i did
#book-recommendations
gray gazelle
fossil island
Thoughts on Gallian Contemporary Abstract Algebra? https://people.clas.ufl.edu/cmcyr/files/Abstract-Algebra-Text_Gallian-e8.pdf
Just sort of going through my future course schedule and past syllabi lol
fluid bay
ugh why are homomorphisms on page 200....
sage python
Yeah this feels way too inefficient
gray gazelle
wow a book that's even slower than dummit and foote
fossil island
Seems like for a first course it goes from ch 1 to ch 12
marble solar
Do you like Diff. Top. by milnor Tterra?
marble solar
So I read it before going to my MS school
Plus a bit of alg. top.
My topology prof just said "I hate diff. top."
So we just did knots instead LOL
marble solar
I like it!
I like short books
It's hard to beat it for how fast it gets things done
gray gazelle
short books are good
whenever im looking at a long book im just like "please get to the point"
marble solar
it's what, a third the size of spivak's calculus on manifolds
gray gazelle
something like that
static crest
make a book with only theorems and non-trivial proofs, no examples or exercises
to the point and short
gray gazelle
marble solar
Huh
marble solar
The point are the examples/exercises
Nobody cares about theorems if they didn't work for a large class of examples
static crest
smh twas a joke
marble solar
It can be a joke
fossil island
everything in prepositional logic
marble solar
but the joke should be
"Just don't prove anything"
"Draw a few diagrams, state the theorems"
"Apply them to examples"
fossil island
"Draw a few diagrams, state the theorems"
category theory...?
marble solar
I'm mainly referring to milnor's diff. topology
static crest
smh why would you need an algebra prof, like y = mx + b, learned that shit in grade 9
why do you need algebra in uni
smhmh
fossil island
^^^^^
marble solar
I figured if I'm applying for PhDs in PDEs/Topology
gray gazelle
347 is going to make me hate algebra
marble solar
I should know what the basics are
gray gazelle
pdes 
marble solar
of both instead of just going off on some tangent of topics that I've been thrust into
gray gazelle
moonbears you should share a cool PDEs fact
fossil island
its engineering
static crest
so funny, I forgot to laugh
gray gazelle
pdes on manifolds
marble solar
If heat flow is zero, then the heat equation is the harmonic equation
gray gazelle
semi related: there seems to be a lot of work done on laplacians on riemannian manifolds, looks neat
marble solar
Also the maximum is at the boundary
That's why when you bake brownies
The ends are burnt
static crest
no way, maximum of a harmonic function is at the boundary
gray gazelle
ah, finally, practical uses for mathematics
static crest
unheard of
static crest
lol, we spent like 3 lectures on harmonic functions
and a bunch of reading from the tb
static crest
rip
gray gazelle
wow what the fuck you're doing more than we are 
marble solar
Holomorphicity is equivalent to harmonicity which is equivalent to analyticity which is equivalent to the integral mean value theorem which is equivalent to the cauchy riemann equations
static crest
wow what the fuck you're doing more than we are <:opencry:586078614865444865>
you guys are probably doing it more rigorously, I know some friends and I go through the tb properly and also refer to ahlfors at times
but most of the class just memorizes specific theorems and use cases
couldn't prove something if their life depended on it
that's probably why our course just zooms through, while you guys take longer
fossil island
you can squish boundaries on integrals and get the same value
that's all i remember
from complex
static crest
in the meanwhile, I coded a complex function plotter
in C++
do help me in my hw
😎
instead of actually just doing the hw
fossil island
do share
static crest
I just recently added the ability to do the nth derivative
and sums
I can model taylor series in it
was a very fun side project
to avoid doing actual work
fossil island
yeah its fun to mess around with all the pretty colored graphs
static crest
they are very colorful
in this one, you can easily see the branch cuts
and the zeros and poles
gray gazelle
static crest
the brightness represents the magnitude, the color represents the argument, the gray lines are lines of constant real or imaginary values (they have to be at right angles if function is analytic)
static crest
sqrt(z^2-1)/(z^2+1)
sterile pelican
Out of curiosity Apostol's Calculus Volume 2 or Spivak's Calculus on Manifolds?
