Mathematics General

What have you guys been up to?

I've mostly been editing papers I need to submit to for refereeing, but I have been looking at Chebyshev polynomials and some of their interesting properties. Anyone ever done anything with them before?

Other urls found in this thread:

youtube.com/watch?v=sqEyWLGvvdw&list=PL0E754696F72137EC&index=1&t=744s
en.wikipedia.org/wiki/Invariant_basis_number
en.wikipedia.org/wiki/Rotation_matrix
en.wikipedia.org/wiki/Scaling_(geometry)#Matrix_representation
en.wikipedia.org/wiki/Proofs_from_THE_BOOK
twitter.com/SFWRedditVideos

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.

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.

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.

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.

Looks interesting, I'll check it out

Triple integrating

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

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.

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

what does anybody think of this lecture series?

it is ok

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.

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

>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.).

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

>not Matveev

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

the poster knows 0 about algebraic topology

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

>tfw no math gf

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.

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

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

Hey mathgen

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

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

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.

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.

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.

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

>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

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.

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?

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

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!

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.

Whoops.

2p kek

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.

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").

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

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)

>being this jealous of Hatcher's bible and taste

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 en.wikipedia.org/wiki/Invariant_basis_number where this kind of thing isn't true in the world of modules

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)

>tfw no gf of any kind

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

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.

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.

It's just... a cursive k?

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?

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

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.

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

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.

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

>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?

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.

Great, I got all my intuition from that guy

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

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

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?

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

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

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

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

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:
en.wikipedia.org/wiki/Scaling_(geometry)#Matrix_representation

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

thanks I will give it some read

Now read Segal - Classifying spaces and spectral sequences

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

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

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

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

Bullying undergrads

Was just reading proofs from The Book.

Rude af

Which one is that?

en.wikipedia.org/wiki/Proofs_from_THE_BOOK

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?

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.

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

...

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.

thank you, I appreciate it

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

I own Gallian, it's great!

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

Over the line!

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.

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.

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

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

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.

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

I get sick of people calling things memes.

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!

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?