/mg/ ∧ /math/ general

"It just keeps happening" edition.
>what are you studying?
>any cool problems?
>any cool theorems or remarks?
>reference suggestions?
>???

Other urls found in this thread:

webusers.imj-prg.fr/~michael.harris/androids.pdf
en.wikipedia.org/wiki/Moving_sofa_problem
arxiv.org/pdf/1706.06630.pdf
arxiv.org/abs/1608.03679
math.stackexchange.com/questions/2211278/riemann-hypothesis-is-bender-brody-müller-hamiltonian-a-new-line-of-attack
ncatlab.org/nlab/show/category theory
twitter.com/NSFWRedditImage

Added to my replied threads.

>DO ANDROIDS PROVE THEOREMS IN THEIR
SLEEP?
webusers.imj-prg.fr/~michael.harris/androids.pdf

So i've been looking for lecture series on modern/abstract algebra

I haven't found any but a series from Harvard ... and but 1/3rd of those are seriously subpar (the prof lets some undergrad do the lecture so...)

:DDDDD
Try READING

pssh nerd

the upper bound for the moving sofa problem apparently got a nice improvement from 2sqrt(2) to 2.37 (lower bound is still 2.2195)

en.wikipedia.org/wiki/Moving_sofa_problem
arxiv.org/pdf/1706.06630.pdf

Let [math]\Sigma[/math] be a 2-surface. Poincare duality [math]H^1(\Sigma) \sim H^1(\Sigma)^*[/math] allows us to define a nondegenerate bilinear 2-form [eqn]\omega(\alpha,\beta) = -\frac{1}{8\pi^2}\int_\Sigma \operatorname{tr}(\alpha \wedge \beta)[/eqn] on the vector space [math]\mathcal{A}_{\Sigma} \ni \alpha, \beta[/math] of [math]G[/math] connections on [math]\Sigma[/math]. This makes[math]\mathcal{A}_\Sigma[/math] into a symplectic manifold.
Let the 3-manifold [math]M[/math] be such that [math]\partial M = \Sigma[/math] and let [math]\tilde{g}, A[/math] be the extension of [math]g,a[/math] to [math]M[/math] respectively, then the vector space [math]L_{\Sigma,a} \subset \operatorname{Hom}(\mathcal{G}_\Sigma,\mathbb{C})[/math] defined by the set of maps [math]f[/math] such that [math]f(eg) = c(a,g)f(e),\quad g,e \in \mathcal{G}_\Sigma[/math], where [eqn]c(a,g) = \exp\left[2\pi i \left(\frac{1}{8\pi^2}\int_\Sigma\operatorname{tr}(g^{-1}ag \wedge g^{-1}dg) - \int_M g^* \sigma\right)\right] = \exp\left[2\pi i (C_M(\tilde{g}^*A) -C_M(A)\right][/eqn] has a natural Hermitian structure.
The bundle [math]L_\Sigma = \bigcup_{a\in A_\Sigma}L_{\Sigma,a}[/math] is then a Hermitian line bundle over the symplectic space [math]\mathcal{A}_\Sigma[/math] of [math]G[/math] connections over [math]\Sigma[/math]. If [math]G[/math] is simply connected and compact, then the symplectic form [math]\omega[/math] satisfies the first integrality condition of quantization a la Dirac. This proves that the Chern-Simons theory defined by the action [eqn]C_M = \frac{1}{8\pi^2}\int_M \operatorname{tr}\left(A \wedge dA + \frac{2}{3}A\wedge A\wedge A\right)[/eqn] is a quantum field theory.

Physics is much too hard for physicists.

I'm studying calc 3 and I just learned line integrals.

When can I apply them to number theory?

But physicists are solving problems in math now.
arxiv.org/abs/1608.03679
>math is too hard for mathematicians
>physics is too hard for physicists
What does this mean?

Didn't that paper get torn apart on stackexchange? I can't understand the fine details but it sounds like it was very overhyped/possibly contained some errors

math.stackexchange.com/questions/2211278/riemann-hypothesis-is-bender-brody-müller-hamiltonian-a-new-line-of-attack

>In conclusion, the sloppyness of the definitions used but the authors leads to a complete mess. Nothing is correct in this paper.