marble solar
It depends on what you're looking for
If you don't have computational background in calc 3, Apostol's Calculus Volume 2
If you have already learned a computational version of calc 3, then spivak's calculus on manifolds combines calc 3 + linear algebra
To do some cool stuff
sterile pelican
Hmm this is the assumption that the background is Spivak's Single variable Calculus and H&K's LA
marble solar
Ok
What I'd do is get some source of standard calc 3 problems
like open stax calc 3
Learn those
And do spivak's calculus on manifold sthen
Cuz the calc 3 integrals are standard curricula, so ya probably need to learn it
marble solar
I mean be my guest
I got lucky there's no subject GRE this year
: D
Hopefully they kill it with fire
marble solar
I just want it gone
sage python
I mean yeah it's fucking stupid
marble solar
"Oh hey I got 3 publications but 75 percentile"
sage python
I'm just saying that it's more sneaky single-variable calc than multi
marble solar
"Try again next time"
sage python
Lol I tell myself that the only reason I didn't get UCLA/Michigan was because of the subject GRE, 790/77th
marble solar
That is true
I've listened to admissions faculty
They say if it's below 80-85th range
They look slant eyed at it
sage python
Since someone on reddit who I know had an overall weaker app was told "Yeah 800+ plz"
marble solar
My friend got 84th percentile exactly and got into 9/10 schools
sage python
Not too salty about UCLA since it's apparently no bueno but I'm a biiiit salty about Michigan tbh
marble solar
The only school he got rejected from was UChicago
Which was....the only school that literally invited him to apply
sage python
All of them
sterile pelican
Cuz the calc 3 integrals are standard curricula, so ya probably need to learn it
I am not exactly trying to learn standard calc 3 but rather wanted to understand what Spivak meant on the preface as he stated, "The formal prerequisites include only a term of linear algebra, a nodding acquaintance with the notation of set theory, and a respectable first-year calculus course (one which at least mentions the least upper bound (sup) and greatest lower bound (inf) of a set of real numbers)" my assumption is his single variable calculus and any rigorous LA, either axler or H&K.
sterile pelican
So the more logical sense is Apostol's Volume 2 before an attempt to CoM?
tired cloud
kind flower
i'm currently vibing with my formal lang theory textbook (hopcroft and ullman) but i wanted to do like. more advanced stuff
does anyone have recommendations 🥺
molten wave
more advanced how
molten wave
specifically languages, or computation?
kind flower
languages specifically but computation is also cool
molten wave
hmm idk if really know any literature on, like, language theory specifically. You'd probably have to start reading papers
it's all finicky stuff about grammars that isn't like well rounded enough to be in a textbook I'd say
maybe Parsing Beyond Context-Free Grammars
kind flower
ooo ok
yeah i mean i'm also doing a fair amount of linguistic syntax and with cfgs you get a bunch of overlap
do you have any suggestions for general theory of comp?
storm sleet
Ullman is solid so far for me, but I also wouldn't mind a more advanced book
Likely most of what you'd be looking for would be axiomatic logic, programming language texts, or books about computability
Because thats where most applications of formal languages are (in my experience)
molten wave
I mean I do type theory
not exactly language, not exactly computation
couldn't care less about NBQEXPSPACE and whatnot
storm sleet
One thats recommended often is Types and Programming Languages, as well as the subsequent Advanced topics in Types and Programming Languages
If you are feeling more adventurous and willing to dive deeper, homotopy type theory is a somewhat "newer" branch
molten wave
is that at me lol
storm sleet
Oh I was confused lol
I thought you meant you wanted type theory. Its 1am I need sleep
molten wave
nah I know literature in my field lol
obsidian basalt
Is it worthwhile to take the time to read another book of a certain topic of math (example linear algebra) even if you already did read one an went through what college gave you for such topic? What i mean is, it is going to help you to expertize a topic, little by little, reading different books for the same thing even if you already know the contents from reading other books and/or college? Or is not going to make any considerable difference?
molten wave
defo yes
I recently skimmed through a lin alg book (as a refresher to do numerics) and identified many gaps in the curriculum that I was exposed to
generally anything you learn in early undergrad is going to be sufficiently simplified and disconnected from complicated concepts, that when you do eventually learn those concepts you will have to relearn the basic topics to recognize new connections that you can now make
hence grad level books in algebra, analysis etc
velvet briar
Lin Alg in particular is famous for being taught badly, haha
That is, a lot of STEM get half-introductions to the course
molten wave
I don't think badly is the right word
if you're not doing pure math the "good" introduction is not worth the effort
velvet briar
I mean - even for STEM it could be taught better. A lot of people come out of these courses without knowing what a basis is
Or at least that's what I've seen. Yeah you definitely don't really need to know what the dual space of a vector space is, ect
molten wave
kinda depends tbh
velvet briar
I could be wrong of course and I don't know how much you in particular know lin alg @obsidian basalt
molten wave
if you're doing physics that has any sort of mathematicalness to it then you do
but then again physicists have their own mechanisms for coping with it
"transforms like a duck"
velvet briar
Lol is that a duck that transforms like a duck?
