is merely a part of real analysis, developed by the Nicolas Bourbaki group.

There is really a calculus, and these people are using it, but it is just a part of the filed they use. Calculus is the computation process: the set of rules and formulae that allow the mathematical mind to derive numerical formulae from other numerical formulae. The computation process is an essential part of a branch of mathematics, but useless by itself; it can only function in the context of a complete number field.
Calculus is normally used in combination with the real number field, its topology and its measured space: the whole system is basically real numbers with analytical methods and properties added, or real analysis.
All the so called calculus problems are really problems of real analysis.

>> No.9404009

This is overcomplicating things.

Those are a finite amount of finite tools trying to separate the inseparable in various ways.

>> No.9404012

it's a meme you dip

>> No.9404397

If you stand on a railroad and look straight ahead, you will see that the rails "meet" a long distance away in the horizon. Projective geometry deals with this notion

>> No.9404404

Yes, simply consider all functions instead of just linear transformations.

>> No.9404405
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i see.

>> No.9404424

Please refrain from posting black people here.

>> No.9404427
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>> No.9404429

IMO calculus is much more about linear transformations than real analysis.

Equations like:

[eqn]\int^b_a f(x) g(x) dx + \int^b_a k(x, y) h(x) dx = 0[/eqn]

are basically similar to tensor equations like:

[eqn]A_{\alpha} B_{\alpha} + C_{\alpha\beta} D_{\alpha}[/eqn]

>> No.9404437

But it's always true for finitely dimensional spaces

>> No.9404439

Are functions of the form:

[eqn] \sum^{\infty}_{k = 0} c_k \frac{\Gamma(x + k - a)}{\Gamma(x - a)} [/eqn]

real analytic?

Is there a formula for converting between this form and a polynomial series?

>> No.9404443

The Schwartz kernel theorem also shows the same thing for integral transforms.

>> No.9404462

watch n j wildberger math history projective geometry very helpful and intuitive

>> No.9404526

invariance under action
estupido, this is one of many consequences
low iq

>> No.9404591

all me

>> No.9404620
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>tfw really struggling with the theory of L^p spaces.

Should I just end it all? I get physically sick when doing relevant exercises and it's anything more advanced than some super simple Hölder application.

>> No.9404632


>> No.9404781


>> No.9404805
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I'm in highschool, which math topics should i focus on other than Calculus?
Discrete math?

>> No.9404806

Introduction to mathematical proofs and reasoning

>> No.9404809

What's discrete math?
I would say trigonometry or algebra in general. Trying linear algebra will put you ahead, it's really easy.

>> No.9404814


>> No.9404818

You don't need proofs unless you're gonna major in math.

>> No.9404866

redpill me on topos

>> No.9404891
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>> No.9404895

Just keep practicing, that stuff gets easy. Just remember Holder, Egorov, Jensen, convolutions, things like [math] |g| = |g|^{1/p} |g|^{1/q}[/math], and that if you want to show that a sequence goes to zero a.e., you can show that the integral of its infinite sum is finite. Among other things.

>> No.9404898

What are the simplest examples of [math]\infty[/math]-groupoids?

>> No.9404908

underage b&
Start with some linear algebra and basic proofs

>> No.9404918
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Discrete math is topics like introductory Logic, Set Theory, Combinatorics and so on
I plan on majoring in math, but I'm not 100% sure, might go for compsci or physics
Isn't linear algebra very hard?

>> No.9404935

That's some nice point of view - actually, linear ODEs behave like systems of linear equations..
But dealing with integral and differential equations also relies heavily on topological (resp. "analytical") arguments for many finite-dimensional arguments don't work any more...

>> No.9405007

>Discrete "math"
It's not a field of math strictly speaking, much like "combinatorics". It's best to ignore it entirely and study logic and set theory in their own right.
>Isn't linear algebra very hard?
No. You'll have to learn it anyway since it's used everywhere even outside of mathematics.

>> No.9405008


>> No.9405202

Linear Algebra for sure.

>> No.9405206

Is blocks the correct term? I don't remember. I am not anglo.

