Mathematics General

Prospective PhD student

What are some interesting research areas to go into if I like (universal) algebra and topology? Any schools to apply for (U.S. here)?

Sheaf theory looks interesting, but seems a bit esoteric.

>>8466380
What about algebra and topology do you like? There is always the fine blend of the two in algebraic topology, but I'd like to know what you like specifically about the two before I can make any decent suggesstions.

>>8466387
Algebra appeals to me, because I like studying structure and finding similarities in structures. The idea of homomorphisms (of all kinds) is just fascinating to me.

I can't really explain why I like topology, I guess the geometry aspect is cool. I think I like looking at it more from the point of view of a poset of open sets though.

I don't know much about algebraic topology, but the idea always seemed interesting.

>>8466415
Try cracking open a book on algebraic topology open (like Hatcher's, which is free online). Take a look around. Also perhaps look closer at homotopy theory.

>>8466422
Looks interesting, I'll check it out

Triple integrating

It's really not that hard at all. Why did /sci/ meme this meme into existence?

>>8466371
Undergrad here, question.

So, I went to community college my first year, saved money, got into a better universe that I could have before, and got almost all of my gen eds down. But this resulted in my being behind in math and physics as they are tiered, obviously.

I want to speed this shit up. Would I be completely fucking myself by taking differential equations and linear algebra at the same time? I'll also be taking elementary modern physics and computational physics.

I'm confident that I could do it, but I'd like to hear what people who have taken them would say.

https://www.youtube.com/watch?v=sqEyWLGvvdw&list=PL0E754696F72137EC&index=1&t=744s

what does anybody think of this lecture series?

>>8466588
it is ok

>>8466460
If you are at least partly math oriented, intro linear algebra is pretty damn easy. I only took 4 courses this semester myself: 3 math and 1 comsci. By far linear algebra was the easiest course. I will say that I am pretty dead with the other math courses and what not stacked together. If your uni has a drop out of a course without penalty period, give it a go, and if you see course work building up, drop one of them. I had to do this because, even though intro stats is a joke, my cosc and proofs course take up absurd amounts of time.

>>8466460
Linear algebra gets used in differential equations, and neither are difficult.

>>8466422
>Hatcher's
What the fuck?
That book is horrible and I discarded it in my third year o physics.
I opted for Munkres - Topology (if you want to understand something about the subject, as opposed to H.).

>>8467135
>That book is horrible
What's wrong with Hatcher? It's super clear imo.

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>>8467141
>>8467135
>>8466422
>>8466415
>not Matveev

>>8467144
Too little content for someone actually wanting to learn the stuff.

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>>8467146
the poster knows 0 about algebraic topology

and it's a perfect introduction, concise, loaded with well-made illustrations, examples, exercises with solutions

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>tfw no math gf

>>8467154
I agree, it's definitely nice as a flick through and belongs in one of those preliminary sections of some different book.
But for a first time learner? I honestly don't think it gives you enough to actually understand the content.

>>8467144
>Not Grothendieck.
That's enough 4chin for today.

>>8467158
fair enough

i would consider it as analogous to a pre-calc book i guess, where you get the kind of practice in calculations that you need to be comfortable with

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>>8466371
Hey mathgen

What does this k symbol mean?, can't find shit on it.

>>8467251
why dont you post context? how is anyone supposed to answer that?

>>8467135
Munkres topology doesn't do anything with algebraic topology. His book on algebraic topology does, but Hatcher is about algebraic topology, not general topology. No wonder you didn't like it.

>>8466371
Yeah, chebyshev polynomials are dope as fuck. It turns out that the matching polynomials for finite undirected tree graphs (which are also the characteristic polynomials in the case of trees) are precisely the chebyshev polynomials of the first kind, and the matching polynomials for cycles are precisely the chebyshev polynomials of the second kind. This is cool, because all of the completely real algebraic numbers are eigenvalues of the characteristic polynomials of trees, and thus the roots of chebyshev polynomials of the first kind. Due to some of the recurrence relations between chebyshev polynomials of the first and second kind, this gives a very nice characterization of a graph's matching polynomial precisely in terms of the matching polynomials of trees and cycles.