hasty turret
Yes
molten wave
it makes sense to them idk
there's probably a rigorous formal system you could build around it
hasty turret
Physicists are mysterious
velvet briar
I got talking a little too much, haha. Yeah revisiting different books is usually fruitful.
obsidian basalt
Honestly i am a bit addicted to learning from books since this year. I feel i don't even want to do practices from college anymlre and just use bibliography (recommended by the pure math career i am following and/or by people here). I have this feeling that learning through books, even if it takes twice for me, it is more fullfilling. I mean, i want some honest answer, would be better for me to attend to classes from my university (considering it is not even top 100 in the world) or just keep working with pure math books and go that way?
molten wave
do you think you are able to critically judge your own output
obsidian basalt
That is a really good question you made
velvet briar
Are you currently in a pure math course, and at the mercy of your prof's marks regardless of your choice?
molten wave
if an exercise asks you to prove a theorem, how confident are you that you won't make some mistake without noticing
it takes a certain level of mathematical maturity to get past that
if you're there then go ahead (but of course not at the expense of your grades at uni)
(I have the privilege of being able to skip most classes provided I deliver homework, your situation may be different)
velvet briar
I'm a firm believer that you can learn from a book and get competent enough to be certain of your proofs. Despite that, if you are supposed to be attending classes, building rapport with your profs is very important
molten wave
@velvet briar I think initially everyone needs external governance
the question is for how long
velvet briar
It's never a bad thing to have your work checked ofc
molten wave
no like
you could be so unaware of potential pitfalls
that you fall into them
without noticing
I have a favorite exercise in this regard
I would ask people to rigorously prove that limit of sum is sum of limits
not a particularly difficult theorem, but many fail
velvet briar
I think I even got this one wrong yesterday
I don't really know how to handle the case when the sum is infinite
molten wave
nah for finite sum
of two things
if the sum is infinite it's not even true in general
which is a pitfall which you need to be aware of
obsidian basalt
My situation its kinda strange. I started pure math this year, but honestly I am not very sure how much I want or not to attend classes now. A part of me it's sure that i should stick with attending to classes but other part of me wants to just keep reading the books and doing practice from them with all the patience from the world (like managing by myself how much time i want to put into a course without feeling rush, i never like to feel i am rushing math, if that makes sense. I like to do all the exercises from a book, day by day, with all the patience of the world, etc...)
But about having someone external to check if i am doing things the correct way always it's a really important thing to consider for myself.
molten wave
anyway back to books
obsidian basalt
Also i realized that abstract algebra it's being a bit harder for me sometimes, so I probably should stick with formal studies until I can build some maturity as you said
You made me reconsider a lot how i want to manage this, but in a good way
sturdy sail
Guys, what books on measure theory you guys recommend as a first introduction to the subject?
Tao's book? Schilling's book which focuses a lot on the stochastic aspects of the theory? Any other recommendations?
stray veldt
my recommendation is introduction to measure theory and integration by ambrosio, de prato and mennucci
ambrosio is a famous italian geometric measure theorist and the book are basically the notes he used for years in his classes
sturdy sail
I see
Oh well
I think I actually should make the scope of the recommendations a little bit more restrictive.
Because I have a preference for books with a lot of exercises in it.
I kind of only use lecture notes or short books with not much exercises whatsover mostly like a way to revisit the topic when I feel that I need it.
stray veldt
the book still has ~10-20 exercises per chapter
sturdy sail
Of course you could "You could just try proving the propositions in the book as exercises"
I do that when I feel that it is definitely possible
stray veldt
and you can always use exercises from other books, but ok
sturdy sail
I do that when I feel that it is definitely possible
But it is still good to have a section only intended to problem solving.
sturdy sail
and you can always use exercises from other books, but ok
Yes
You are right
Maybe I will read this one
But still
And then just check for exercises in other books
It is definitely possible, but it still would be good to have some recommendations of books with tons of good problems.
gray gazelle
Guys/girls, what comes after calculus III?
hasty turret
calculus IV
gray gazelle
Does that exist?
hasty turret
Depends on the institute
gray gazelle
Interesting
I bought a calculus I and II for dummies book and in the book it says there is even a calculus III book.
hasty turret
You buy books?
gray gazelle
Yup
sturdy sail
You know these all these "Calculus ***" mostly just exist to organize the curriculum of a given institute.