>> No.9405309

>study logic and set theory

>> No.9405317

'Coset' is the word you're looking for.

>> No.9405394

But, isn't coset a term for groups only?

>> No.9405418

A topos is the natural model for intuitionist higher-order logic.

>> No.9405420

The fundamental infinity-groupoid.

>> No.9405426

Vector spaces are groups under addition.

If you just have a set, then the word should be "partition".

>> No.9405446

Not exactly. The partition is the set of all equivalence classes
You can call them equivalence classes/classes/blocks. But cosets are a special case.

>> No.9405497

Nop, whole books just use linear algebra,few calculus and some easy combinatorics.

As textbook algebraic geometry hartshorne will be more hard book for students.

>> No.9405599
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>[math]V_n^{**} = V_n[/math]
Only if [math]V_n[/math] is finite dimensional. Also replace [math]n[/math] by [math]x[/math] to denote tangent vector spaces [math]T_xM[/math] at the point [math]{\bf x}[/math], on which this [math](r,s)[/math]-tensor business makes sense.
>[math]{\bf e}_i[/math]
What space is this basis for? [math]T[/math] should act on the basis for [math]V_n[/math] and the basis for [math]V_n^*[/math] at the same time, not separately. You need to put what you've written down together.
>[math]{\bf T} \in \operatorname{End}(V_n)
\cup \operatorname{End}(V_n^*)[/math]
No. You need the exterior algebra.
>How is this possible?
Have you never looked at a photograph in your life before?
General Volterra equations of the first kind can't be cast into an ODE tho. One way to construct solutions is via the resolvent map, which exists depending on the topology of your function space.
Expand [math]\frac{1}{\Gamma}[/math] in its Laurent series and you'll see that at specific points [math]x \in \mathbb{R}[/math] it acquires infinitely many negative powers.
>Is there a formula for converting between this form and a polynomial series?
You mean formal Laurent series? Use the Cauchy integral formula.

>> No.9405604
File: 1.08 MB, 947x941, unknown.png [View same] [iqdb] [saucenao] [google] [report]

>>[math]{\bf T} \in \operatorname{End}(V_n)
>\cup \operatorname{End}(V_n^*)[/math]
Meant [math]{\bf T} \in \operatorname{End}(V_n)\cup \operatorname{End}(V_n^*)[/math].

>> No.9405620

Every infinity groupoid is equivalent to a fundamental infinity groupoid.

>> No.9405666

True, but I was talking about a general algebraic structure.

>> No.9405693
File: 32 KB, 600x600, quantum-systems-channels-information.jpg [View same] [iqdb] [saucenao] [google] [report]

I'm studying quantum information theory using pic related. Anyone ever study this before? Not sure where to go. Would like to perhaps study more general C*Algebras as I have covered a lot of that topic.

I've also heard of quantum information having ties to geometry.

>> No.9405698

this picture is funny because it's true

>> No.9405757

You are comparing notation for abstract theory and notation for computation.

Topologists are as messy when it comes to computation (i.e. spectral sequences)

>> No.9405924


>> No.9405932

but that's affine and not linear transformation

>> No.9405933

[math]\[\left| - \right|:\operatorname{Kan} \rightleftarrows \operatorname{Top} :\operatorname{Sing} \][/math] is a Quillen equivalence in the standard model structures of the two categories.

>> No.9405935

[math]\left| - \right|:\operatorname{Kan} \rightleftarrows \operatorname{Top} :\operatorname{Sing} [/math]

>> No.9405951

norman pls go

>> No.9405952

It's not untrue, it's just redundant since module over a field is always free

>> No.9405955

>it's just redundant
No, it's simply incorrect.

>> No.9405961

It's actually not a bad definition. It allows you to do linear algebra without globally assuming the axiom of choice. He's still being an ass about it though.

>> No.9405965

>It's actually not a bad definition. It allows you to do linear algebra without globally assuming the axiom of choice.
But he/she did assume the axiom of choice.

>> No.9405977

>It's actually not a bad definition.
It's bad because now it becomes extremely difficult to tell whether certain modules over a field are vector spaces or not.

