book-recommendations

[marble solar]

Like it's best to do a little bit of two or three things every day for like 2-3 months

Than it is to cram everything in under one month

Especially if you're not already at the research level

[gray gazelle]

Ok I will take your advice. I will let the exercises pace me

[marble solar]

Whenever I feel that I'm in too much of a grind trying to "push out problems" quickly or looking excessively online for hints/solutions

I just stop, take a two week break or so

Come back to it fresh

There was a PDE problem I got stuck on in august for like a week

Looked up solution, and thought I was dumb

Stopped doing PDE for a month or two, came back to problem

[gray gazelle]

I am sort of forced to confront the topic because in my algebra class I believe we are going to spend the rest of the semester on com alg

[marble solar]

I could do it without looking at anything

And I had forgotten what the online post was, I just followed the definitions. So I got better at math by literally stopping math for a set amount of time

and coming back to it

I'm not in grad school right now, so I have the luxury to do things at my own pace

when I feel like it

[dense wren]

@marble solar

[marble solar]

It's given me a healthier, broader, and deeper perspective on how I should approach math work/research

e.g. for Research I perform well under 2-3 month deep dives

But for problems & exercises I want those in longer term memory

[gray gazelle]

I want to be able to take long breaks but time isnt exactly on my side. I want to go to grad school so there is pressure to learn more and get better. As an aside I normally am okay with one or two day breaks from a subject. I have three math classes and I have homework constantly, so I can only afford 1-2 day breaks regardless. Althought, I do feel like your advice is beneficial to my mental health and maturity as a future mathematician.

[marble solar]

Yeah ~ it's a dual problem of what a university schedule looks like

versus what is best for you

When I was in grad school, I frequently took 3-4 classes with problem sets and went to seminars

It was a lot, but now I can take it more slowly. I'm hoping to be back in grad school next fall

[gray gazelle]

Do you recommend taking it slow or do you get used to fast pacing?

Im only taking three grad classes now and the work feels manageable if I were to take a fourth.

But the mental stress might be higher than I am expecting

[marble solar]

I really don't recommend doing 4 grad classes at once

Well I did 4 grad classes + 20hrs of work /week

then I did 3 grad classes + 55 hrs of work/week

[gray gazelle]

😱

[marble solar]

So maybe if you're not working it's more manageable

[gray gazelle]

Im only working twenty hour weeks

The last scenario sounds hellish

[marble solar]

yUh. I should have taken a semester off but uhh I didn't wanna give up math, and I didn't want my family out on the street or in a lot of debt

So I just had to work

[marble solar]

Mr. Berg

[spice sentinel]

Hi, I would like to ask whether anybody might know any good books about the history and development of mathematics

Or a subfield

I’d like to read about how mathematicians developed their work

[steady mango]

Mindyourdecisions

[coral narwhal]

Loch has good recommendations he gave me a good book for history of abstract algebra called "The Genesis of the abstract group concept" by Hans Wussing
same author has other books about history of math but loch said they are not translated
but look up "Mathematics 6000 years" by Hans Wussing if you can find translated version or you can read german

[marble solar]

Boyer's history of Calculus

That's a good one, also Stillwell A history of Mathematics

[spice sentinel]

Thanks for the recommendations!

I’m especially interested in looking at Euler’s work

[dapper root]

There’s an Euler foundation I believe

Which collects stuff regarding him and his work

Maybe try to poke around there and see if they have any books on a website or something

[gray gazelle]

A O P S

[brisk ice]

How many classes / credit hours would you think someone should / could take if they are full time student and no work (undergrad).

Can go up to 18 credit hours but after that I need approval from school office I guess

[marble solar]

I'd say do what's ideal for you

I had friends that took like 27 units in undergrad per term

I have friends that took the bare minimum and focused on succeeding and learning everything possible for that class

[brisk ice]

and how did that turn out?

[marble solar]

they're doing a PhD

So I guess fine

[brisk ice]

well what is the max allowed before approval or something if that is a thing at the school?

[marble solar]

I dunno

[brisk ice]

so how soon did they finish

and in what field?

[marble solar]

jesus

20 questions huh, Physics + Math, they finished in the normal time frame

The MORAL is

[brisk ice]

sorry :/

[marble solar]

Find a system that works for you

If you like taking a lot, and only perform well when you're always busy

Then take many units

If you like taking it slow, then take it slow

[misty wyvern]

you should be able to hit the credit max for undergrad courses because the material is easy

you may not get a good score because many profs like to put meme questions on tests but exposure is important

for grad courses i believe 1-2 classes max

dont pile grad courses on like me, cause then youll have to relearn everything later

[brisk ice]

well there is a credit max (18) and then there is "overload" which is anything above that. I am currently interested in possibly doing 21 credits because one of the courses is a programming course and it sounds basically the same as what I have done other than it is in C++ instead of java

so it would be 6 courses instead of 5

[cursive orbit]

Would you say graduating early or just spreading undergrad out is preferable to taking a lot of grad courses then?

[brisk ice]

plus it is online so it would be flexable

[marble solar]

I tend to focus on learning things deeply

[misty wyvern]

no taking grad courses eaarly is preferable to boost your admissions for phd, but for learning limit the amount you take

[marble solar]

Rather than doing a lot of things all over the place

[brisk ice]

did you also always take summer classes? or just fall and spring

[misty wyvern]

for admissions dump on as many as you can imaginably do well in

[marble solar]

e.g. you can finish 7 grad courses before you graduate with your undergrad

[misty wyvern]

no summer courses, v expensiv

[marble solar]

but maybe only doing 4 would allow you to spend more time on each topic

[misty wyvern]

yeah but do 7

[marble solar]

lol

[misty wyvern]

for admissions

you have the time to learn it properly later

[marble solar]

I think it's hard to say which approach is better

[misty wyvern]

admissions is a dumb rush

[marble solar]

Cuz if you get wrekt in the grad courses cuz you piled it on

Then that will hurt more than just doing a few well

[misty wyvern]

most grad courses are easy As

[marble solar]

Well, some are

[misty wyvern]

less work than ugrad cause grad students are busy

[marble solar]

And then there are profs that say only 10% of class gets As

[misty wyvern]

and grad courses center on that

yeah for those profs just accept that its not worth the trouble to get an a or wait

[marble solar]

Some schools even reach out and ask if some grad courses even had assignments

When you apply

That happened to a few of my friends

So it's hit or miss

[cursive orbit]

Huh?

[brisk ice]

i think for me they are at the same rate. (but the selection is limited)

[cursive orbit]

So do a bunch of grad courses, but not for learning?

[misty wyvern]

yes theyre for phd programs or job opportunities

all my job.4/academic successes came from taking grad courses actually

profs picked me up as an undergrad researcher, which turned into job opportunities and pubs for phds

much better network than school alumni garbage

[cursive orbit]

I see

[pine trellis]

What is a good source for learning homology theory

I'm doing an alg top reading course and I've done homotopy through the 1st chapter of Hatcher, but I kinda want a more abstract/category theory approach to homology

[sturdy sail]

I think Rotman does a good job at presenting (co)homology. It doesn't go straight into the Eilenberg-Steenrod axioms, since it is intended as a first course, but it does a really good job at presenting the material in a very didatic way. It develops simplicial and cellular (co)homology and computes a nice variety of examples before going into the more abstract singular (co)homology theory and presents the Eilenberg-Steenrod a little bit after that.