>>8467280
Oops, path graphs, not trees for the first part of that. And I may have confused the first and second kind in this explanation, I can't quite recall. But there's a great book by Godsil covering the relationships of a variety of cool families of polynomials to invariants of graphs, if you're interested in learning more.

>>8467266
The second part of the book is about algebraic topology.
Have you ever opened it?
Hatcher's books is for brainlets.

>>8467299
>Hatcher is for brainlets
>opts for Munkres Topology instead, which barely covers any significant amount of algebraic topology at all

At least use Munkres' Algebraic Topology book, Munkres Topology as an Algebraic Topology book is actually for brainlets

>>8467299
It has the fundamental group and covering spaces. It is not even complete enough to do half a course on algebraic topology. It has no homology or cohomology. No chains. Munkres himself has stated that at the time of writing it he was only just learning about algebraic topology. As a result, it is written as someone who is only familiar with general topology, trying to learn algebraic topology. It is incomplete. No wonder you didn't like Hatcher's book, it's not even the same material.

>>8467280
>>8467287
I never read about the graphing properties. I will check out Godsil, thanks!

Where can I find a flawlessly rigorous construction of the Lebesgue integral? I just can't wait for a measure theory course in my school.

is there any promise in doing research for combining multidimensional system theory with partial differential equations?

>>8467529
pretty much any book that has 'measure theory' in the title

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Hmm. I am working on lifting K-theory from Abelian categories (and stable ∞-categories) to general AT categories and AT ∞-categories. K-theory takes an abelian higher category and spits out an abelian group, and should also take a higher topos and spit out a commutative ring. I plan to use this to formalize the Goodwillie calculus. Also, I found a correspondence between group presentations and "shadows" of varieties (general forms in the language of polynomial rings that yield varieties when applied to fixed polynomial rings; perhaps they can be formulated as certain colimits in the category of endofunctors on commutative rings or something). Using this tool, I have found some connections that extend the Goodwillie calculus to new contexts, relating to deep theorems such as Bott periodicity. I hope to apply these tools to homotopy theory, where we can safely track what happens when we pass to G-equivariant stable homotopy from classical homotopy theory. Also, I'm trying to broaden my understanding of how quotienting an algebra by some sort of ideal corresponds to an extension of the dual space. My new language of apparatuses is making it ever easier to translate the "macrocosmic" setting of this into a "microcosmic" one.

Anybody working on similar stuff? I feel like I am part of a simultaneity phenomenon wherein some other researcher is doing the same work. How funny!

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Can someone help me out here?

I'd like to show what's boxed in red here, but am struggling. And there's very little about this stuff online for some reason.

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>>8467759
Whoops.

>>8467251
2p kek

>>8467644
How do you know if what you are doing is too difficult or simple? I was looking at this one open problem awhile back and it was simple enough to state the conjecture, but it has been darn near impossible.

>>8468208
I suppose you just develop a good intuition for when certain problems are too tough. Sometimes you start to get why they are too tough, and it suggests new lines of attack that male ot easier.

I have definitely run into my fair share of questions to which I cannot produce an answer. One of them is the question, "does every adjunction factor into the composite of three adjunctions, either of the form of a localization, then a globalization, then a localization, or as a globalization followed by a localization followed by another globalization?"

Every localization and globalization clearly enjoys this property, where we can actually forget the other two adjunctions and let it factor itself. A step out, we get idempotent monads, which can actually be characterized by the fact that they can always be factored as either a localization followed by a globalization or the other way around. So my question amounts to asking if adjunctions giving rise to non-idempotent monads can still be factored. Maybe there is a grading on adjunctions by how few localizations and globalizations can be used to factor them.

The point is, everybody runs into really tough problems. You just eventually get good at detecting them early on (I always say, "this one smells fishy").

>>8468290
Thanks, good to know I am not just a total dumb-ass!

>>8468625
Jacob Lurie says 90% of his investigations are dead-ends! Don't be disheartened.