>> No.9405982
File: 79 KB, 700x700, nomizi.jpg [View same] [iqdb] [saucenao] [google] [report]

There's countably many finite cardinalities (0,1,...).

Are there countable many infinite cardinalities? (ordered by power sets?)

>> No.9405988
File: 84 KB, 378x252, contradict.png [View same] [iqdb] [saucenao] [google] [report]

There are countable many countable infinite cardinalities but uncountably many infinite cardinalities in general.
Uhh I mean
>infinite cardinality
No such thing exists xDdxXDd

>> No.9405992

>There are countable many countable infinite cardinalities
Isn't there only one countable infinity?

>> No.9405996

[math]\omega + 1,\omega + 2 ,\dots[/math]

>> No.9406004
File: 27 KB, 1280x720, maxresdefault.jpg [View same] [iqdb] [saucenao] [google] [report]

Do those not have the same cardinality?

>> No.9406012
File: 98 KB, 250x312, file.png [View same] [iqdb] [saucenao] [google] [report]

Cardinals, ordinals, what's tha differenece???

>> No.9406015

they have

>> No.9406235

Finite sets are countable, little one.

>> No.9406834
File: 42 KB, 645x729, tfw no brain.png [View same] [iqdb] [saucenao] [google] [report]


>> No.9406989

Just don't use arbitrary modules over a field, that should be fine for you and your kind since you have already decided to lose generality by working over a field in the first place.

>> No.9406991

>But he/she did assume the axiom of choice.
I didn't though. You could even assume its negation and that would still be the correct definition of "vector space".

>> No.9407061

>it becomes extremely difficult to tell whether certain modules over a field are vector spaces or not

Yes, which is something you would need to prove explicitly in a constructive universe. This is the point of constructivism, it forces you to prove things in a constructive/more informative way.

>> No.9407347

So I just learned about [math] Li(x) [/math] from my number theory book. It's applications to number theory are fun, but what I want to know now is what new integrals can I find antiderivatives for using this beast?

>> No.9407485

How can I get into relation theory? Something better than Halmo's pls

>> No.9407594
File: 87 KB, 1200x692, 1514477693529.jpg [View same] [iqdb] [saucenao] [google] [report]

H-how do you know so much of so many different fields of maths? Or am I just retarded?

>> No.9407638

Finishing college

>> No.9407640

Not him but that's pretty standard info for undergrads and first year grad students in math. Basic facts about manifolds and dual vector spaces are encountered at the undergrad and grad level, so is complex analysis, the only one that isn't standard is the ODE stuff, unless you're doing applied math

>> No.9407647


>> No.9407715

How much maths should I know to stop hating myself for being retarded?

>> No.9407729

Jesus fucking Christ

>> No.9407731

>Jesus fucking Christ
Are you okay?

>> No.9407739

No. I want this meme to stop. We know you are just pretending to not be male. I bet you aren't even actually trans in real life.

>> No.9407741

>No. I want this meme to stop. We know you are just pretending to not be male. I bet you aren't even actually trans in real life.
But this isn't about me, it's about her.

>> No.9407747


>> No.9407748


>> No.9407753

>I bet you aren't even actually trans in real life.
He is pretty pathetic, so I wouldn't be surprised at him being "tr*ns".

>> No.9407757


>> No.9407763

That's a bit of a bizarre question.

>> No.9407768

Here's a problem that I'm struggling with:
find absolutely integrable function f: R->R, such that f is zero outside the interval [-1, 1], and the convolution of f with itself is constant on interval [-1, 1].
Is there any such function except for zero?

>> No.9407796
File: 22 KB, 600x450, azu64.jpg [View same] [iqdb] [saucenao] [google] [report]

No amount will suffice. It is not related to anything mathematical.

>Jesus fucking Christ
Why the blasphemy?

>> No.9407827

>No amount will suffice. It is not related to anything mathematical.
How about the amount of maths I need to know so that I don't fear people considering me a fucking retard?