Even if it doesn't go right into the more abstract foundations of (co)homology, it's still a great reference.

[pine trellis]

this book? http://www.ugr.es/~acegarra/Rotman.pdf

[sturdy sail]

Yeah!

This is precisely the one I was referring to

I went to sleep as soon as I sent that message, sorry lol.

[analog pollen]

Yes?

[rain hound]

Does an applied mathematician ever need to know analysis as a higher level than what’s in Courant and Fritz’s books?

[misty wyvern]

yes lmao

i think the point you start losing applied mathe,aticians is in post-grothendieck ag

everything else has a cottage industry of applied people working in it, to varying degrees of seriousness

[wintry lotus]

What's the best place to learn singular homology as a beginner?

Something that explains stuff well, gives intuition, etc.

[hasty eagle]

Hausdorff

[slim peak]

you can think about exterior algebra of differential forms, with smooth functions

[wintry lotus]

Ooh, I'll have to learn about it

[gray gazelle]

[slim peak]

in dimension 3, the exterior derivatives is either , the gradient, the curl, the divergence

composition of two of it gives 0, thanks to Schwartz Lemma for smooth functions (derivatives commutes on smooth functions)

[wintry lotus]

Interesting

[full linden]

hey @marble solar , I see that you're always referencing Spivak Calculus, would I skip something important by starting at the section of limits?

[quick hornet]

as long as you know the prerequisite proofs and logic stuff already, thats fine

maybe start a bit before then though

wherever he talks about the archimedean principle

[full linden]

got it, thanks tho

[marble solar]

I think chapter 2 can be useful for combinatorial stuff

But overall I agree with Namington's assessment

[misty wyvern]

You could just skip per your judgement and work backwards if you're missing something.

That's basically how I read all books nowadays

I didn't even have to read Harry Potter, I could just read the last page and know what happened in the prior 3000

[marble solar]

I think the advice for a grad student is different from someone that's getting grounded w/ proofs & analysis/calculus

[sturdy sail]

Is Partial Differential Equations by Lawrence Evans a good introduction to PDEs?

[misty wyvern]

It's fine yeah, I prefer Taylor vol. 1.

Vol. 2 and 3 are natural followups and good references, though in Taylor's usual fashion very dense and challenging

[silver kindle]

Principles and Techniques in Combinatorics by Chen. I did not know much about combinatorics either when I took a course called Theory of Combinatorics. I did however have a solid knowledge of logic, proofs, analysis, Elementary Number Theory, etc. So I do not remember if it is a from scratch book or not. Definitely give it a try though, because I did not even know basic combinatorics before reading it, and now I am confident in it

[rocky nebula]

any book to learn binomial expressions?

[gray gazelle]

I don’t think you need a whole book to understand it. Maybe the counting chapter in book of proof by Hammock. I think I said his name properly.

[rocky nebula]

alright thank you

richard hammack?

[gray gazelle]

Yeah

The book is free in his website.

[rocky nebula]

i got it, thanks!

[inner ledge]

Any good books for statistics?

[gray gazelle]

Cengage lol

[frosty girder]

i doubt if cengage will help here

[karmic thorn]

Mathematical Statistics by Hogg might be worth looking into

[jolly quarry]

hey

does anyone have any books on Algebra 1

guys anyone pls???

[analog pollen]

Just use khan academy

[jolly quarry]

i am

but i am seeing if there are any books

on Algebra 1

<@&286206848099549185>

pls

so any books

?

[stray veldt]

does your school not recommend a book

[jolly quarry]

bruh no

i am in 8th bro

and in india so

i am trying to find some books on algebra 1

btw I do algebra 1 in khan academy too

[stray veldt]

does india not have school books?
anyways, there is a book by israel gelfand, but it doesnt really matter which book you take, there are hundreds of highschool algebra books and i assume they are more or less equivalent

[jolly quarry]

sry bro

[gray gazelle]

[jolly quarry]

sry

[stray veldt]

all good, but please read up #❓how-to-get-help as to when its ok to ping helpers

[jolly quarry]

i did

i pinged after 15 min

[gray gazelle]

but you technically did not apply it

and you abused helpers ping

here

[stray veldt]

(the reason nobody answered you is that nobody here will have a strong opinion on highschool algebra books, it really doesnt matter what book you take)

[jolly quarry]

ya but i cant find any books

[stray veldt]

[gray gazelle]

amazon.com

hs books don't even help me understand algebra

even calculus

[stray veldt]

oh

you can check https://openstax.org/subjects/math

[gray gazelle]

[jolly quarry]

?

[stray veldt]

more or less any highschool algebra book

[jolly quarry]

?

[stray veldt]

just check this

[jolly quarry]

ok

u know what forget it

i will just do khan

[stray veldt]

[gray gazelle]

I think Gelfand might be a good book suiting your needs.

[frosty girder]

we do

[frosty girder]

u can try seeing ncert books
its basic, and does a good job of explaining

[delicate shore]

I highly recommend the works of Franz Kafka

[gray gazelle]

I thought someone actually pinged me here

[uncut zealot]

Does anyone know any good books on, I guess it would be called Complex Dynamics? One of my professors asked me to code a program that takes every point on the complex plane and then color it according to which root it tends towards using Newton's Method, and for polynomials of degree greater than 2, you get this weird fractal pattern. When I asked my professor why this happens, he said it's because every point on a boundary between two colours has to be on the boundary between every colour, and when I asked why that happens, he shrugged.

Basically, are there books that explain that phenomenon, the Mandelbrot set, etc., requiring no more than, say, undergraduate complex analysis?

[prisma snow]

@uncut zealot I don't have a book recommendation, but 3b1b's last 2 videos is on this topic, so I recommend you watch that, even if it's just for the pretty graphics

[frosty girder]

the video covers exactly what u said, so it will help provide some insight at the very least

[uncut zealot]

I'll go watch them. Still, 3b1b isn't really what I'd go to if my goal is to actually learn a subject; he's good for helping intuition and as a supplement to a course but he's not very rigorous.

I just checked on his website. His visualization tool is actually somewhat more limited than the one I did (mine does 100 iterations and up to degree 10 polynomials, but can take like 30 seconds to draw), but otherwise it definitely looks like the same thing. Unfortunately, he doesn't cite any books on his website.

He cites some papers in the video descriptions on YouTube, but they're all either quite advanced or also on the more intuitive side of things.

[dark orbit]

i haven't really seen any holomorphic dynamics textbooks floating around exactly though i haven't looked very hard

your best bet might be to look for literature reviews of some kind, stuff aimed at incoming grad students

probably I've missed something, though, who knows

oh nevermind, looks like Springer GTM has "Iteration of Rational Functions", though it ends on Mandelbrot

[gray gazelle]

Hi, I have been reading a book about algorithms and realized I also need an understanding of maths to be able to solve that algorithms, can you recommend me a book that emphasizes the math needed for solving programming algorithms?

[rain hound]

Concrete Mathematics by Donald Knuth

[gray gazelle]

thanks

[uncut zealot]

That sounds like it might be what I'm looking for.

[timber mesa]

to be fair I'm not entirely sure why that happens either, though I've taken a course on holomorphic dynamics before

Beardon's Iteration of Rational Functions is a great book, it's the simpler one out of the usual references

(the other usual ones being Milnor and Carleson-Gamelin)

https://arxiv.org/abs/1506.07113 is a nice and very short introduction though it doesn't cover much

[latent pulsar]

whats a good textbook for undergrad probability theory?