Spent the last three days trying to find out whether [math]GL_n(k) \simeq GL_m(k) \rightarrow n = m[/math] (turns out that it does and the proof is elementary, but it requires some work, especially in characteristic 2)

>>8467299
>being this jealous of Hatcher's bible and taste

>>8469024
do you mean an isomorphism of algebras? just comparing the dimensions should give you that immediately

sort of a side tangent but you might be interested in https://en.wikipedia.org/wiki/Invariant_basis_number where this kind of thing isn't true in the world of modules

>>8469144
Nah I meant an isomorphism of groups, it's not trivial (the dimension argument only tells you that you cannot have a continuous/rational isomorphism between GLm(k) and GLn(k) unless m = n, but it's not enough). Still, it can be completely solved with some linear algebra

Thanks for the link, I was just thinking about whether it extends to rings (to integral domains, yes, because it follows from the proof for fields but in more general cases I don't really see any argument)

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>>8467156

>tfw no gf of any kind

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I've noticed a strange similarity between fuctors between abelian categories and homotopy of continuous maps!
Functors <-> continuous maps
Natural transformations <-> homotopies
>tfw you can do category theory by thinking things through as if they were topology and checking a few facts afterwards

>>8466371
Yes in my undergrad borderlining grad studies we had Chebychev polynomials in some introductory signals theory course (EE). But I haven't done much interesting with them.

>>8469836
I like that more general algebras can be represented by matrices of simpler algebras. I haven't even had a serious introductory abstract algebra class, but thanks to those properties with matrix representations I can still apply such things somewhat.

>>8467251
It's just... a cursive k?

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>>8470031
That's actually pretty nice. It simplifies things nicely, too, and makes complex looking claims more reasonable (and so easier to prove). Did you notice this by yourself?

>>8470494
have you solved my KR-theory problem yet from the other thread

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>>8470525
I'm not sure I know what you mean, but I am sure I have not. I know practically no KX-theory where X is a letter or nothing.

>>8470685
Were you note the avatarfag who claimed that Atiyah's proof held up to rigour?

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>>8470709
That would be someone else. I only commented on Atiyah that he is a cool guy. His proof (assuming it is correct) is very elegant looking, though. I can't say it's elegant with my insufficient knowledge of what he refers to, but it looks elegant.

>>8466371
Can someone post a breakdown of the absolute best books on Linear Algebra?

>>8470766
>best
By what standard? What courses have you already taken, what kind of linear algebra are you looking for (applied or pure or what?), and are you in an engineering program or like a theoretical program?

>>8466588
I found it enjoyable and informative (I used it as an extension of my real analysis class, because real analysis at my uni is a fucking joke). It would be nice, though, if the quality weren't absolute potato-tier and the camera person could keep up with the lecturer.

>>8466588
Great, I got all my intuition from that guy

How do I visualize functions defined from R^3 ->R^3

>>8471108
Simple. Just visualize a function from R^n->R^m, then set n=m=3.

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>>8466371
I'm looking through Woodhouse's Geometric Quantization book and I've noticed that it puts a lot of emphasis on the existence of polarizations upon reduction/quantization.
I realize that the existence of an integrable polarization on a symplectic manifold guarantees the consistency of Legendre transformation and some other nice properties. I believe that we can always make sure that a polarization on the Hilbert space of sections can be inherited from that of the symplextic manifold after prequantization. The trouble seems to come from quantization where we trim the Hilbert space, which can lead to the Hamitonian flow no longer preserving the leaves of the inherited polarization. But the existence of a polarization is so important why don't we not do quantization? Why is it so important to have a physically "small" Hilbert space?

>>8471108
Imagine them as vector fields, ie. a vector planted
at every point (corresponding to the value of your function at that point)

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so I have a chart with x,y points on, what kind of equation can I use so it converts all the points as if the chart was rotated 45 degrees and could fit inside the original chart? also the 0,0 point could be anywhere

>>8471803
You can apply a rotation matrix to each of your points:
https://en.wikipedia.org/wiki/Rotation_matrix

>>8471811
thanks!
is there a way to also make the rotated points fit inside the first chart or would the corners be floating out?