It's not like it makes sense in general
gray gazelle
Oh, that is interesting
sturdy sail
"Oh what comes after Calc III?" Idk, what have you studied in Calc III? Differential equations? Differential and Integral calculus of functions from R^n to R^m (some people call it vector/multivariate calculus)
gray gazelle
My next question, what topic should I learn after calculus
sturdy sail
So it really depends
It also depends on what are your goals in the future
If you want to go deep in math
The it's pretty natural to study real analysis after calculus
And if you haven't studied linear algebra before
You definitely should already have studied it by now
hollow current
somehow when i see book "... for dummies" i understand that it is bad
sturdy sail
And then, after analysis, it's pretty natural to go after studying point-set topology and basic abstract algebra.
Since these are very important subjects which are the basis of a lot of different fields in math.
hollow current
somehow when i see book "... for dummies" i understand that it is bad
i mean no offense but these books are usually really crap
sturdy sail
Ranging from analysis, to geometry.
Topology and Abstract are pretty much the substrate of fields like differential geometry, differential topology, Riemannian Geometry, complex manifolds anything dealing with manifolds in general.
While topology is a pretty natural language to study continuity of functions in analysis to define various other concepts.
Algebra is also linked with pretty much anything related to number theory.
So yeah, I guess that algebra and topology are the most important topics you should study after analysis/calculus.
Because they are used widely in various other fields, pretty much any field in modern mathematics.
And then I guess that after that, you are kind of more free to choose which topics to study.
You can focus more on number theory related subjects, more analysis related subjects, more algebra related ones, etc...
You don't have a linear path to guide yourself anymore.
hollow current
ari please take english name
sturdy sail
Yup
calm crane
ari please take english name
aiya can do @ ariana right
sturdy sail
You are right
valid moth
my math journey:
calc 1 -> hs algebra -> 3 years break
comp LA -> get on this server and learn group theory by osmosis -> algebra (from actual book) -> LA (abstract) -> etc
sturdy sail
I guess that it's pretty much "pick your favorite" after some point
calm crane
yup
but ig basics are like real analysis & point set topology complex analysis group/ring/module/galois
valid moth
@calm crane smh, that has too much analysis
static crest
my journey's been
calc 1/2 -> lin alg + calc 3 -> odes -> probability -> complex + pdes + signals* -> control theory* -> (planned future) -> groups/fields/rings -> galois theory
- courses are more eng than math, but I put them there anyways
hollow current
my curr math journey
calc 1 + hs algebra -> calc 2 -> a bit of discrete math -> basics of computation theory -> a bit of abstract algebra -> real analysis -> a bit of abstract algebra -> a bit of topology (really a bit, just very basics) -> linalg (i am here now)
valid moth
yeah ssee that's like a more typical sequence i think poros
i think the usual sequence is like calc 1, 2, 3, LA, DEs
hollow current
and stuff from set theory and logic taken by the way
valid moth
and since i didnt know any other paths, i was going down that initially
calm crane
<@!223340518619217920> smh, that has too much analysis
real and complex isnt a lot🤔
hollow current
oh ye i also have ODE now
calm crane
my math journey is very wtf
valid moth
look, i don't even know real analysis 😔
calm crane
i started a rh crank
valid moth
lmaooo
valid moth
i basically started doing pure math in december 2019
static crest
look, i don't even know real analysis 😔
just look at real numbers? easy?
calm crane
reals is competition of Q in usual boring metric
static crest
do complex analaysis and replace all $z \in \bC$ with $x \in \bR$, boom real analysis
hasty eagleBOT
PorosInMyAshe
hollow current
oh also i had probability theory but this is just abuse of counting methods
valid moth
@calm crane wrong
R is the terminal coalgebra of a ... im too lazy to type this out again
you know, it's the usual joke
laugh
@static crest complex analysis??? just learn calc 3, it's jjust the dim=2 case 😎
static crest
true?