All simply a matter of finding a more suitable Hamiltonian. Transform a math problem into a physics problem and the physicists can make up their lack of rigor with what the mathematicians lack: intuition.

and people say math isn't practical....

Thank you, I extremely like your post. If you can tell more about topics connected to physics, I will be waiting for your stories. %%It's difficult, but I'm trying hard to understand it lol.%%

Thanks user. Posting here is a way for me to organize my thoughts on some interesting relationships and connections between things I've read. I'm glad people are enjoying them too.

Is this a joke? Have "people" not solved it yet?

Top kek

Looking for a recc for a subject to learn. It will be through self-study, so any texts would also be welcome.

I know the basics of algebra, analysis & topology, with a little more depth in commutative algebra and algebraic number theory.
(i.e. taken pretty much all the typical "undergrad" courses)

What should I check out next?
I'm taking an alg. geometry class next semester so I figure I may as well just wait to learn that.

Why are you making a duplicate thread?

I've been studying chemistry/physics for fun for quite some time now. I'm gonna become a Junior this year and try to take some actual courses.

Some statements: Should I get into organic chemistry(or is it sonething every chemist should know about)? Opinion on Biology? How can we have creativity play a bigger role in schools; Assuming only someone with high-levels of creativity could find something new in science?

Science is an important part of my life, and without it I would've ended up in a miserable situation. It just hurts that i come off as unrelatable to some others and I get less motivated to study more. How do i fix that?

Here's another extremely interesting thing.
The Chern-Simons action above can be used to define Witten's tangle operator formally via [eqn]Z_k(M) = \int_{\mathcal{A}_M/\mathcal{G}}\mathcal{D}[A]\exp\left(2\pi i k C_M(A)\right)[/eqn] as a Feynman integral. And if we define holonomy representations [math]R_i[/math] on the components [math]L_i[/math], [math]1 \leq i \leq n[/math] of the link [math]L[/math] one can define the Wilson loop operators [math]W_i(A) = \operatorname{tr}_{R_i}\operatorname{Hol}_{L_i} A[/math] such that [eqn]Z_k(M,L) = \int_{\mathcal{A}_M} \mathcal{D}[A] \prod_{i=1}^n W_i(A)\exp\left(2\pi i k C_M(A)\right).[/eqn] The exciting thing is that the quantities [eqn]\langle \psi | R_i(L_i)|\psi \rangle = \int \mathcal{D}[A] W_i(A)\exp\left(2\pi i k C_M(A)\right),[/eqn] where [math]\psi \in \mathcal{H}[/math] is a Hermitian section of [math]\Sigma[/math], actually coincides with the representation of quantum braid groups a la Nayak. In fact the moduli space [math]\mathcal{M} = \operatorname{Hom}(\pi_1(\Sigma),G)/G[/math] (obtained via the Marsden-Weinstein quotient [math]\mathcal{L} = \mathcal{A}_M // \mathcal{G}_\Sigma[/math]) can be equipped with a complex structure such that the Kahler polarization of it gives a Hermitian section [math]\mathcal{H}_\Sigma[/math] *equivalent* to the space of conformal blocks on [math]\Sigma[/math]. This indicates a connection between gauge theory and CFT that echoes my old suspicion regarding unifying strongly-coupled phenomena with category theory.
This may indicate that if I can establish an equivalence (or duality) between TQFT and CFT that I've talked about recently, then I may use this in conjunction with the equivalence of principal bundle and holonomy approaches to gauge theory to really pin down the quantum braiding found in strongly-coupled phenomena.
>"Intuition" is often faulty.
That's too bad for you then, sweetie.

K-theory.

What's so funny? Integrals helped me in number theory. But I google applications of line integrals in number theory and I find nothing.

Has there been no theorem linking line integrals with arithmetic functions?

Do we really need to spread this shit thin over 2 threads because some autist had to meme his anime pictures?

Are you saying mathematicians are the servants of physicists with that image?

Samefagging while ban evading?

Are they not?

Analytic Number Theory involves a lot of complex analysis and complex analysis involves saying shit follows from Cauchy's theorem every 5 seconds.