>> No.9407849
File: 330 KB, 719x512, haha.png [View same] [iqdb] [saucenao] [google] [report]

That amount is obviously related to the population you surround yourself with. On the other hand, the fact that you are even asking a question like this makes me think you will most likely be permanently percieved as a retard.

>> No.9407851

Why the vulgarity?

>> No.9407853
File: 15 KB, 662x69, vice corrects pronouns.jpg [View same] [iqdb] [saucenao] [google] [report]

When did you ask for my pronouns shitlord?

>> No.9407859
File: 245 KB, 512x512, weird toy.png [View same] [iqdb] [saucenao] [google] [report]

A finished course in Algebraic Geometry, then you're slightly less retarded.

IUT if you want the hate to stop.

>> No.9407904

>On the other hand, the fact that you are even asking a question like this makes me think you will most likely be permanently percieved as a retard.
How so?

>> No.9407917
File: 50 KB, 540x720, f316d662.jpg [View same] [iqdb] [saucenao] [google] [report]

Because the question was ill-formulated. The reason was given in my post. Btw look at this cat inside this sock. It is cute.

>> No.9407925

I don't understand why people will perceive me as a retard.

>> No.9407930
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>> No.9407981

Fourier transform

>> No.9407989

What would be a good math textbook for an of age highschooler taking AP Calc BC? The one I use for my class is very problem heavy- and while that's great and all, I feel like my understanding of the topic is pretty shit.

>> No.9408048


>> No.9408058

>not libgen.io

>> No.9408062

But piracy is I L L E G A L!!!

>> No.9408064

>But piracy is I L L E G A L!!!
Which memecountry do you live in?

>> No.9408071

Oh, I live in the memiest of them all. In which there are no restrictions placed on libgen either. I was just helping that person justify linking books on sale insteadd of ones free for everyone to take.

>> No.9408106

And if you are a weeb


>> No.9408123
File: 256 KB, 516x604, 1503523267785.jpg [View same] [iqdb] [saucenao] [google] [report]

confidence is low lads

long story short: i'm 30 years old and going to college in a few weeks. my placement test is in about a week and depending on how i score i'll be placed in either pre-college(non-credit) courses or college level courses. there's a lot of pressure

the goal is to graduate with a STEM degree so knowing maths is a must. it's been 12 years since i've been in an academic setting and my math skills are as bad as you can imagine. i've been studying with khan academy for three weeks and i'm up to algebra(factoring quadratics, polynomials, nested fractions and the like).

to get to my point, i'm struggling, and i'm beginning to doubt i'm smart enough for STEM. getting every problem is really wearing me out. i'm learning but it's going very slowly. word problems are what i'm struggling with the most(write an equation to solve x word problem). i fucking hate word problems. i feel a lot dumber than i used to be, and the hopes of placing into college level courses are non-existent. i've accepted that i'll have to spend a semester learning the basics. i just hope that i'm smart enough to get into a proper physics/mathematics course by the fall semester.

i do enjoy doing math which is surprising. i thought i'd hate it. solving a hard problem is very satisfying. i'd like to make a career out of it; astrophysics or something along that vein, but i realize that's a long ways down the road. i'll probably be close to 40 by the time i graduate but i honestly don't care anymore.

anyway, i guess i'm posting here to see if anyone else is in a similar situation, and if being in a classroom setting with a proper teacher will make it easier to learn, and if my intellect will ever return

>> No.9408124

>tfw no manga guide to IUT

>> No.9408181
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>> No.9408244

>Btw look at this cat inside this sock. It is cute.
Does me finding the cat cute require the axiom of choice, categories can get a little hairy like that.
Depending on the degree the workload can get pretty heavy, I'd say have a back up to a stem degree that you have confidence in getting.

>> No.9408250
File: 31 KB, 1381x661, karvinen abstraktio.png [View same] [iqdb] [saucenao] [google] [report]

No. It only requires a little spark of animal love.

>> No.9408255

>Does me finding the cat cute require the axiom of choice
No, the category of cats having enough projectives suffices.