[rain hound]

good question. For the most part, they don't exist. The only text I've seen that doesn't make me want to completely gouge out my eyes is Feller's books.

[coarse grove]

What you think about the book

Mind for Number

I bought it from Amazon cuz it's on discount

[orchid musk]

Introduction to Probability by David Anderson from the Cambridge Mathematical Textbooks series is a nice introduction

[gray gazelle]

does anyone know a rigorous differential equations book with proofs and theorems and stuff?

[jolly quarry]

ok thx

[timber mesa]

maybe Hirsch-Smale-Devaney

Differential Equations, Dynamical Systems and An Introduction to Chaos

[gray gazelle]

it seems good

thanks

[dapper zenith]

do i need to know something before reading concrete maths as i m finding it quite difficult

[jagged spruce]

Can anyone recommend books for learning number theory

[gray gazelle]

What's your background?

[hybrid flicker]

Anyone know the huge russian math workbook its like MEGA thick?
The author or the name?

[iron granite]

Anyone knows a good book on axiomatic set theory?

[gray jungle]

kenneth kunen set theory is quite good

[iron granite]

Thanks, will check it out.

[still jay]

You may also want to look at Introduction to Set Theory by Hrbacek and Jech.

[cloud bolt]

how many books does someone self-studying typically work through at one time? I read somewhere 4-5 is the max but that seems high to me, it seems more like 2-3 is better for absorbing info

[gray jungle]

personally 3 is the limit for my self-study and even then i make a week for each subject

[iron granite]

The most I have used are 2 books for Calc 1, and numerous online sources.

Though, I guess it depends on the book?

books**

[cloud bolt]

I think my question is actually about # of different topics -- I think using a couple of different sources for the same topic is all fine

[gray jungle]

3 books sounds alright to me
some topics are too light to even worry about than others

you just gotta find what works for you

[cloud bolt]

cool, thanks. was just wondering if I should push myself to do 5 or if that was as crazy as it initially sounded to me

[gray gazelle]

Any good problembook for functional analysis (operator theory)?

[sage python]

Idk any dedicated problem books, I think Brezis has good problems and it's tbh the main book I know for functional lmao

Though Brezis has more of an emphasis on the functional that builds toward PDE, eg it only does spectral theory for compact operators

Maybe @slim peak would be good to ask here

[slim peak]

Operator Theory on Hilbert/Banach spaces ?

If yes, Spectral Theory by Raymond and Cheverry is a very good book

More focused on Spectral Theory as the title said, but it deals with operator theory and functional calculus, and it contains a lot of exercises

[gray gazelle]

Yeah

Alright, I'll check that out

Thanks

@slim peak this seems to be a textbook, I was looking for a more problem oriented book

Or you think that trying to prove those theorems on my own would be a good practice?

[slim peak]

There is a lot of exercises in it

Not problems, but a lot of exercises

[slim peak]

zn yes, except the part talking about Grushin formalism, and the last one, this could be really good to practice

[gray gazelle]

There is also this C_p theory book, I think I'll go to a cave next year for a semester and just solve this book then 😄

[marble solar]

@slim peak What you think of Lax's functional book?

[misty wyvern]

It's fine. Lax is a big name.

Why, are you shopping for a functional book?

Cause my pick would be Rudin, unironically.

[sage python]

C_p meaning \mathbb{C}_p?

@gray gazelle

[marble solar]

You know I've been burned on that recommendation twice teafortwo

Fool me once, shame on you; fool me twice, shame on me; fool me thrice, well now

[marble solar]

I'm pretty sure I'm starting my PhD next fall. I'm planning on taking PDEs, Functional, and Diff. Geometry my first term

I will have been out of school for like 2 years by the time I start

[misty wyvern]

My main issue is Lax doesn't sit well on my metaphorical shelf, since I barely ever reference it. As a pedagogical book it's definitely fine though.

If I was going for pedagogy I might recommend Stein. What did you find bad about Rudin?

So Brezis is another good recommendation.

He does functional analysis with a hard PDEs bent.

@marble solar

[marble solar]

I have stein and shakarchi volume 4

It's not quite functional analysis, it's more of analysis of functions

What I don't like about Rudin is that the exercises don't really follow that well from the material in the text

I like something I can follow because I'm dumb

Everyone says Brezis, but I'm not a huge fan of paperbacks

[misty wyvern]

Well, this is a little bit of a meme recommendation but the appendix of Taylor's PDE's vol. 1 is an entire course in functional analysis.

We have a functional prof here who teaches like that lol

Grats on your PhD admissions

[marble solar]

well

I haven't been admitted yet

But my research advisor said and I quote "It is likely you will get in, unless you're just very unlucky"

[slim peak]

I never read it

so I can't say

The Table of content shows me that it seems to be very complete

[frigid comet]

Lax is my FA reference.

Has a lot in it compared to something like Rudins FA.

I have also used Reed-Simon vol1 for stuff like Frechet and LF spaces.

[misty wyvern]

I cant use Lax as a reference, I cant find shit in it lol

its a messy book

[frigid comet]

I find the order of main topics fairly sensible, but some niche things could be hard to find I suppose.

[cunning stirrup]

Omniscient Reader is pretty good

[broken meadow]

ange

[gray gazelle]

No, it's C(X, \mathbb{R}) with topology of pointwise convergence

[orchid musk]

whats a good book on spectral theory after undergrad abstract linear?

[slim peak]

Spectral theory on matrices is almost trivial, and is done in every good enough book about linear algebra. Spectral theory for other stuff than the finite dimensional setting like on Hilbert spaces, requires much more technology, like Lectures on complete vector spaces etc.

If you are an undergrad, you probably haven't a good enough overview on Banach spaces to check the latter

[orchid musk]

oh im doing an independent study in banach spaces, should i just keep pushing that then?

[slim peak]

yes

[orchid musk]

nice ty

[hollow peak]

mazel tov

[remote ginkgo]

let me

suggest Kreyszig for fun anal

[dapper root]

Does anyone have experience ordering a book from Springer MyCopy? In particular, is it a normal paperback with normal binding? Does it seem lower quality than any other paperback textbook?

[karmic thorn]

I think @slim peak ?

[slim peak]

I have plenty of it, and it's standard textbook quality

it looks very similar to the books you can find in your Uni's library

[iron granite]

Dokja dies too many times, it ruins some shit imo.

[cunning stirrup]

he does? damn thanks for the spoilers I only read the manhwa

[iron granite]

Sorry my dude.

[cunning stirrup]

he died once in the manhwa

[iron granite]

Thought you read till the end.

[cunning stirrup]

I mean there's the webnovel and the manhwa

manhwa is still airing

[iron granite]

Manhwa got me into the webnovel.

And I read all of the webnovel.

Decent read, but too many Korean themes and I couldn't keep up with character names.

Lord of the Mysteries is a much better webnovel imo.

[cunning stirrup]

I don't like novels except Art of War 😤

[iron granite]

It got a 'manhwa' too

• Sun Tzu, The Art Of War

[cunning stirrup]

WHAT IT HAS A MANHWA

[marble solar]

I've heard good things

[green estuary]

Any good problem book on limits that has hard problems? I'm looking for a book that only uses elementary functions, and I would like it to contain some cool arctan limits. I guess this is too specific but any suggestion is appreciated.