calm crane
oh you learnt at before👀
sturdy sail
Mine is like
Differential and Integral calculus of real valued functions -> Linear Algebra -> Differential and Integral calculus of functions from R^n to R^m ---> ODES and a bit of PDES ---> set theory (I've learnt from a lot of sources, Halmos, the beginning of Munkres' Topology book, YouTube etc) and logic (Ebbinghaus) ---> real analysis ---> Point set topology ---> Groups/Rings/Fields/Modules ---> (VERY non linear path)
calm crane
ah
valid moth
nah i didnt learn that much
but like just some introductory topics
and some basic cat theo
i think AT is a good context for first learning cat theo
sturdy sail
I guess that I will focus more on manifold related topics, algebraic topology and category theory
These are the topics I'm mostly interested in
This year was a quite weird one
Because I've learnt so much stuff in a really non linear fashion
calm crane
mine is
wikipedia ->
reading random online notes on calc and analysis ->
(physics -> lin alg and DE -> manifolds and rep theory) and
(crypto -> elliptic curves -> basic algebra -> com alg -> alg geo)
and random
point set -> a bit AT and geometricization conj
somewhere somewhere i read first part of jech as well
calm crane
self-study paths are so weird lol
lol ikr
valid moth
math professor parents
calm crane
valid moth
there arre kids who like
sturdy sail
I have pretty much took a glance at a lot of math subjects all at once, just to see which ones I'd have the most interest in.
sturdy sail
So it was really really weird
valid moth
can you imagine
calm crane
insane luck
sturdy sail
self-study paths are so weird lol
Yup
calm crane
damn
sturdy sail
But I guess that at some point you will get yourself down the track
calm crane
no i have like
valid moth
yeah, he's also self-studying ahead but i think his path is less random
sturdy sail
Damn Ari gwusheuduwix
calm crane
lol
sturdy sail
I guess I will do what I just said and study cat, algebraic geometry and riemmanian geometry
But starting next year
valid moth
calm crane
osmosis it
sturdy sail
I've read a little bit of MacLane
calm crane
ahh
valid moth
yeah i mean you should like
sturdy sail
It really is quite a pain
calm crane
my cat is through osmosis
valid moth
know several different areas of math first
b4 cat theo
that's really how it makes sense imo
valid moth
you see a definition and you think "oh, here are examples of things i already know"
valid moth
hm? im not saying you need to know things in depth necessarily
calm crane
and it starts appearing everywhere
valid moth
but like a bank of objects that can serve as examples
because if you dont know any math
and you just start doing cat theo at first
it's not really gonna be grounded
valid moth
like you're not gonna have any examples
calm crane
nlab
valid moth
like if you're learning cat theo for the firs ttime and ytotu know like, LA, calc, group theory: even just that let's say
you already have examples you can come up with
valid moth
"hmm... vector spaces and linear maps... continuous maps... homomorphisms..."
sturdy sail
Yeah, cat theory shouldn't be studied before at least have studied a bit of different algebraic structures (like groups, rings, modules, vector spaces) and topology. I guess these are the prerequisites imho.
calm crane
you dont rlly need topology just basic point set knowledge
sturdy sail
What cat does is pretty much just "hey, you know functions from set theory, let's call it morphisms from now on, you know homomorphism from X algebraic structure, let's call it morphisms from now on, you know continuous maps from topology? Let's call it morphisms from now on, you know smooth maps from manifold theory? Let's call it morphisms from now on."
So it's pretty much a HUGE generalization of a bunch of different math fields
sturdy sail
And if you haven't studied any math field in depth which deals with structures and structure preserving maps, you don't need to study cat theory at all.
calm crane
i.e. a topology, a smooth structure, a ring structure etc.
calm crane
And if you haven't studied any math field in depth which deals with structures a...
yesh
sturdy sail
Because that's pretty much what cat theory is all about.
valid moth
yeah you dont need to know AT already
like i said imo
first learning AT
is a grreat context i think for first learning cat theo
sturdy sail
noy rlly set theory
It's because there's also the category of sets, so that's why I gave this example.
valid moth
there's even like a recent book that does cat theo + point set at the same time
gray gazelle
Okay tf is going on with your name ariana
calm crane
Okay tf is going on with your name ariana
it is a story
sturdy sail
there's even like a recent book that does cat theo + point set at the same time
OOOOOH
It's topology from a categorical point of view.