>>8471819
You can make them fit inside the first chart as long as you scale everything to fit inside the first chart (as you have done in the second picture).
This time you can apply a scaling matrix to each of your points:
https://en.wikipedia.org/wiki/Scaling_(geometry)#Matrix_representation

>>8471819
nvm I just realize I just have multiply it with the new smaller line height

>>8471827
thanks I will give it some read

>>8469836
Now read Segal - Classifying spaces and spectral sequences

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>>8471909
I think I will when I have time to do so. Thanks!

What is everyone up to today?

Ive been a complete dumbfuck at math my whole life but now that im out of highschool I want to be good at it

where do i start? Only got past Algebra 2 in HS

>>8472793
What do you remember doing last? And do you want to brush up on the stuff before it?

>>8472816
Algebra 2
im retaking basic algebra in community college but i wanna catch up as fast as possible

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>>8472764
A bit of fooling around with some categories and rings. How about you?

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>>8472764
Bullying undergrads

>>8472932
Was just reading proofs from The Book.

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>>8473160
Rude af

>>8474059
Which one is that?

>>8474361
https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK

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>>8474435
Thanks!

Hi mathfags.

Brainlet chemfag here, tired of getting fucked by babby-tier orgo and I'd like to try my hand at some higher math. I've done all the typical physical science calculus, linear algebra, and differential equations requirements, so I'm curious as to where I could start. Abstract algebra? Group theory?

>>8474792
Abstract algebra is my personal favorite. A good introduction is by Joseph Gallian, Contemporary Abstract Algebra. I think it would be a good intro for someone not specializing in math. If you want something more serious, go with Dummit & Foote, but I wouldn't recommend this with your background.
Also make sure you have a good foundation of proofs.

>>8474792
mah nigga, ex medfag here got to 4 years till m.d. now playing around with EE and its dank af

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>>8469222

I want to be a better shooter. What maths do I need to know (along with practice obviously) in order to shoot better? Not joking.

>>8474818
thank you, I appreciate it

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>>8474361
Nice face. Almost as blotchy as the categories you play with

>>8474818
I own Gallian, it's great!

>feeling bad about probably doing poorly on my last Abstract Algebra exam.

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>>8475627
Over the line!

>>8475083
Ergodic theory, dynamic systems, chaos theory, linalg.

Why algebraic topology and category theory are being memed so hard right for the past few years, especially in social networks (including fucking 4chin)? I feel like it's undeserved, and unfair to other fields. Let me clarify.

I have been asking quite a few people (age 20-30 mostly) from several countries and universities (France, USA and Russia). The majority doesn't know some rather basic shit from other fields, like Riemannian geometry, differential equations, functional analysis, etc. Most importantly, that shit is absolutely beautiful!

I concur there are some neat tricks you could do with, say, algebraic topology, but to ignore so much at the same time? I obviously can't say for you, anons, but I hope you knowing maths doesn't begin and end at the "hip" subfields.

>>8475845
People are riding categories so hard because they're perceived as very abstract (perhaps rightly so) and for some reason retards equate abstraction with mathematical quality. The most abstract thing is the hardest and it's what will make you look the smartest.

So you people like the faggots in this thread very loudly repeating
>DUDE look at all my categories so abstract and pure I love le pure mathematics
Incidentally this is the same reason undergrads meme Grootendick to death.

>>8475821

Thank you. I will look into this stuff as soon as my power comes back on. Any recommendations for books (that I can torrent)?

>>8469836
Anon, this is closely related to the macrocosm principle and the general Dold-Kan correspondence. Very cool stuff!

>>8472932
Hey Animenon, speaking of rings... I am examining some really cool shit right now. I'm trying to generalize Freyd's result on AT categories to ∞-category theory, and I am drawing some super deep connections between the Goodwillie calculus, tangent cohesion, abelianization, Pontrjagin duality, and the two primary contexts for the six operations (Grothendieck and Wirthmüller). My work isn't done yet, but basically stabilization in homotopy is maximal abelianization, which is why stable homotopy only works with pointed spaces. Essentially, stabilization/abelianization must freely correct the obstruction to something receiving a morphism from the field on one element. Topoi are categorified commutative rings, and abelian categories are categorified abelian groups! I smell some connections to Freyd-Mitchell, which is why I wanted to let you know.