So yes, line/contour integrals are involved in number theory.

Analytic Number Theory

Basically Riemann Zeta Function

You made your bed, now sleep in it.

What ordinary mathematicians like to talk about? Aren't they obsessed by their theorems and proofs 24/7?

Yeah man I think their obsession with rigor has been holding them back for centuries desu

>>what are you studying?

psychohistory.

>any cool problems?

too numerous to count.

>any cool theorems or remarks?

A few, but nothing I'd like to share at the preliminary stage... analytical sociology is a a tad on the complex side.

Well unless they're talking about math, ordinary stuff usually: sports, current events, work, stuff like that. They're normal people, you know. They just happen to be good at math and tend to talk a lot about math.

So to apply them I need complex analysis? Complex anal is a junior course here so I guess next year.

Math noob here

What the hell is this and why is it in an introductory book to algebra?

To number theory, yes.

What kinds of things can we say about a matrix [math]PU[/math] where [math]P[/math] is a projection and [math]U[/math] is unitary, in particular about the eigenvalues [math]\{\lambda \}[/math] of [math]PU[/math]?

Clearly [math]\lambda\le 1[/math]. If [math]\lambda\ne 0[/math], then the corresponding eigenvector [math]v[/math] satisfies [math]Pv=v[/math]. Anything else?

What exactly are you confused by? It's just an exercise involving fractions, it certainly fits the criteria of elementary algebra

>What the hell is this
Looks like a fun exercise for noobs

> why is it in an introductory book to algebra
Idk, rational numbers are probably the most intuitive field to work in though.

This just seems like elementary proof writing if you ask me.

a) Suppose one can be simplified, lets say [math]\frac{a}{b}[/math] and lets say this maximum common factor is k. Then [math]
\frac{a}{b} = \frac{e}{f}[/math] where e=a/k and f=b/k, but are still integers. Then note that ed - fc will be an integer, but ed - fc = [math]\frac{ad - bc}{k} = \frac{1}{k} [/math] which isn't an integer.

Yeah, seems like your basic freshman level "apply your definitions" textbook. Makes sense, I guess. When we studied fields back in the day we did an in depth study of the rationals. Very quick though.

>What kinds of things can we say about a matrix
We can definitely say that you're a subhuman tripfag.

what's wrong with tripfriends?

Does it even make sense to ask "what's wrong with animals?"?

>Does it even make sense to ask "what's wrong with animals?"?
No, I'm still confused what your point is

Why are you asking me if there's something wrong with animals? Animals are just animals, they aren't really capable of thought in the way you and I are capable of it. There isn't anything "wrong" with them, they just need to know their place.

i saw ed witten give a talk about the chern-simons action a few years ago. i'm less interested in pure research than i used to be, is there any use of me keeping up with this sort of mathematics?

i'm still a student. i'm asking because i want to know if any of these maths are used in any sort of industry.

>Why are you asking me if there's something wrong with animals?
I'm not, that's why I said it didn't make sense to ask, you seem confused

Alternatively for a), suppose the fractions are neighbours and [math]\frac{a}{b}[/math] can be simplified. Then there is an integer [math]k\neq\pm1[/math] such that [math]a=ke, b=kf, e, f\in\mathbb{Z}[/math]. Then [math]\pm1=ad-bc=ked-kfc=k(ed-fc)\neq\pm1[/math], which is a contradiction. Therefore neither of them can be simplified.

What's the intended meaning of your posts then?

He means that tripsfags are subhuman.

>What's the intended meaning of your posts then?
To ask what's wrong with tripfriends
see:

see:

Hey man, I like that picture. Do you have more math anime girls? Could you post them all? I want to save them and have math anime girls to use them when I post in the /SQT/. Maybe make it my phone's background if I can. Thank you.

...

You're still doing stuck on this exercise? (I assume you're that guy from the previous thread, right? user...)