[marble solar]

Spivak's calculus chapter 5

should have some cool limit problems

[dapper root]

🤩

Thx!

[tropic lion]

What do y'all think of this book? https://arxiv.org/pdf/2108.04902.pdf

[sturdy sail]

Is A Primer on Mapping Class Groups one of the few book references on Mapping Class Groups out there? I can't find anything else.

[sudden kindle]

its supposed to be really good

according to a friend

I havnt read it

[sturdy sail]

Yeah, I might have to read it at some point...

Waaay later lmao

[marble solar]

It's an amazing book

I've read the first three or four chapters

[spice jungle]

hey guys, i learnt pre-algebra, then i need a book for algebra❓

[gray gazelle]

Go for Gelfand's Algebra

[spice jungle]

thnks

[gray gazelle]

👍

[livid ermine]

Do anyone know what are the prerequisites for "Hodge Theory and Complex Algebraic Geometry" by Claire Voisin?

Is decent grasp of complex analysis (say alfhors) and a decent bit of commutative algebra enough?

[azure ice]

Idk maybe some familiarity with alg. geo. and diff. top. is also needed? Chapter 1 seems to introduce all the prerequisites but it’s fairly quick imo. I have saved it for after I will learn more alg. geo.

[analog pollen]

What do you guys think of schaums outline linear algebra, ill be using this book for my next semester

[iron granite]

Hmm, I've only ever heard of Schaum's Outline being used as a supplement and not the main book.

Though maybe you can use it.

[analog pollen]

This is my class book

Was just asking what you guys think of it

[iron granite]

Oh.

Yeah, it's probably the class book for a reason.

The Schaum Outline series is an outstanding series of books.

[marble solar]

I've seen some classes use it as the main book

Schaum's outline to linear algebra is a great book

[analog pollen]

oh bet

[sage kelp]

Hey, what's an honest opinion of Pugh's real analysis book? How hard is it? When should I go for it? In the preface he writes that the target audience is college juniors and seniors, but just wondering if this is actually true since I have heard/read different comments about it

[marble solar]

It's very good, but it is pretty difficult. I'd recommend reading spivak's calculus or a cognate text before Pugh's real analysis

[sage kelp]

So would you say Pugh can be used as a second/intermediate course on real anal?

[marble solar]

It's an advanced course on real analysis

Usually for classes at Berkeley or UCLA honors real analysis

[sage kelp]

Okay, makes sense and it's what I have heard. Thank you

[marble solar]

Yuh

[misty wyvern]

If Pugh is advanced what's not advanced?

[marble solar]

Abott, Ross, and other cognates of the bad real analysis books

[iron granite]

I heard Tom M. Apostol also wrote a book on Analysis.

Anyone read it?

[sudden kindle]

I read part of Vol 1

[iron granite]

I think you are confusing his Calculus books with his Analysis Book.

It's not the one with the red cover.

[prisma snow]

@frosty girder is reading it rn

[iron granite]

I'll ask him his opinion on it then.

[frosty girder]

yup

[marble solar]

I read it

[frosty girder]

i think its a pretty good book
however it contains stuff that isnt in a usual analysis course

[iron granite]

Such as?

[marble solar]

Fourier stuff, lots of special function, some complex stuff

[frosty girder]

Stuff after the 9th chapter

its a little measure theory, multivariable, and fourier stuff

[marble solar]

It's good stuff

Apostol's book is like Rudin with extra exposition

[frosty girder]

im still on the second chapter but its nice

[marble solar]

is that the topology one

I don't have my copy of apostol on me

I have a first edition

[iron granite]

Rudin's proof for the uniqueness of the n-th root is really unmotivated.

[frosty girder]

its the set theory one

the third one is the topology chapter

[marble solar]

I learned that theorem by heart

Thinkin "Oh yeah good test question"

[frosty girder]

i have the second edition

[marble solar]

Wasn't on the test at all

[iron granite]

Stuff just comes out of nowhere.

[frosty girder]

i tried reading rudins book
but it was too scary

[marble solar]

I learned every exercise and every theorem of chapters 1 & 2 of baby rudin for my real analysis class

and I still got a 55/100

[iron granite]

RIP

[frosty girder]

[marble solar]

on the first midterm. I talked to people and everyone was like

"Oh for that prof that's like an A+"

I was like ok I'm chillin'

7 people got perfect scores

I got a B-

[iron granite]

Did you know what even went wrong? lol

[marble solar]

yeah we had a 4 question test for 50 minutes

and I made minor mistakes

I used an induction proof when an estimate would have worked. I messed up a detail on defining the bijective function between (0,1) and [0,1]

I blew the topology question because it was phrased in a slightly different manner than what I was used to

Is there an infinite closed set that isn't an interval which has a non-empty derived set

Then there was a totally ordered field question asking about an upper bound property, and the idea was a < a+b/2 < b

But I had gotten burned on 2 = 0 in a field for linear algebra

and I didn't wanna divide by zero. Forgot that 0 < 1 < 1+1 in a totally ordered field

So I blew a question I should have known, and minor mistakes on 3 questions sank me

[iron granite]

Higher Math tests seem VERY different than the ones in HS. Damn.

We recently had a MCQ test with 40 limit questions.

Of course I Lopital'd that mf.

Is 4 questions - 50 minute test the norm, or do the finals have more questions and time?

[misty wyvern]

I gave up on prepping for exams halfway through undergrad, astonishingly it didn't majorly impact my grades.

If there's such a thing as a time waster, it's exams.

[iron granite]

@misty wyvern What is the grade distribution for undergrad?

[misty wyvern]

IDK, I ended up in an upper percentile despite not doing well on many exams.

[iron granite]

Seems like you were lucky.

[misty wyvern]

I doubt it, I took too many classes to get lucky. Maxed out credits-per-semester, every semester basically. I think school ends up rewarding people who study the material in their free time consistently, so the grades you lose from not studying exam-specific tricks is made up elsewhere.

[iron granite]

Hmm, seems reasonable

[marble solar]

After a certain point 1 hour math tests aren't a good filter

Or math tests in general aren't a good filter for people. It's more important that you learn to think deeply about things

[misty wyvern]

I don't like the "think deeply about things" take either even though it's literally true, simply because when you're really thinking about things, realizations happen after extended mulling, the kind you do when you're picking apart details in your favorite novel or whatnot.

[marble solar]

I mean I count that as a part of thinking deeply

Taking a break, going to a film, talking with your friends about random stuff

[misty wyvern]

Yeah, like I said it's true in a literal sense.

I just mean it creates this impression that it's somehow a different mental experience from, say

mastering a video game.

[iron granite]

More insight into a problem had hit me not doing math than doing math.

[marble solar]

It's the exact same process

I guess a lot of people that doesn't dawn on them

for a lot of people

my hands are cold

It's raining outside

and my university is running the AC

[iron granite]

Claire de Lune is playing?

[misty wyvern]

Anyways I'm teaching spectral theory to a bunch of early PhD students for the next week. I'm super excited but wondering if I should mention projection-valued measures and unbounded operators.

This is like, my favorite topic.

[marble solar]

unbounded operators are good to expose students to

So much of early functional stuff is all bounded

Which is of course the nice scenarios

[azure ice]

They are also important in QM

[misty wyvern]

I try to avoid corrupting the purity of mathematicians by letting them know their work is applicable.