>>8475845
>>8475857
You don't have to like category theory, but you can't deny the incredibly deep connections turned up by it. We wouldn't have modern algebraic geometry or algebraic topology without it, and there have been new developments in fields from combinatorics (Joyal's species) to differential geometry (generalized characteristic classes, index theory, et cetera), to computer science (internal logics, deeper understanding of monads, homotopy type theory).

Whenever people call it a meme field, I can't help but wonder if they have even bothered to look into what it has to offer. There are surely some memesters studying for the "prestige," but that's not indicative of the field's worth.

Homogenization of randomised elliptic equation on time-dependant domain mostly

>>8475952
I get sick of people calling things memes.

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>>8475930
I will look into this with more depth. I have this Morita-like equivalence constructed as an idea, but the details are still hazy. Maybe cosmic thinking is what I need, thanks!

>>8475939
Nice. I hope you are succesful with your project! What do you think of the idea of using the 0-object of an abelian category as a basepoint analogue?

>>8476134
Regarding your second comment, it's precisely the intuition to be employed! The idea is that stabilizing/abelianizing gives you enrichment over the category of higher modules on some sufficiently abelian thing (be it Z for abelian groups or the sphere spectrum for spectra), and since this thing has an identity, you have a trivial module acting as an identity for the algebra of modules (corresponding to a basepoint geometrically), and then giving something enrichment over these modules must mean that every hom space has an identity corresponding to the existence of a trivial module. Having pointed hom objects automatically gives you a zero object, the "point" of the category.

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>>8476143
That is pleasant to hear, since it is precisely the idea I have been using, too.

im trying to make sense of tyhoyse dirty tricks one uses to compute limits using logarithms in elementary calculus, and "change of variables"
I have come to this
let X,Y,Z be metric spaces
let f map X into Y
let g be a bijection from Y to Z
let a,b,c be contained in X,Y,Z respectively

then: if
lim (x->a) (g o f)(x)=c
and
limit (z->c) g^-1(z) =b

then if lim x->a f(x) exists, it is equal to b


is this right? can the conditions be relaxed a bit?

>>8476370
my proof is the following
[math] let \epsilon >0
then \exists \delta_1 such that
d(z,c)<\delta_1\Rightarrow d(g^{-1}(z),b)<\epsilon
and \exists \delta_2 such that
d(x,a)<\delta_1\Rightarrow d(g[f(x)],c)<\delta_1
\Rightarrow

d(x,a)<\delta_1\Rightarrow d(f(x),b)<\epsilon
[/math]

does this prove that the limit exists? or only that if it exists it is equal to b?

>>8476444
I fucked up, but I guess you guys get the idea

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>>8475845
It's like philosophy. It allows people to sound deep without actually doing any of the heavy lifting.
Category theory is the cancer to mathematics that makes it stray further away from the light of God (i.e. physics).

>>8475952
Perhaps I did not word my post in a clear way. It is not the field or its applications I have a quarrel with, but rather people who don't have a grasp on basic things, and act like they know the Maths as It Was Meant To Be Known. While it is an example of common hubris, my point is that it is far more frequently comes up with people reading up on those True and Enlightening hip fields.

Got here from the front page, not a /sci/ regular.

You should all feel good about yourselves.

>>8476676
I guess I have to agree with you on the fact that people who pursue category theory, algebraic topology, algebraic geometry, or any other "hip" field, just to say that they are doing "real math," are irritating. I actually do, however, hold the view that categories are the most "moral" position from which to view a lot of mathematics, which is why they are so useful. It's not to say that doing math without categories or the nPOV is bad or anything, just that I prefer those lenses and they are more enlightening to me personally.

>>8476370
I saw the chain rule proved in a calc 3 class I followed. Ive always wondered about the magic behind the math. Tell me more anon.

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>>8466371

Stalled out on a little personal history project. I wrote up a treatment of the ideas over a week ago, and I've since been reviewing the field axioms and other foundational bits, in order to set up the treatment in the way that I would like. I re-proved a lemma about depressed polynomials though (it's just two bits that vanish, but to really see that you have to go to the trouble of setting the thing up).