1. The exercise has a typo:
Assume, to the contrary, that [math] \frac{a+b}{c+d} [/math] is a neighbour fraction to [math] \frac{a}{b} [/math].
Then [math] a(c+d) - b(a+b) = ac+ad-ab-b^2+bc-bc=\pm 1+(a+b)c-(a+b)b= \pm 1 [/math] whence [math] (a+b)(c-b) = 0 [/math]
But if either of [math] a+b [/math] and [math] c-d [/math] is 0, then [math] 1 [/math] can be written as the product of two integers (since [math] ad-bc=\pm 1 [/math]). This is absurd.

The correct neighbour fraction for (b) is [math] \frac{a+c}{b+d} [/math]. (Should be obvious why.)

2. This should have been piss easy but here's how you do (c):
Assume, to the contrary, that there exists [math]\frac{e}{f} [/math] such that (wlog) [math] \frac{a}{b} < \frac{e}{f} < \frac{c}{d} [/math].
Then [math] ( \frac{e}{f} - \frac{a}{b} ) + ( \frac{c}{d} - \frac{e}{f}) = \frac{1}{bd} [/math].
But [math] \frac{e}{f} - \frac{a}{b} = \frac{eb-fa}{fb} \geq \frac{1}{bf} [/math] and similarly [math] \frac{c}{d} - \frac{e}{f} \geq \frac{1}{df} [/math], whereby [math] \frac{1}{bd} \geq \frac{b+d}{bdf} [/math], which contradicts the fact that [math] f < b+d [/math].

Yep, I knew it. I fucked even this one up. Been brainfarting all day.
That should be [math] c-b=0 [/math] not [math] c-d=0 [/math]. You get no contradiction like this but the exercise is still poorly contructed since a neighbour fraction like that is less general, only holds if the numbers are of the form [math] ad = (b-1)(b+1) [/math], while [math] \frac{a+c}{b+d} [/math] is always a neighbour fraction whenever [math] \frac{a}{b} [/math] and [math] \frac{c}{d} [/math] are neighbour fractions.

I'm studying nonstop for my algebra qual. Dear god please save meh.

Algebra is pretty easy
Rule 1: a(bc) = (ab)c
Rule 2: Sometimes ab=ba

>he limits himself to associative algebra

Oh shit, he got me. I have been outed as the algebra 1 group theory kiddy I am. I must now commit sudoku.

I'm studying a construction of the fractional Brownian motion through correlated random walks on [math]\mathbb{Z}[/math]. Does anyone have suggestions for good resources?

What's the point of non associative algebra? Like what the hell is left if you drop associativity?

lie algebras

That does not really count, because although Lie algebras are not associative, you still have a something similar, i.e. the Jacobi identity. It turns this is so strong in fact that you can construct for each Lie algebra its universal envelopping algebra, which is associative again.

>Lie algebras don't count as non-associative because you can do something to them to get a new object which is associative

WHO HERE /AUTISTIC NUMBER THEORY/?

[math] g(m) = \sum{f(d)}\ \ \ \ \longleftrightarrow \ \ \ f(m) = \sum{\mu(d)g(\frac{m}{d})}[/math]

>autistic number theory
>posts the most elementary result about arithmetic functions

????????????????????

It wouldn't be autistic if it was interesting now would it?

Autistic number theory should be the cutting edge shit that is completely detached from anything sane.

The inversion formula is even taught to kids.

I got it from Concrete Mathematics on the 4th chapter so eat my dick

kek

bump

hey /math/
what are some good resources for getting into category theory and what are the prerequisites?
thanked

I found an RNG algorithm by randomly generating assembly basically

gonna use it to calculate random numbers in an OpenGL shader

top is a bitmap made by 1 bit per iteration bottom is what it looks like with sorted pixels

>MacLane, Categories for the working mathematician
>ncatlab
>Abstract and concrete categories

Although you pretty much don't need to know any prequesites except maybe some basic set theory I would argue that it is very useful to know

- basic linear algebra
- basic topology up to homotopies
- some basic algebra

so one can appreciate why one should even do category theory.

thank you. I'll look into them.
do these books cover some motivations and uses in maths of categories?
I haven't yet read something on them, but I never understood the underlying principle behind them.
could you care to explain to me, shortly ofc, what's the fuss regarding categories all about?
thanks

if you have enough time and interest, study also algebraic topology along the way. the best way to learn category theory is to use category theory.