[gray gazelle]

Boooo

[green estuary]

I did some limits from spivak's calculus, I found them a bit easy, so I'm looking for something more advanced, any suggestions?

[marble solar]

really

You did all of chapter 5

[green estuary]

Of course no, I looked at the problems and did a few.

[analog pollen]

wait ill send a problem for u

[cloud bolt]

anyone else not like Strang's linear algebra? his video lectures on MIT OCW are nice, but the book feels way too... casual? informal? it feels like a friend is explaining it to me and it's weird

[analog pollen]

Dont use l’hop @green estuary

And aa = a^2

[analog pollen]

i haven't read the book myself but ive seen alot of reviews that say the same

[green estuary]

Ok, I think I got it thanks for the problem it's pretty cool only uses conjugates. (Sorry for this small conversation it shouldn't belong to this channel)

[analog pollen]

no problem, why you so interested in limits tho?

[green estuary]

I have an exam coming up that's mostly limits, they are also interesting that's why.

[analog pollen]

ah ic

[analog pollen]

if you find spivak ez and this i think youll ace ur test

[novel obsidian]

Has anyone read 'The Cauchy-Schwarz Masterclass'? A book about inequalities in preperation for real analysis. 3b1b reccomends it (and I would never question our lord and saviour) but I was wondering if anyone else has an opinion on it.

[misty wyvern]

I really liked it.

Its proofs are a mix of clever and well-motivated.

And they're useful!

[novel obsidian]

What pre-reqs would you say are required?

I assume calc,

[misty wyvern]

Basically nothing. IDK, maybe analysis at most.

[analog pollen]

how is analysis needed if its a prep

[novel obsidian]

mhm

[analog pollen]

lol

[novel obsidian]

could you do it after a pretty rigorous multi-var calc course?

[misty wyvern]

whats a prep

[novel obsidian]

before analysis

[misty wyvern]

I think you can just try reading it and see if you get stopped by a lack of knowledge.

[novel obsidian]

good idea

aight ty

[marble solar]

So why don't you try like 2/3rds of them

[solemn panther]

Any book recommendations for trigonometry from basics to advanced for pure self study?

[sage kelp]

Why do you say those books are bad? Lol

[marble solar]

because spivak's calculus does the same thing but better at an easier level

(and other cognates of spivak's calc: apostol, courant, salis-hille-etgen, etc.)

and other books get the more advanced material better (like rudin, pugh, apostol, etc.)

[sturdy sail]

Guys, I need to review some measure theory. Do you prefer Stein Shakarchi or Michael E. Taylor?

[sturdy sail]

Shakarchi's chapter 6 actually looks like a great way to review baby measure theory. And I think I will be reading Michael E. Taylor more carefully in the long run.

[solemn panther]

?

[marble solar]

Stein and Shakarchi is good because the most important case is the lebesgue case and it devotes a lot of time to that

Although there are things that aren't true in general

[peak trout]

Anyone know how i can find a pdf file of ICE-EM Year 10 Mathematics textbook

[gray gazelle]

<@&268886789983436800>

[tulip blade]

Kahn academy

[forest ridge]

Any book recomemdations for learning linear algebra, from the very basics?

[cursive orbit]

Axler

[forest ridge]

Are you referring to books by Sheldon Axler?

[marble solar]

That's a bad book from the very basics

for from the very basics

[cursive orbit]

I read it as my first introduction to linear algebra

[marble solar]

Survivorship bias

[cursive orbit]

Fair lol

[sudden kindle]

Its a good book

[marble solar]

I'm not claiming otherwise, it's just bad for a first book

[forest ridge]

What's the exact book title?

[marble solar]

Linear Algebra Done Right

You could literally search "Linear Algebra Axler" and it'd come up tho

[forest ridge]

Thanks, I'll check it out.

[gray gazelle]

Is elementary analysis by Ross a great introduction to analysis?

also same with a book called basic topology by Armstrong but for well... topology

[sharp latch]

Most people recommend Rudin for analysis and Munkres for topology

[gray gazelle]

don't really have access to those in my library 😦

[prisma snow]

I've heard of Armstrong

Can't remember if it was a good or a bad review though

[gray gazelle]

[prisma snow]

I think it's fine. Try it and let us know

Munkres too long anyway

[ornate shuttle]

have you not used library genesis?

also rudin is impenetrable if you're learning analysis for the first time

[gray gazelle]

I like physical copies better

mhmm, that's what I've read too

[ornate shuttle]

you can definitely give it a try
you can find a copy on amazon for like 10 or 15 dollars

i've heard understanding analysis by Abbot is good but i haven't read it

[gray gazelle]

I mean I'm fine with pdfs but these are like the only books my library has so just wanna know if they're good or not

[ornate shuttle]

rudin is the gold standard so yeah its good

[gray gazelle]

not rudin lol

[gray gazelle]

these two

[ornate shuttle]

this guy can tell you more than me
https://www.youtube.com/watch?v=42IP_U4NnBc

[gray gazelle]

👍

[gray gazelle]

Thanks a ton

[ornate shuttle]

mhm

[misty wyvern]

All of math is an appendix in one of the volumes of Taylor's PDEs.

[gray gazelle]

[marble solar]

No

Ross is bad

[lapis sundial]

Best intro to analysis is

1. tissue for tears
2. baby rudin

[summer shuttle]

Can anyone reccommend some books on how to create cool patterns mathmatically and geometry in 2D an 3D? anywhere from beginner to advance

[orchid musk]

You could look into geometric algebra for computer scientists

[summer shuttle]

sadge, all those books look really dry I should write a book that's abit more artistically focused haha

thanks though I'll look into it

[orchid musk]

It’s not too dry actually

Strays away from the typical math format and is used mainly for people in computer graphics

[summer shuttle]

this one?

https://www.amazon.com/Geometric-Algebra-Computer-Science-Revised/dp/0123749425

[orchid musk]

Yeah

[summer shuttle]

I'll grab it thanks

[gray gazelle]

[iron granite]

Rudin's existence**** of n-th roots proof gave me mathematical insecurity.

[sage python]

@iron granite you mean existence and uniqueness?

[iron granite]

yes, that one.

Uniqueness is intuitive, but the existence portion comes out of nowhere.

Like, how tf are you supposed to know how to do any of that.

[sage python]

Sorry I was out a while

The idea should be that like

So fixing some real number x, we want to find some y such that y^n = x

Well, you take the set of real numbers whose nth power is less than x

If you show that's bounded and non-empty, it has a supremum

You wanna say that this supremum is your desired y

[iron granite]

I get the idea behind that but not how you arrive at those specific inequalities.

For me, it's unmotivated.

[gray jungle]

i know how you feel i felt the same once i 1st dealt with it give it some time take it slow and you'll be fine ,the book is not meant to be rushed its a bit dry and challenging but its just a book ,one thing that helped me especially through chp 2 is using other references for certain ideas thay was left unmotivated

[iron granite]

I'll take your advice into consideration.

[marble solar]

I think the proof that root 2 is irrational is incorrect. It was either Ross or Abbott that had an atrocious proof

[marble solar]

Divine desperation

[ornate shuttle]

how do you mess up the proof root 2 is irrational lol

do you remember what it looked like?

