Mathematics - 2025-02-19

Jakobian

01:25

@BenSteffan @Thorgott I had this brilliant idea to ask this question. What, in your opinion, would be like this ideal way to learn algebraic topology to the point that you are experienced enough that you can further branch off in some direction. I'm not asking because I want to learn algebraic topology, but I know that there exist books such as Hatcher, Spanier, Dieck and so on, and I am curious what path would a student have to take

Ben Steffan

Picking up one of the standard texts and working through it as about the best you can do.

There's a question of how well this really prepares you for "branching off" in some direction, but there is a relatively standard "core" that's contained in these books that forms the bedrock of everything.

For a better preparation there's a few more standard topics that these books don't really treat, like the Serre spectral sequence and theory of bundles and characteristic classes.

Thorgott

I wish I had a better answer to that question

Ben Steffan

yeah, me too

there certainly remains a gap even if you know all of these things, and the gap can be quite large depending on what direction you go

Jakobian

gap in what sense, between basics and things that are more advanced?

Ben Steffan

Yes. Gap in the sense of "the standard resources for a given field may assume you're familiar with more than this teaches you"

if there are standard resources to begin with

Thorgott

01:39

it's a somewhat disparate field and for a lot of things, I feel like there is not a single reference that explains them optimally

like, if someone comes up to me and asks me how to understand the theory of classifying spaces and bundles, I will tell them to read
-Mitchell's notes to understand how to work with them
-tom Dieck's book for the optimal construction, but this should really be phrased in terms of simplicial spaces for which I will further refer you to a book of May and a paper of Segal
-Milnor's paper for the original construction, another paper of tom Dieck for its relation to other constructions and May's book for the bar resolution construction (which doesn't always work)

Jakobian

Sounds like there is room open for a new book about algebraic topology then, if anyone wants to take up this challenge

Ben Steffan

A book that's several thousand pages long

Or several new standard texts

The latter would definitely be good

Thorgott

@BenSteffan Friedl has you covered :)

Ben Steffan

The lecture notes from 3-manifolds this term actually cited Friedl for something

first time I've seen anybody do that

But yeah, my kingdom for a good standard reference for stable homotopy theory

Jakobian

oh, I remember I wanted to read Friedl's notes from topology. But I stick to more set-theoretic things (at least for now)

Ben Steffan

01:48

One that's not 50 years old like the blue book

Jakobian

I didn't actually know he was somewhat famous

Ben Steffan

The notes are famous for being, well, long

how many pages is it now? 4000+?

that's his only claim to fame afaik

Jakobian

I'm not sure if I'm looking at the latest version, but the one from topology has 3,700 pages

Ben Steffan

well, something on that order of magnitude

Thorgott

I saw a paper of Friedl referenced recently

@BenSteffan isn't that what Higher Algebra is :^)

Ben Steffan

01:57

@Thorgott I wish I could say you were wrong...

there is uncertainty as to where one should draw the dividing line between higher algebra (as in the field, not the book) and stable homotopy theory

that's perhaps one thing that's keeping people from writing books, given how terrible the higher algebra side of things is

one potato two potato

03:25

It turns out that the uniformization stuff I said before here is total bullshit.

I thought the uniformization does not hold in general surface but only for compact Riemann surface (or surface with finitely many punctures)

The misunderstanding is based on the fact that the proof I saw uses some PDE method involving the integral over the given surface.

one potato two potato

04:08

The problem with the Riemann Roch theorem is that it only counts meromorphic functions with the prescribed poles in "at most" sense. It really doesn't tell how many meromorphic functions actually exist.

PM 2Ring

05:40

If anyone's currently lurking, the spam wave has splashed over to Math.SE. Please consider visiting the main site and casting a few spam flags. Your assistance will be appreciated.

Don't bother downvoting or close voting. Spam flags automatically cast a downvote, and if a post gets enough spam flags, the post is automatically deleted

one potato two potato

I just flagged one post and it disappeared immediately

PM 2Ring

Yep. I think it takes 4 flags to delete. It used to be 6, but they dropped it a little while ago because the network's been getting severely hit by spammers.

leslie townes

06:00

i'm a little confused. i just checked main and i see a lot of low quality posts from new accounts, but not 'spam' per se in the sense of the spam flag (described as "Exists only to promote a product or service, does not disclose the author's affiliation"). am i missing something?

i.e. if something appears to be a chatterbot that is just posting garbage, is spam the right flag for that? is there actual spam (in the sense of the flag description) also appearing on main?

one potato two potato

06:15

There was actual spam on the main. The post is not about math but some strange mail something.

I didn't flag for low quality question posts.

leslie townes

ah, ok. some of it must be being deleted so fast i'm not seeing it

PM 2Ring

There's a lot of airline-related spam. For examples, take a look at superuser.com & serverfault.com which are the current main targets.

@leslietownes If it's just garbage (like cat walking on keyboard random text), flag it as rude / abusive.