>> No.9408403

The wiki suggests A Transition to Advanced Mathematics by Smith, Eggen, and St. Andre in the computer science for intro to proofs, but it doesn't include number theory which is something that I'm interested in.

Is there another comparable book that touches on number theory?

>> No.9408465

So, what do we call the category of cats...[math]CAT[/math]?
>Is there another comparable book that touches on number theory?
Are you specifically interested in number theory as related to compsci? Knuth's art of programming has the relevant number theory

>> No.9408496

Wtf? What took you so long to go to college?

Sorry, I know that's not helpful. I don't know what to tell you, because you're not really being specific. Word problems are typically pretty easy to translate into equations, but I don't think I've had to do such problems since I was in elementary school or something.

Do you have a decent grasp of algebra? Trigonometry?

>> No.9408685


>> No.9408841
File: 51 KB, 736x736, cute.jpg [View same] [iqdb] [saucenao] [google] [report]

Unless you have a good plan, try my experimental study guide :3
1: Finish Khan Academy math subjects up to Algebra 2 and have done the tests
2. Pick up Serge Lang's "Basic Mathematics" and do everything except Trigonometry chapter. Now you will have clear and rigorous grasp on College Algebra and able to prove few things and learn bonus things like Binomial Theorem
3. Go back to Khan Academy and only do Trigonometry chapter. Because trig is fairly easy and at college level it's nothing but memorizing identities and formulas, you can learn more about trig later in calculus. ALSO, Khan Academy exercises are very intuitive so I recommend going back to Lang's book to do the trigonometry exercises there.
4. Now you are well prepaired to do Calculus. Depending on which letter you choose to major in STEM, you can choose a Calculus book on appropriet difficulty. So you can pick up a soft Calculus book for E like Stewart's Early Transcendentals or if you choose to go for a better major like M you can pick up Spivak's Calculus which is rigorous.
4.5. At this point you're doing well so you should do another book to help you through problem-solving and proving things to ease your mind during difficult exercises and heavy texts like Spivak's. I recommend "How to prove it" by Velleman and/or "How to Solve it" by Polya. Polya's is important if you need help solving things in general, while Velleman's is great foundation for later studies in university like Analysis and Linear Algebra.

>> No.9408844
File: 28 KB, 532x565, cuter.jpg [View same] [iqdb] [saucenao] [google] [report]

5. At this point you should be very familiar with Calculus and easily get into university if you do entrance exam applications. Now you can go find your major, and do some headstart studies by checking the curriculum of your class. Or alternatively you could do things like:
Re-visit and skim through some other calculus books to get a better and clear grasp on things and be polished.
Try more rigorous books and topics like Linear Algebra, Number Theory and Multivariable Calculus.
Master problem-solving by doing things like archived Olympiad problems and some difficult ones on Brilliant.org
Whatever you choose to do is up to you and about how much you like mathematics or if you need to get better understanding of things. The point is that it's important to keep doing math and not take too long breaks.
6. Make your own study guide and plan ahead :3

>> No.9408923

Please elaborate, i thought about fourier transform but it didnt help.

>> No.9408934 [DELETED] 

Yeah actually after thinking about it, I realized that it didn't actually solve the problem. You could have deduced what you want under the stronger assumption that f*f is constant everywhere (because then the fourier transform of f is continuous, and its square is a multiple of the dirac mass at 0, hence zero)
I'm not sure what to do about this

>> No.9408936

The only "number theory" in a discrete math courses is Fermat's Little Theorem / Euler's Theorem and Euclid's Algorithm.

Just read one of the three intro books here:

>> No.9408979

What is the Fourier transform of a function constant on [-1,1] and zero everywhere else?

>> No.9409013

It was actually more complicated than I thought, I'll have to keep thinking about it

It's not assumed to be zero outside of [-1,1]

>> No.9409148

Assume f(x) is zero inside [-1,1]. Then rect(x)*f(x) = 0 everywhere, so sinc(w) convolve F(w) = 0 everywhere, i.e. the integral of sinc(w) F(a-w) is 0 for all a. Perhaps differentiate the integral wrt a?