[marble solar]

I think the goal was to make it more "intuitive". It's actually quite difficult to write a long book without some mistakes

Especially if you're trying to go for a consistent style of proof

Just try to latex up your homework sets consistently. The longer and longer it gets, the higher probability you have of muddling something

Or making it not fit neatly into the surrounding text

[ornate shuttle]

i see

[marble solar]

Even better analysis books like Rudin muddles the fourier series, multivariable, and measure theory stuff

So Rudin is only really good for chapters 1-7. Even then chapter 2 could be improved substantially

So then someone writes a book with more exposition at the same level, like Pugh, but then some other things get muddled like what a dedekind cut is

[iron granite]

Pugh's book seems really neat. Like, the exposition looks like I can eat off it.

[marble solar]

I agree that Pugh is a superior real analysis book, it's probably my favorite even though there are aspects to it that could be improved

If you want to go into analysis at least, I couldn't think of a better text

At that level

[iron granite]

Yeah, I'm starting to think buying Rudin off the bat was a mistake.

[karmic thorn]

You probably didn't lose much (possibly other than a chapter or two from your copy) if you bought the International/Indian edition.

[iron granite]

It's not about losing material, it's just that I feel like I could have bought a better book imo.

[karmic thorn]

Yeah fair

I have several textbooks at this point which I felt were rush of blood purchases

[iron granite]

That's my entire expenditure at this point. I'm going to sunk-cost fallacy myself into a math career lol.

[karmic thorn]

Lmfaooooo

I don't buy books that are horrendously expensive but I still buy about 2/month on average

[iron granite]

Bruh, most math books are too expensive here in India.

They're trying to sell me "How to Solve It" by Polya for 1500.

[marble solar]

I don't think Rudin is too bad of a book to buy

It's still an excellent text to learn analysis from

Sometimes the slickness of Rudin is useful in narrowing your point of view to what the core is and what's just details

[iron granite]

I don't doubt Rudin, I doubt myself lol.

[karmic thorn]

There exist methods to get printed copies delivered to your home over here, if you can access the digital copies legally ofcourse. :)

[iron granite]

Sure there are methods, but I feel like I'm being ripped off. Though conveniently, the 1990 edition of How To Solve It is 450 rupees.

[ornate shuttle]

if you have the time, it's really satisfying to try and find alternative proofs for some of the theorems in rudin
it took me about 20 hours, but I managed to find my own proof for the cauchy schwarz inequality for complex numbers
that felt nice

[karmic thorn]

Another plus with an ancient canon text like Rudin might be that you can easily find solutions, problem sets, notes, etc. for it.

Like OCW Analysis 1 follows it I think

So the problem sets listed there are exercises from Rudin

That can also help you narrow down problems to a necessary minimum

[iron granite]

Compelling reasons.

I mean, I've already bought the book so why not lol

[karmic thorn]

Yeah definitely

[gray gazelle]

I am just on that part for the proof lmao

[karmic thorn]

Meanwhile I'll drop these notes with exercises on introductory real analysis here, they seem to be very clean and self-contained: http://www.math.louisville.edu/~lee/RealAnalysis/IntroRealAnal.pdf

[iron granite]

Much thanks.

[ornate shuttle]

i don't think some of the solutions for rudin's problems are correct lol
exercise 1.6 in particular
where it asks you to prove b^(x+y) = b^x b^y for arbitrary real numbers
i don't think the solutions are correct
they say things like sup{b^t : t in Q & t <= x+y} = sup{b^t : t in Q & t < x +y}
and b^y = 1/(b^(-y))
but i'm pretty sure you can't prove these two things without using the fact that there is always a rational power of b between two real numbers

that has to get proven separately

[iron granite]

This seems like my time to log off.

[marble solar]

https://www.youtube.com/watch?v=D8BDdJ_PPwc&list=PL54Pt_mZzBqjcodJ_GqXxqG1WCCmq433k&ab_channel=ArtemChernikov

These seem to be pretty good

[karmic thorn]

I have this playlist saved

I will get back to this next year definitely, looks very good

[marble solar]

It's the first good looking lecture series I've seen on intro real

On youtube

[karmic thorn]

Oh wait

I had mistaken it for Chernikov's Mathematical Logic series

That Jesse shared before

220A

[marble solar]

yUh, it's real analysis 131A

[gray gazelle]

@marble solar

his proof is like since +-1 and +-2 are the only rational numbers that could've possibly been the roots of x^2 - 2 = 0 but are not and since root 2 is a solution, it cannot be a rational number

[marble solar]

Wouldn't it be x^2 - 2 = 0

[gray gazelle]

yea that

mb

[marble solar]

No worries, I think there was issue in the specific wording or what was done

Rather than the approach

It was either Ross or Abbott

I think of them as equally bad

[karmic thorn]

At this point I'd say both are better than Tao for a self-learner seeing analysis for the first time

[marble solar]

Tao's text isn't meant to be a text

It's meant to be a lecture

[karmic thorn]

Yeah, it has that feeling

[iron granite]

Tao is more exposition than much else.

[gray gazelle]

[karmic thorn]

That book could be OG if only Tao bothered adding a few more problems everywhere

And showed how stuff works in the wild

[iron granite]

He asks a lot of whys.

tho

[karmic thorn]

That's a plus

[gray gazelle]

you could do that

[marble solar]

I think the issue is Tao gets bogged down in detail

And rigor

[karmic thorn]

I have worked through Analysis 1 almost cover to cover and I didn't learn much

[karmic thorn]

I agree

[marble solar]

I still think if you're learning baby analysis, spivak's calculus is the best way

Since it's the gentlest while also not sacrificing good exposition and challenging problems

[karmic thorn]

I think Tao was trying to show a bunch of people how rigour and precision in math works with analysis serving as the medium

[gray gazelle]

can't find that in my library

so I'm stuck with Ross

[karmic thorn]

But got too bogged down in that goal

[gray gazelle]

I mean it's not bad I guess

[iron granite]

Z-library it.

[gray gazelle]

already done

but I prefer physical copies

[marble solar]

If there was a path forward to learning analysis I'd probably do something along the lines of the following ordering

[gray gazelle]

would order it but my family wants me to focus on school before ordering books like that lol

[iron granite]

I tell my parents that "it's an investment don't worry".

[gray gazelle]

I mean I probably won't even pursue a major in math (maybe a minor) but it's just interesting

[marble solar]

Spivak's Calculus -> MVC with some theory & baby differential geometry (frenet equations, fundamental forms, etc.) -> Linear Algebra with basic theory and proofs -> Baby point set topology -> Spivak's Calculus on Manifolds -> Pugh

[karmic thorn]

The accusation in my case has changed from "you already have so many books untouched" to "what will you do with these books once you read them completely". Mark of a true sigma grindset.

[karmic thorn]

LA after MVC?

[gray gazelle]

[marble solar]

You can do them in either order

[gray gazelle]

these are books or topics or both

[marble solar]

A mix of both

[karmic thorn]

Hubbard develops them concurrently btw

[iron granite]

Tbh I can sell some of mine at retail.

[karmic thorn]

LA/MVC/basic differential forms stuff

[marble solar]

Yeah, every book that I've seen of concurrent development doesn't do necessarily a great job at it

[karmic thorn]

Few buyers for math books here

[marble solar]

It goes bogged down in details

[karmic thorn]

I think Hubbard is fine

[gray jungle]

you think its that bad?