I've left off on actually understanding the history of the Italian mathematicians in some more detail. Yes we're doing high school algebra all the while but I find the story to be interesting and worth writing up again. The more I read, the more sympathetic Cardano becomes, having credited everybody in what seems to be a proper manner, and actually publishing the damn thing(s).

I appreciate that this will be dull to some, but I like both the history bit as well as the math bit, which is what holds my interest. Basically the next step is to slog through Ars Magna itself, and actually read the historical text. It's not terribly long, though, 260 pp.

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>>8476733
>I actually do, however, hold the view that categories are the most "moral" position from which to view a lot of mathematics

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Can anyone help me out with #2.8? I started working through proofs by induction in the intro part to an abstract algebra book. #2.7 wasn't too hard but I'm completely at a loss for #2.8. It seems like it should be precalc level shit but I don't know how to do it.

>>8477511
Try making up a generating function

>>8466371
I am presenting a poster at WQSQQ 2016 this week relating to chebyshev polynomials.

The random walk operator on a finite lattice with absorbing boundaries can be represented by a matrix.whose characteristic polynomial is a chebyshev polynomial. Since the roots are easily obtained via a cosine substitution we immediately have edge values and eigenvectors of the matrix so that we may describe it long term behavior.

For a quantum walk on a finite lattice with absorbing boundaries, the characteristic polynomial follows a recursion similar to a chebyshev recursion but not quite:
p_n(z)=(z+a)p_{n+1}+bzp_{n-2}(z)
The z coefficient in the second term makes solving for roots analytically impossible. We can approximate top eigenvalues and bound all others however. Turns out the top eigenvectors are approximately equal to those for the classical random walk but other eigenvectors bear no relation.

This is a quantum walk btw
https://m.youtube.com/watch?v=-7NAVvBF4DY

>>8477511
It's basically just an algebraic trick, as far as I know.

Suppose you know all the sum formulas up to k-2, and you're looking for k-1. Then what you do is go up another degree and set
S_k = 1^k+2^k+...n^k
and do some fiddling.
S_k + (n+1)^k = 1^k+2^k+...(n+1)^k
S_k + (n+1)^k = 1 + (1+1)^k + (2+1)^k+(3+1)^k...(n+1)^k

If you expand all the shit on the RHS, what you get is a whole bunch of S_ks of varying degree with some binomial coefficients in front. The big S_k will cancel out and it will let you find k-1 in terms of k-2,k-3...2,1.

They show everybody this at least once in university but I don't think anyone remembers it. I had to check one of my books for the starting idea.

>>8477490
Well memed, newfriend! I can't argue with that.

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>>8477586
>is pretentious enough to call his chosen masturbatory subfield "morally" superior to other mathematics
>thinks he isn't a walking meme

>>8477608
I mean, it is... That's why it's pursued. Morality is a folklore term in the community, like the idea of an elegant proof or a good definition. You've done nothing to make a case against me. There is no "wrong" way to do math, but there are certainly "right" ways to formalize it.

>>8477608
>implying category theorists are self-aware of their own enormous autism and ego

>>8477616
Why is it egotistical to observe that calling a group a "set with a binary operator, equipped with an identity element, satisfying associativity and closure, and such that every element has an inverse," is objectively less enlightening than seeing how it fits into the grand scheme of things by seeing it as a pointed, connected ∞-groupoid? Are you guys shitposting, or unironically arguing that category theory's abstraction makes it exceedingly useful?

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>>8477624
>calling a group a "set with a binary operator, equipped with an identity element, satisfying associativity and closure, and such that every element has an inverse," is objectively less enlightening than seeing how it fits into the grand scheme of things by seeing it as a pointed, connected ∞-groupoid?
category theorists, everyone

>>8477624
>literally proving my point
LMAO. Are you sure you shouldn't be studying engineering instead? Fucking idiot ROFL

>>8477628
>>8477630
Dogmatic traditionalism, everyone. I guess we can't expect everyone to accept progressive ideas and advancements; you remind me of Kronecker shitting on Cantor when he introduced different infinities. Enjoy your closed-mindedness.