A good read about that is ncatlab.org/nlab/show/category theory

some of it may be quite overwhelming for a complete novice. So I guess I'll try my own words:

Category theory is less an actual "theory" and more sort of a "language" or a "philosophy", which seems to inherently gives you the "right" intuition or point of view for many areas of mathematics, especially topology, algebra, algebraic geometry, etc.

The most interesting part of category theory are the definitions and less the actual theorems of category theory, because they serve to unify very different notions in different branches of mathematics back together, because from the viewpoint of category theory they behave very similarly. Thus it is makes it possible to quickly transfer knowledge in one branch to another branch. An example: Products

A product of sets or a product of topological spaces or of groups are defined quite differently, but from the viewpoint of category theory they all behave in the same way. If you know category theory and know how the product behaves, then you should be able to quickly determine what a product of top. spaces, or groups or... is, without having to memorize it for each seperate case.

that's truly interesting. so it's basically an abstraction on the form of some objects and how that abstraction behaves? sort of?

will do. just a small curiosity: does topology and algebraic geometry have any applications in theoretical CS ?

I wish more math questions were written like this.

> so it's basically
Plato's theory of Forms in a purely mathematical context.

Look into computational algebraic geometry. Also, I'm pretty sure algebraic geometry plays a role in coding theory.

Yes, sort of. It is sort of an abstraction of abstractions.

Category theory studies abstract structures, which is a little funny, because a category itself is an abstract structure, so you can use category theory on itself. I have not heard of any useful results obtained that way, though.

As that guy said, ironically category theory is best studied by working with concrete examples of categories. Some categories are very well suited for this like Topology, Algebraic Geometry and so on.

This guy is right. Algebraic topology is a good way to get familiar with the basic concepts, as you will encounter a lot of them. You will have functors whenever you consider homotopy or (co)homology, you will actually both covariant and contravariant functors, you will have adjoints, you will have stuff about (co)limits, you will have pushouts and pullbacks, you will have categories over objects and consider their subcategories, etc. You don't need much abstract nonsense to develop this stuff, but if you study them simultaneously, you will be like "Oh so the category of covering spaces over [math](X, x_0)[/math] is a subcategory of [math]\textbf{Top.}/X[/math] (where [math]\textbf{Top.}[/math] is the category of pointed spaces)", and so on.

You should always try to find some concrete example of those things. Does this concept make sense if the context is that of set theory, or does it make sense if the context is that of topological spaces or group theory? And one more thing, borrowed from the symmetry group of a triangle: certain objects or arrows may get extra properties when you restrict yourself to a subcategory. With the non-abelian group [math]S_3[/math], this means that all the elements commute with one another in any proper subgroup of [math]S_3[/math], and, similarly, in the full subcategory of the category of sets whose objects are all the non-empty sets, singletons are both initial and terminal objects, but in the whole category they are just terminal objects ([math]\{ x\}\to\emptyset[/math] doesn't exist for any [math]x[/math]).

Brainlet here... i don't really know how to approach this.

Can anyone explain to me the motivation for a coset? I'm halfway through Pinter's abstract algebra. Something to do with creating a partition? Not sure why it'd be a useful one, though.

1) Consider the group of integers, and suppose we want to have all the congruence classes of an integer [math]n[/math]. Consider the group [math]n\mathbb{Z}=\{ nk\ |\ k\in\mathbb{Z}\}[/math]. This is a subgroup of [math]\mathbb{Z}[/math], and its cosets are of the form [math]k+n\mathbb{Z}[/math] for [math]k=0, 1, \dots, n-1[/math]. If you look at those [math]k[/math]'s, you will see that they are all the possible remainders of an integer when divided by [math]n[/math], and hence [math]\mathbb{Z}_n=\mathbb{Z}/n\mathbb{Z}[/math] is the additive group of the congruence classes of [math]n[/math].