>> No.9409253
File: 34 KB, 1024x576, w.jpg [View same] [iqdb] [saucenao] [google] [report]

Nice OC. Saved.

>> No.9409278

What is a binary operation? Is it a function from the set AxB to C? Any other properties/definitions?

>> No.9409290

like when youdo operations on digits in base 2

>> No.9409292

A binary operation [math] \rho [/math] on a set S is a subset of [math] S \cross S \cross S [/math] such that [math] (x,y, *) \in \rho \ \forall x,y \in S [/math]. Generalisations to binary operations on multiple (distinct) sets are left as an exercise to the reader.

>> No.9409296

Errata: [math] \rho \subseteq S \times S \times S [/math].

>> No.9409309

Oh, and * is uniquely associated to the pair (x,y).

>> No.9409313

Errata: [math] \forall x,y \in S \exists ! * \in S, \ (x,y,*) \in \rho [/math].

>> No.9409331


Where can I learn about topics related to p-adic 'density'? What I mean is, imagine you have a p-adic rational with infinitely many non-zero digits. You now multiply it by another p-adic rational. How many non-zero digits will the resulting number have?

E.g. in binary, 11111111111..... = -1 in the 2-adics. If you multiply this by 1111111111... you get 1. So you got a very 'sparse' number (1) from multiplying two very dense numbers. What I'm interested in is a theory for when the numbers aren't necessarily algebraic or rational. Will two 'dense' numbers generally speaking multiply to form a sparse number? Will a sparse number and a dense number tend to form another sparse number?

>> No.9409354 [DELETED] 

Why do topologists write [math]\mathbb{C}{\mathbb{P}^n}[/math] but geometers write [math]\mathbb{P}_\mathbb{C}^n[/math] ?

>> No.9409413

Why do topologists write [math]\mathbb{C}{\mathbb{P}^n}[/math] but geometers write [math]\mathbb{P}_\mathbb{C}^n[/math] ?

>> No.9409809
File: 51 KB, 899x544, .png [View same] [iqdb] [saucenao] [google] [report]

Man why math teacher is a literal retard.

>> No.9409817

Can you re-post it but not in Arabic this time?

>> No.9409840
File: 1.54 MB, 230x230, 1511935228934.gif [View same] [iqdb] [saucenao] [google] [report]

Because it's called the [math]\mathbb{C}[/math]omplex [math]\mathbb{P}[/math]rojective space and not the other way around.

>> No.9409841
File: 127 KB, 480x270, 1511608881745.gif [View same] [iqdb] [saucenao] [google] [report]

>xyeet doesn't know german

>> No.9409845

Most of the people currently living in Germany don't either.

>> No.9409851
File: 427 KB, 1920x1080, 1512363672240.jpg [View same] [iqdb] [saucenao] [google] [report]

So we must help our Germanic brethren by preserving their language. Thank you for providing a good reason to know it.

>> No.9409852

Oh sorry, it's just a find the maximum/minimum exercise with [math] { x }_{ i } \ge 0 [/math] and the formula to minimize being [math] 5{ x }_{ 1 }-{ x }_{ 2 }+{ x }_{ 3 }+2{ x }_{ 4 } [/math]. I have no idea why he is fucking around with the stuff in the brackets when he already has 1s everywhere. Seems like a dumbass thing to do but what do I know.

>> No.9409880

or l'espace [math]\mathbb P[/math]rojectif [math]\mathbb C[/math]omplexe

>> No.9409887

Or, you know, projective space over C

>> No.9409910
File: 271 KB, 560x560, 1.jpg [View same] [iqdb] [saucenao] [google] [report]

> l'espace [math]\mathbb P[/math]rojectif [math]\mathbb C[/math]omplexe
How about no.

>> No.9409944

Because [math] \mathbb{P}_\mathbb{C}^n = \operatorname{Proj} \mathbb{C}\left[ {{x_0},...,{x_n}} \right][/math] and [math] \mathbb{C}{\mathbb{P}^n} = \left( {{\mathbb{C}^{n + 1}}\backslash \left\{ 0 \right\}} \right)/ \sim [/math] .