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>>8477640
>dogmatism
meanwhile on your end
>categories are the objectively right way to view mathematics

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>>8477640
>still this fucking stupid and self-unaware
This is why literally no one takes you seriously.

>>8477646
It's not right for every area of math. It's not right for every problem. But it is certainly more natural, and for many problems it is the "right" way.

>>8477658
YOU don't take me seriously. That's fine, but you should probably check your own ego at the door before trying to project it onto everyone researching in an entire branch of mathematics.

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>>8477672
>everyone
Just you, retard, and quite frankly you're making literally anyone in category theory look bad. So good fucking going, dumbass.

>>8477685
I still don't see what you have presented to enforce this idea which you are so obviously bent to push. One of you champions was referring to all category theorists in more than one of your posts, so you weren't just trying to bash me. While you seem to think I am shedding a bad light on my field's contributors, your efforts have amounted to nothing more than a thread derailment.

When you have something tangible to prove your points, you know where to spit your slander.

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>>8477705
>thread derailment
you replied to a hatpost with a passive-aggressive butthurt remark an hour after it was posted

if you didn't have a raging need to respond to every reply like the terminal autist you are nothing would have been said past >>8477490

you can't even pull your head out of your own ass far enough to understand what people are shitting on you for

>>8477723
Huh, ignorance is bliss I guess. I'm not the one shitting on the way other people do math. As usual, you people seem to find the most creative ways to misinterpret what I say here, after yourself initiating the melee. Enjoy your sour little existence.

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>>8477705
>n-not an argument :((((

>>8477734
>I'm right because they called me out for throwing mud and seeing what sticks!
>lol OHP is such a retard!

>>8466371
Finishing up my proof of P=NP.

>>8477705
I think the problem most people have is that they see category theory as syntactic sugar, which by itself is not so bad but should not be seen as the be all end all of math. Maybe categories will replace sets as the foundations of math, good riddance.
It won't change the fact that for a great number of mathematicians (ie. people working on anything quantitative), it will probably not help much. Categorical methods might tell me what the analog of a connection should be on a stack or whatever but it probably won't help me estimate the curvature in any "concrete" situations (ie. questions that people are currently working on)

tl;dr: category theory makes it easier to transfer ideas between fields but you need fields and actual ideas.

Hey guys

I just found out about topological groups, anyone has a good intro book for this or any other reference?

>>8477773
There isn't all that much you can read about topological groups themselves that's worth the time. I'd say learn about H-spaces (Stasheff's book is a good overview) or lie groups, depending on if you prefer homotopy or geometry.

>>8477770
I definitely agree, and categories have always been bad for computational stuff. I think in the future this will improve somewhat, but mathematicians will always get "down and dirty" when they need concrete results. There are useful cases where we can get new formulas for invariants from categorical constructions, such as various higher trace formulas and the like. Category theory can feasibly be used to transfer computational tools from one field to another as well. You make an excellent point, though, and that is precisely the reason that I respect computational methods and understand why category theory is not perfect (there are other reasons for this latter view which I won't expand upon unless asked).

I guess I see category theory as a guiding tool for math, which I basically conflate with philosophy at this point. To me, computation and concrete constructions are happy by-products to doing math. To others, these things are the actual content, and category theory is a scaffolding. It's a difference in opinion that I respect.

>>8477794
Ok, i'll look inyo it. Thanks

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Pls no bully

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

yeah I filter shit with different order Chebyshev

What does the -st stand for in the Laplace transform?

>>8478173
-stupid

>>8478181
thanks you were really helpful

>>8478182
in case you are bring serious, s is a complex variable

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Hpw do I interpret this shit?

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>tfw no gf

>>8479418
I remember saving this more than a year ago.

>>8479418
The sheer use of buzzwords.

>>8476656
wow, nice to see an undergraduate spreading his stupidity

>>8479745
>guys i dont understand so this must be a meme right Xd

>>8476656
Do you know a single thing about category theory?
Just define what a category is and I'll believe you know a single thing about it.
>pro-tip: wikipedia will fail you