[iron granite]

Mfw Spivak Calculus is 20000 rupees (almost 300 dollars) on Amazon.

[gray gazelle]

even fewer sells for decent books here

[marble solar]

I mean the path I advocate for is roughly the path I followed, but I did linear algebra and baby topology before MVC

[gray gazelle]

yo what

it's 104 dollars for me

[karmic thorn]

Shit Luna is here

Tao is the best book guys

[marble solar]

It's ok to be wrong, we don't have to make a joke about it

[karmic thorn]

Maybe I'm downplaying its impact, but at the very least I can say not seeing any problems outside of core theorems did affect how much I learnt

[iron granite]

Anyone else feel like Tao is judging them for lack of rigor when doing the exercises?

[karmic thorn]

All the time

But I continued to write a lot of crank proofs

[marble solar]

Nah

I've been judged by Terry personally for lack of rigor and I just thought "man, why does he care so much about details with this chart argument"

[karmic thorn]

You did grad analysis with him, right?

[marble solar]

I took 246C back in 2018, although half-way thru I did withdrawal

Due to having to be away from class for like 3-4 weeks

[karmic thorn]

Ah, I see

[marble solar]

Eh, saved $50k

[gray gazelle]

[karmic thorn]

[gray jungle]

i felt like tao was a good intro to rigour before something more dry like rudin

[karmic thorn]

By withdrawing from a single class?

[iron granite]

Eh, that's only enough for 177 copies of Spivak Calculus.

[karmic thorn]

I have the same thoughts kinda, like now I feel Tao didn't give me much but the fact stands that learning analysis from any other book was a major pain when I started Tao.

[gray jungle]

like i felt much more capable of doing texts like rudin/h&k after half of tao
it helped so much building some proof maturity

maybe its bad for analysis i suppose

[gray jungle]

ye pretty much

[iron granite]

I wouldn't feel confident claiming 1 + 1 = 2 in front of the guy. That's my 'Calc 2' take.

[gray gazelle]

hmm would be Tao be nice after smth like spivak?

[gray gazelle]

lmao

[gray jungle]

i heard spivak serves good with proof based calc so prolly not

[gray gazelle]

is a first pass reading in calc necessary to tackle Spivak?

[gray gazelle]

maybe a decent understanding?

hmm

[analog pollen]

60 for me

[gray gazelle]

pugh is sus

Figure 148

sus 💪

[sharp latch]

Yeah Tao is good if you don’t have a great background in sets/logic/functions/etc but if you do it’s super not fun. But it’s self contained which is nice

[surreal phoenix]

I'm going though Understanding Analysis right now, would Tao's be good after that or should I try my hand at something tougher?

[gray gazelle]

Kolmogorov and Fomin

[quick hornet]

no

it helps

its not necessary

[misty wyvern]

name a more iconic duo than asking if you have the prereqs for spivak and not reading spivak

[cursive orbit]

lmao

[trail kernel]

not a book but this yt channel explains a lot of very advanced math concepts in a relatively clear and understandable way

https://www.youtube.com/c/RichardSouthwell/videos

extremely underrated

[sudden kindle]

Where can I learn about spectra of compact riemann surfaces?

@sage python

[timber mesa]

asking for book prereqs and then reading them without the prereqs anyway

[swift sleet]

Any good recommendations for learning about all different types of interpolation and when to use each type?

[misty wyvern]

harmonic analysis is all about interpolation i recommend grafakos

[sage kelp]

What other analysis books are similar to or little less terse than Rudin besides Pugh?

[iron granite]

Apostol Analysis or Pugh give me an answer

[karmic thorn]

Pugh

Although Apostol also does some complex memes

Smh all of you asking for first analysis textbook just read those notes I sent yesterday

[manic fox]

what notes

[atomic venture]

looking for basic trigonometry books or any other resources

[frosty girder]

as far as i can tell, only in the first chapter

[iron granite]

So I only have an older version of Pugh Analysis available. @karmic thorn

There isn't much difference b/w editions, is there?

[karmic thorn]

Isn't the last part devoted entirely to complex?

[frosty girder]

i havent gotten that far so

[karmic thorn]

I'm not sure, you should check out the preface of the latest edition

[frosty girder]

also u dont need to do that much
its outside of the scope of a normal intro analysis course

(^ur fav emoji manan)

[karmic thorn]

http://www.math.louisville.edu/~lee/RealAnalysis/IntroRealAnal.pdf

[frosty girder]

lets make everyone grind these notes

[karmic thorn]

Yes

Everyone finish chapter 1 by tomorrow

Analysis Reading Group lesgooo

[frosty girder]

1 chapter a day should be enough

[karmic thorn]

Time to rename LRG to ARG

[frosty girder]

and whoever fails to finish
gets shot :catgun:

[frosty girder]

:nice:

[marble solar]

no thanks

ok first thing wrong is

glb instead of sup and inf

[karmic thorn]

[iron granite]

better than glub

or lub

[frosty girder]

[iron granite]

@karmic thorn There are only 2 editions for Pugh. lol

[karmic thorn]

I'm not sure, that might be the case

[iron granite]

I'm not losing much, first ed is only 350.

Guess I'm getting that.

[karmic thorn]

Use it a bit before buying lol

[iron granite]

That goes against my hoarder instincts.

Well, I did read about 10-15 pages of Second Ed online.

So I'm comfortable with my purchase.

[sharp latch]

Can't wait till Reese Witherspoon's book club does Introduction to Real Analysis Course Notes from Louisville University

[karmic thorn]

[sage python]

@sudden kindle Bergeron is good

[gray gazelle]

im looking for math practice problems

i know theres khanacademy

its good

but i was wondering if there was more websites

[iron granite]

What kind of problems do you want to practice?

[gray gazelle]

algebra or calculus

[iron granite]

hmmm there are problem books online which you might like

you can dl for free using some specific websites

(Zlibrary)

[gray gazelle]

im on it

but im not sure what to do next

[iron granite]

well first of all do delete the screenshot as there's risk of discord T&S

you need to first look up problem books online

search google for some

when you have the name, use the website to search for the book

and dl

[analog pollen]

Look up a pdf of a book like stewart calculus and make problems there

[iron granite]

You can also use https://tutorial.math.lamar.edu/.

[analog pollen]

Do those have problems?

Damn

[iron granite]

Some solved problems

[analog pollen]

Oh

[gray gazelle]

thx

also stewart calculus has price

ill check paul online math

[iron granite]

that's whats the website is for

[gray gazelle]

no piracy please

i found free stewart

spivaks calc > bb rudin for an intro to analysis?

[iron granite]

It depends?

Why don't you sample each of them first?

[quick hornet]

spivak is barely an analysis text

[iron granite]

^ that too

[quick hornet]

that isnt to say its unsuitable though

but theres a pretty big philosophical difference between the two

you cant compare them with "better" or "worse" really

like, i'd have no real reason to crack open spivak nowadays but rudin is still a useful reference

[iron granite]

I don't have a lot of words to say about Rudin

Except that it's terse

[quick hornet]

its a fantastic reference text since the proofs are succinct, the exposition minimal, the results as general as possible (given the material), and the statements precise and direct

this also makes it pretty pedagogically unsound

like if you cut the intro paragraph and all the fluff sections from a wikipedia article

you CAN still learn from it, and if you wanna do metric space stuff ASAP for some reason, its still the best choice

[gray gazelle]

I'm looking for more of an intro and I've heard Rudin can be a bit difficult for that

[quick hornet]

but itll be harder than any other popular text

[slow matrix]

how about pugh? some people are saying it a better and modern rudin. but is it good as reference?