Both spaces are interesting to a geometer, but only the latter to a topologist (because the Zariski topology is boring from a topological perspective).

>> No.9410302
File: 560 KB, 1366x768, chiaki_self_energy.png [View same] [iqdb] [saucenao] [google] [report]

Then enjoy being treated like a brainlet for the rest of your life.

>> No.9411197

Your usage of "dense" and "sparse" is non-standard. As to your particular question, I have no idea.

>> No.9411261
File: 2.00 MB, 4032x3024, 760EE043-1D62-493B-AFFD-F7623157D532.jpg [View same] [iqdb] [saucenao] [google] [report]

>This article does not cite any sources

>> No.9411289
File: 323 KB, 1080x1080, gns88jf4no701.jpg [View same] [iqdb] [saucenao] [google] [report]

Most likely you're going to get placed at the level you need to be at in order to have a fair shot at progressing it's not really worth stressing too much about just keep chugging through problems. I wouldn't isolate myself to khan academy either sometimes it is bit unhelpful. Studying math on your own is hard compared to with a class with an experienced professor to guide you I wouldn't get too down about the frustration you might be feeling now.

People like to be all up their own ass on this board about math but at least for where you are skill is a byproduct of work. The more time you spend with a pencil to a page working problems or taking notes the more your skills will improve.

>> No.9411485

Samuel has a nice amerimutt face.

>> No.9411494

be the change you want to see in the world, anon

>> No.9411659
File: 20 KB, 347x334, elliptic.png [View same] [iqdb] [saucenao] [google] [report]

i think /sci/ is a hellhole but langlands is my guy
what are you all up to? i left for software last year and i have this silly hope that in my spare time i can toy with the conjectures i was too cowardly to chase while in academia
i'm a few years out of date on bsd and that's the project for january

>> No.9411667

>i have this silly hope that in my spare time i can toy with the conjectures i was too cowardly to chase while in academia

Go for it!

What kind of software are you doing?

>> No.9411693

"mobility" for an automaker. it's not as exciting as one might think but my only relevant experience was as a hobbyist, so i'm quite grateful

>> No.9412193

How do I visualize an Euclidean 4D space

>> No.9412228
File: 254 KB, 2048x1536, este.jpg [View same] [iqdb] [saucenao] [google] [report]

why is this wrong?

>> No.9412235
File: 161 KB, 223x309, complacency.png [View same] [iqdb] [saucenao] [google] [report]

Visualize an n-space and set n = 4.

>> No.9412256

Can you Latex it? Your disgusting handwriting is illegible.

>> No.9412258

ups basic mistake sorry for the brainletism

>> No.9412296

apart from the arctg (which is merely different) it's not bad

>> No.9412670 [DELETED] 

Is there a Stable Hurewicz Theorem?

i.e. Given an (n-1)-connected spectrum [math]E[/math] , is [math]{\pi _k}\left( E \right) \to {\pi _k}\left( {H\mathbb{Z} \wedge E} \right)[/math] an isomorphism for k<=n ?

I can't find a reference.

>> No.9412815

>look at how retarded I am

>> No.9412818

>>look at how retarded I am
Who are you quoting?

>> No.9413023

Are you assuming choice?

>> No.9413041

So you know how in microsoft paint or adobe photo shop you have the color scale, let's say from red to blue, well consider this, you know how to visualize 3d space, then at each point imagine being able to assign this red-blue color scale. In effect what you've created is a 4d space where the fourth dimension is represented by color (since you can associated a color to a numerical quantity based on it's placement on the scale, you could say the edges of the scale are assigned to plus and minus infinity so that you never actually reach the edges of the scale). From there it's easy to gain some intuition about the properties of objects in 4d space.

>> No.9413045

>you know how to visualize 3d space
I don't...

>> No.9413130


Pretty much all of modern mathematics uses sets as mathematical objects; its pretty useful to know their properties.

Logic is the formalism by which all proofs are written. Understanding logic makes prof writing/reading much easier.