The system treats R/A & spam flags as almost the same, so 4 of either delete the post. But spammers get an extra penalty, and added to the spammer database.

Posting spam incurs a -100 rep penalty. But of course that's irrelevant to these 1 rep spammers.

Here's an example

-5

Q: How do I change my ticket on Expedia?

If you need to change a flight after you've already booked it with Expedia, you'll have to call their customer support team at (+1-844-636-7402) (inside the U.S.) or (+1-844-636-7402) (international). Can I change my name on a flight ticket with Expedia? Yes, you can change the name on an Ex...

brownian-motion

-4

Q: ¿Cómo cambiar tu nombre en un billete de Delta?

Sí, para cambiar el nombre en su boleto de avión, visite el sitio web de Delta Airlines y seleccione la opción "Administrar mi reserva", llame a atención al cliente o comuníquese con un +1-888-529-5402 (US) (OTA) o + 52||800||953||0585..

power-series

leslie townes

06:49

i was just wondering how to change my ticket on expedia, thanks!

also thanks for that info re flagging :)

PM 2Ring

07:01

Fun fact: Most spam removal on the network is done by users. The Stack Exchange system software does have some spam monitoring and automated removal capability, but most of the work is done by the community. See charcoal-se.org

Normally, most spam gets detected and deleted fairly quickly. But we've had a bit of a spike this month. metasmoke.erwaysoftware.com/graphs

one potato two potato

14:20

34

Q: H-principle and PDE's

According to Wikipedia: "In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations (PDRs)". I'd like to know if h-principle and theory from M. Gromov's "Partial Differential Rela...

ap.analysis-of-pdes homotopy-theory h-principle

Interesting work

psie

14:59

I may have asked something similar before, but just out of curiosity, from which book did you'all learn linear algebra from? Would you recommend the book you learnt from? Second, do you think a treatment of determinants like in Axler's book is pedagogically sound (i.e. putting them in the very last chapter and not really using them as a means to prove some of the theorems in a second course in linear algebra)?

Side note; I'm not very familiar with Axler's treatment as a whole, so my claim that he is not really using them may be inaccurate. I'm only familiar with his formatting really :) and I think it is different from other math texts to say the least.

Soumik Mukherjee

15:14

I learned from Friedberg et al. and Hoffman Kunze. Imo LA is not something to worry much about how to learn it, just learn it and move on.

psie

@SoumikMukherjee ok 👍 thanks for sharing. How much of these books did you read?

Like did you read the full book or just some chapters?

psie

15:28

@SoumikMukherjee which edition of Friedberg et al.? :) they have a couple of editions

nickbros123

16:26

@psie axler is amazing imo

leslie townes

17:35

psie to your second point it depends on the purpose of the study but generally i think axler's approach is pedagogically fine (i think the amount of material about matrices, which was initially almost nothing, has varied slightly in subsequent editions). also pedagogically fine to have a book which uses determinants left right and center and nobody knows abstract linear algebra

i mean, if at some point the goal is to develop a working understanding of some body of thematic material, and not to prepare for or pass some test that covers specific techniques, then the techniques you use to approach that material basically don't matter

at the extremes, you might end up with a limited perspective if you studied linear algebra without ever learning much that matrix computations exist, or if you studied linear algebra and never learned how to handle anything outside of a matrix computation framework, but i don't think any of the well known textbooks, like, force their readers into that kind of mindset

axler specifically is written for a [maybe USA-specific] context where it can be safely assumed that most students using his book [in a US university classroom] have had a class on matrix calculation already

ie he's not assuming that his audience would only ever learn linear algebra from that book

Soumik Mukherjee

18:06

@psie most of them, but not like start to finish

@psie 4th

psie

18:25

alright, thanks for the comments

leslie townes

19:21

several times in grad school, i heard people making jokes to the effect of, the more "elite" the undergrad institution you went to, the less likely you were to be able to do any matrix calculations :) nothing to do with axler specifically (not as popular of a textbook when it was new, and not enough of a 'pure algebra' book to reflect the backgrounds of these people)

Ben Steffan

what's a matrix? :(

rschwieb

The answer is out there, Ben. It's looking for you, and it will find you if you want it to

leslie townes

yeah, nobody can be told what the matrix is, you have to experience it for yourself

Ben Steffan

that sounds like a threat

the matrix is not flesh. not yet

psie

22:31

I struggle with understanding the assumptions of the converse with what Rudin has established prior to this theorem. Why does it suffice to consider $m=1$? Nowhere has he proven yet that $f:\mathbb R^n\to\mathbb R^m$ is differentiable iff each of its components are. I think this is key in his assumption in this theorem, or?

He does have a statement concerning the continuity of a parameter dependent matrix; it's continuous iff its entries are.

But we need to know that $f:\mathbb R^n\to\mathbb R^m$ is differentiable iff each of its components are first, then we can apply the statement about parameter dependent matrices and continuity.

He has introduced the Jacobian matrix. I don't know if that counts as proving that statement which I'm after.

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