[quick hornet]

(idk why youd wanna do metric space stuff ASAP though, its basically just ℝ^n but without heine borel)

not familiar with pugh but its not popular as a reference

so take that as you may

[gray gazelle]

I mean I'm reading through spivak rn, and I've liked it so far

[iron granite]

Rudin references the word ordered set, field, irrationality, sequence and approximation of root 2 by finite decimals in the first page.

Doesn't seem that heavy at all but then I flipped to the next page and had my confidence as a math student robbed

[gray gazelle]

[coral narwhal]

this has me interested, @quick hornet what are your recommendations for analysis(intro)

[quick hornet]

no comment

[slow matrix]

thankfully I'm physics student, I don't need confidence in math.

[quick hornet]

never taught it

and not very familiar with the texts

[gray gazelle]

[iron granite]

Some say Pugh, some say Apostol and some say Rudin.

No general consensus

[gray gazelle]

why not just read all of em

[slow matrix]

yeah, those three are most of what I heard too.

[coral narwhal]

book recommendations argues which [undergrad course] book is better every other day

[iron granite]

Buy one that you like and use the rest as pdf for reference?

Seems like an optimal solution.

[gray gazelle]

I mean I benefit a lot from it tbh

like now

since we are on the topic of intro books, what about topology?

[iron granite]

Munkres

and I haven't even started it.

Man is neat.

[gray gazelle]

lmao

[iron granite]

I mean, it's the only topology book that is consistently recommended online.

And it looks dope too so

there's that.

[stray veldt]

it ok, but a bit long

[iron granite]

that's usually a good thing, I find

[stray veldt]

wait until you do point set lmao

[iron granite]

That is needed in Analysis, right?

[stray veldt]

a bit

any analysis book will introduce the topology of R^n

[iron granite]

Then I'll probably get to it in a few months

[stray veldt]

general topology is ... a bit gross

[quick hornet]

idk about gross

its very nice and very boring

[iron granite]

what even is topology? No one has given me a concise explanation yet lol.

[stray veldt]

topologies are too general, they lead to gross behavior

[quick hornet]

like intro group theory, except most of intro group theory is relevant later on

[stray veldt]

you have to impose a bunch of extra stuff to do actual nice stuff

[quick hornet]

which is not the case for sufficient conditions for your topological space to be locally regular

[quick hornet]

the study of topological spaces

at least, thats what point set topology is

[stray veldt]

also terminology is gross

[iron granite]

I'm afraid to ask but, what is a topological space?

[quick hornet]

terminology is a fair point

[stray veldt]

why does my book call compact spaces quasi-compact

[quick hornet]

a set with a topology on it

dont worry this is the last layer of pointlessly vague definition

[iron granite]

now that I understand

[quick hornet]

a topology is a particular collection of subsets that satisfy some closure properties

[iron granite]

closed under infinite union

and finite intersection

?

[quick hornet]

yes

this may seem like a very general and pointless formalism

and well

it kinda is

[stray veldt]

i mean if you know what a metric space is, they induce a topology by the open balls

[quick hornet]

we usually add some extra sauce

[stray veldt]

which is the first picture that you should have in mind tbh

[quick hornet]

metric spaces have a LOT of extra sauce going on

theyre very well-behaved

even just hausdorfness is enough for 99% of use cases though

(screw off zariski)

[iron granite]

I'm afraid to ask this, especially in a math discord server, but why is a topology relevant?

[stray veldt]

its very general

you can define topologies on any object and suddenly have many tools developed by topologists at hand

[quick hornet]

also when the definitions are nice, theyre really nice

like the definition of continuity

compared to the one from analysis

[iron granite]

something to do with connectedness?

[mellow wren]

qualitative geometry is a cool way I heard of describing it

[quick hornet]

this simplifies a lot of proofs in practice

[quick hornet]

a continuous function is just a map (between topological spaces) where preimages of open sets are open

thats it

[iron granite]

I'm going to do analysis first, this is just an unnecessary headache for now.

[quick hornet]

no need to define screwey limits or whatever

or talk about neighbourhoods and epsilonics and shit

this definition is short and easy to work with

[stray veldt]

you can work in analysis with sequences to define continuity

[quick hornet]

(though applying it to specific topological spaces often forces you to get 'down in the weeds' a bit)

[stray veldt]

and thats so useful that you want to do it on topological spaces often as well

[iron granite]

Any whackiness in Topological spaces, like how AoC leads to Banach Tarski?

[quick hornet]

not really sure how to answer that

there certainly exist counterintuitive results in topology

like path connected does not imply connected

er

other way around lmao

connected does not imply path connected

geez i need to sleep

[stray veldt]

its not that weird once you know the definitions tbh

turns out intuition for connected is probably path connected

at least when you first do topology

[mellow wren]

In a topology the empty set and the entire space are both closed and open at the same time

[quick hornet]

thats just definitional though

[mellow wren]

Sounds funny tho

[quick hornet]

replace closed with "co-open" or "antiopen" or whatever and it makes more sense

"closed" is just a more convenient word

(and agrees with our intuition about ℝ)

idk, most of point-set topology is fairly... i wouldnt say intuitive

but like

you dont get anything that defies expectations per se

its more like, you dont know wtf your expectations should be

what does a nonhausdorf space even look like

fuck if i know

bad

thats what

you cant distinguish your points by looking at open sets so its like

open sets dont give you a geometry?

which means ????

i have no clue what this means but

its not good

i know that much

[iron granite]

I just looked up the definition of a Hausdorf space and

yeah, how can a space be non-hausdorf?

[stray veldt]

hausdorff used to be part of the definition of topology

until zariski

(ok, i dont know whose fault it is but probably zariski)

[slow matrix]

there's book counterexamples in topology, maybe it can help

[quick hornet]

consider the topology {{}, {1, 2, 3}, {1, 2}} on the set {1, 2, 3}

[iron granite]

Fuck it, I'm doing Topology first.

[quick hornet]

note that 1 and 2 violate the hausdorff condition

they belong to the same open sets

namely {1, 2} and {1, 2, 3}

i cant type

[gray gazelle]

[quick hornet]

now this isnt really a useful topological space

besides for week 1 examples

but there arent many useful nonhausdorff spaces in general

the only one i can think of is the zariski topology

[stray veldt]

its a horrible example nami

[quick hornet]

which is VERY useful but i struggle to explain it here

[stray veldt]

even for week 1

like wtf is this

better example: line with two origins

i guess you need to introduce more stuff formally before it makes sense

but when it does, its something tangible

[iron granite]

Why does Topology seem more tangible than Analysis...

[stray veldt]

because you dont know topology

[iron granite]

I don't know Analysis either.

[quick hornet]

i mean loch it just feels weird to define hausdorff before you can define counterexamples

thats fine in a grad course

but its a bit of a leap in intro topology

[stray veldt]

ok true

introduce them in week 3 when you do separation stuff

and then you forget 3/4 of the definitions

[quick hornet]

that implies we're doing separation stuff for some reason

[gray gazelle]

[stray veldt]

you should know they exist i guess

separation axioms that is

[karmic thorn]

Are seperation axioms that T1 T2 stuff

[stray veldt]

yes