2) For any group homomorphism [math]f\colon G\to H[/math], you have a subgroup [math]\text{ker }f=\{ g\in G\ |\ f(g)=e_H\}[/math] of [math]G[/math]. No matter if [math]G[/math] is abelian or not, that is always a normal subgroup, so you can take the quotient group [math]G/\text{ker }f[/math]. Let [math][g][/math] denote the equivalence class of an element [math]g\in G[/math] in the quotient group, and define two homomorphisms: [math]\pi\colon G\to G/text{ker }f, g\mapsto [g][/math] and [math]\overline{f}\colon G/\text{ker }f\to H, [g]\mapsto f(g)[/math]. That the first one is a well defined homomorphism of groups is easy to check, but the second one is not that obvious, so I'll do it for you. We need to have [math]g, h\in [g]\Rightarrow f(g)=f(h)[/math] for all [math]g, h\in G[/math] for the function to be well defined, so suppose [math]g, h\in [g][/math]. Then, [math]gh^{-1}\in\text{ker }f\Rightarrow f(g)f(h)^{-1}=f(g)f(h^{-1})=f(gh^{-1})=e_H \Rightarrow f(g)=f(h)[/math], and [math]\overline{f}[/math] is well defined. That it is a homomorphism follows from this: [math]\overline{f}([g][h])=\overline{f}([gh])=f(gh)=f(g)f(h)=\overline{f}([g])\overline{f}([h])[/math] for all elements in the group. Moreover, [math]\overline{f}[/math] is injective, and so, by quotienting away the kernel of a homomorphism, we can use that to construct and injective homomorphism.

[math]G/textker f = G/\text{ker }f[/math]
typo

Yes exactly, think about equivalence classes.

If you take a set, then you can "divide out" an equivalence relation, which gives you a partition of that set into equivalence classes. The useful property is then that you can consider the set of equivalence classes. This is exactly the set that you get if you start with a set and then "quotient out" an equivalence relations, which gives a this new set.

For groups the really cool thing is, that under some assumptions on the subgroup (it needs to be normal), we get that not only do the cosets form a partition (and thus give rise to an equivalence relation on the group), but also that the cosets form a new group, which we call the quotient group or factor group.

Oops, that example didn't quite work out LOL. If you consider the category of pointed sets, then it singletons are initial. Sorry, my bad. You do get different behaviour, though, when restricting to a suitable subcategory. If you take the category of groups, and restrict it to the full subcategory of abelian groups, you get an abelian category which the category of groups is not.

>I have not heard of any useful results obtained that way, though.

Studying the Category of Categories is not necessarily interesting, however there are categories of objects that built over categories which are interesting.

For instance Algebraic Stacks. When you study Schemes (which are built over sets) you pretty much always study them relative to a base. i.e. Instead of just [math]X[/math] you look at [math]X \to S[/math]

Similariy, for an Algebraic Stack you fix a base scheme [math]S[/math] and you stack is some category [math]\mathfrak{X}[/math] together with a functor [math]\mathfrak{X} \to {S_{Et}}[/math] to the big etale site of the base scheme. You require the functor to have certain properties, that sort of shape the overlying category into something useful. i.e. Fibered in Groupoids, Effective Descent, etc.
The point is categories can be used for more than just studying categories of more classical mathematical objects, but can be used to build interesting objects in their own right.

thanks, that makes things clearer. also, i'm really enjoying this book, it's tasty stuff.

just started the chapter on homomorphisms, but i'll take another look once i've read it. ta

claudia has bigger tits

Should I study some "standard" math such as calculus if I'm mainly interested in things related to logic and category theory?

What have you been doing so far? Are you in a uni? Do you need "standard" math for something? If you have no reason to restrict yourself to "standard" math, then, please, study the material you are interested in.

>getting memed so hard that you go around saying you're interested in category theory before you know how to take a derivative
Veeky Forums, folks
not even once

Underrated post.

No. You will end up not understanding anything because you have absolutely no motivation for any of the definitions.

>What have you been doing so far?
As in my background? I've been self-studying programming language theory and now I find that math itself is more interesting to me.
>Are you in a uni?
Not yet.
>If you have no reason to restrict yourself to "standard" math, then, please, study the material you are interested in.
I guess that's what I'll do for now then. Thanks.

Are you okay? Do you need to talk to someone?

I guess that makes sense.