>> No.9413133


Depends, formal logic can cease to be useful to most areas of mathematics. Once you go down the set theorist or logician path you will never come back.

>> No.9413139

>Pretty much all of modern mathematics uses sets as mathematical objects; its pretty useful to know their properties.
>Logic is the formalism by which all proofs are written. Understanding logic makes prof writing/reading much easier.
The post that was replying to said to study them "in their own right". Anything beyond the most elementary of properties of set theory and logic is irrelevant to most mathematics

>> No.9413252
File: 104 KB, 320x425, Hypersphere_coord.png [View same] [iqdb] [saucenao] [google] [report]

Tried your method but still can't visualize a 3-sphere in my head

>> No.9413318

you don't, only through methods in R-3 can we approximate what manifolds might look like in higher dimensions.

>> No.9413335

Their existence is only a conjecture at this point.

>> No.9413406

That doof forgot to mention that a single point could have a line extending from red to blue, which still may give you some intuition about 4d, but is not so easy to visualize.

>> No.9413513

If you want, here's a playlist that tries to explain how to visualize higher dimensions through various methods, for the 3-sphere in particular they employ the hopf fibration. They form a nice little series (a couple of hours long but it's fun so who cares).

>> No.9413579

pls go away

>> No.9413891

What are you trying to say?

>> No.9413896

fuck you and fuck off

>> No.9413928
File: 1.04 MB, 1366x768, 1448724538618.png [View same] [iqdb] [saucenao] [google] [report]

You are very rude.

>> No.9413931

Why the vulgarity?

>> No.9413986

Calculus is shit desu

>> No.9413996

I'm doing an HNC in Electronics Engineering, hadn't done maths since GCSEs and was rubbish at it then, got a C and struggled and now my course is pretty much all maths.

It's a cunt but I'm getting through it with help. I found Khan Academy takes too long, I think once you get back into maths some of the stuff comes back to you.

You might be slow though, it takes me longer to memorise things and actually understand them. So far I've got the top grade on both my assignments so far though so I'm getting through it.

>> No.9414318

expect he's 100% right, so fuck off.

>> No.9414323

>expect he's 100% right, so fuck off.
Do you need to swear?

>> No.9414332


>> No.9414339


>> No.9414345


>> No.9414378
File: 559 KB, 1024x595, 1506967902611.png [View same] [iqdb] [saucenao] [google] [report]

>tfw will NEVER be as smart as my proffs

>> No.9414473


>> No.9415071

Can someone give me a quick rundown on the current state of math ? I'm curious as to what are the important fields of mathematics today, and what problem they're trying to solve.

>> No.9415089

>Can someone give me a quick rundown on the current state of math ?
Too broad of a question, but if you look at https://arxiv.org/ a lot of research falls under the headings of:
Algebraic Geometry; Algebraic Topology; Analysis of PDEs; Category Theory; Classical Analysis and ODEs; Combinatorics; Commutative Algebra; Complex Variables; Differential Geometry; Dynamical Systems; Functional Analysis; General Mathematics; General Topology; Geometric Topology; Group Theory; History and Overview; Information Theory; K-Theory and Homology; Logic; Mathematical Physics; Metric Geometry; Number Theory; Numerical Analysis; Operator Algebras; Optimization and Control; Probability; Quantum Algebra; Representation Theory; Rings and Algebras; Spectral Theory; Statistics Theory; Symplectic Geometry

>> No.9415248

Thanks, do people just specialize themselves or is it possible to, for example, learn every branch of Algebra.

>> No.9415287

Depends on what you mean by learning. Whatever you do, you should finish your undergraduate with some knowledge in as many of the areas listed above as possible and, if you can, take a wide range of graduate classes.
The reason for this is that your interests might change and that you never know where a problem could take you.
That being said, people do specialize, but algebraists usually have (at least) some working knowledge of most other parts of algebra.

>> No.9416336

should proof of academic enrollment in as a math/physics/chemistry major be requirement for browsing /sci/

>> No.9416360


>> No.9416362

why was hilbert a brainlet

>> No.9416368

killing this thread with no survivors

>> No.9416373


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