I know it
Too bad I can't buy it
Because I live in the middle of nowhere in Brazil
valid moth
amigo mio, libgen
sturdy sail
Math twitter was shilling this book a lot
Wow
It's already on libgen?
It's a pretty new book
gray gazelle
Math twitter was shilling this book a lot
math twitter...... lbgt and ugct do sound pretty close.....
calm crane
it explicitly says it isnt a intro
calm crane
honestly dont bother with it
sturdy sail
I just didn't give myself the time to search it on libgen.
calm crane
go read a normal book
sturdy sail
math twitter...... lbgt and ugct do sound pretty close.....
Math twitter is actually a thing lmao
calm crane
sturdy sail
Since the first day I discovered it
gray gazelle
someone find the pin about trans math twitter and category theory pls
calm crane
lmao what
sturdy sail
Thanks to math twitter now I believe that every mathematician in existence is a beautiful trans/non-binary person
calm crane
there is a trans math twitter?
valid moth
sturdy sail
A THING
calm crane
oh cool
gray gazelle
thanks arch
gray gazelle
gray gazelle
based ari
sturdy sail
Math twitter is all about trans, category theory and algebraic geometry
I mean
That's all it is about
I am not even kidding
molten wave
do complex analaysis and replace all $z \in \bC$ with $x \in \bR$, boom real ana...
all differentiable functions are smooth
calm crane
so i went to more spicy parts of twitter
calm crane
not saying👀👀👀
gray gazelle
kpop twitter
fast portal
hmmst
static crest
all differentiable functions are smooth
all functions are smooth
molten wave
that's when you replace with $x : \bR$
hasty eagleBOT
mniip
gray gazelle
expand your mind to different topologies
hmmmm
sorry i'm not ready to accept |x| as a smooth function
gray gazelle
expand your mind to different topologies
Is it some kind of College level math joke I'm too schooler to understand?
Anyways, should I do Principles and Techniques in Combinatorics or A path to Combinatorics for Olympiad prep?
karmic thorn
A Walk Through Combinatorics by Miklos Bona.
Not sure if it's specifically good for olympiads but it's a great book on combinatorics.
The undergrad level.
gray gazelle
oh ok. Any idea between these two?
if you're aiming to do olympiad stuff then you should probably look at things meant for that
otherwise just do a regular textbook
karmic thorn
Haven't used either, sorry. But I'd say you should just grab any book and try a variety of problems, probably from multiple books.
gray gazelle
i say probably since i'm not an olympiad and never was
valid moth
hey what's up, i'm the peruvian math olympiad
gray gazelle
if you're aiming to do olympiad stuff then you should probably look at things me...
That's why I'm asking as these two are the two most recommended ones. But nevermind.
gray gazelle
yeah, i am a competition
You are a competition
valid moth
yep
valid moth
@gray gazelle i mean, i probably do have them stored in my non-working memory
let me clear the cobwebs
gray gazelle
give me an interesting analysis problem
valid moth
okay i remember one
a generating sequence is one with first term a_0 = 1 or 2, and a_{n+1} is obtained from a_n by either putting a 1 to the leftmost of the digits of a_n or a 2 to the rightmost of the digits of a_n (this is all decimal). question: is there an (infinite) generating sequence with no terms divisible by 7?
this is from like the peruvian junior math olympiad
i dont know what year
or what round
gray gazelle
yeah it looks like one of those olympiad problems lol
valid moth
it's pretty easy
gray gazelle
molten linden
where do i find the calculus book
only one i found on the internet was calculus of variations
calm crane
what is this
molten linden
gelfand's algebra
molten linden
okay okay
calm crane
https://cdn.discordapp.com/attachments/740692549110595587/783736750983872522/unk...
oh it says future books
so maybe like it hasnt been published b4
molten linden
hm maybe
gray gazelle
I don't think the calculus one was published
hollow peak
give me an interesting analysis problem <:hmmm:773801269532819466>
prove that the p-adics have countable basis
gray gazelle
@ merosity
calm crane
prove that the p-adics have countable basis
how are you defining p adics
it can either be extremely trivial or slightly harder
gray gazelle
hollow peak
I didn't know there was an alternate definition
Completion of the rationals using the p-adic valuation
flint forge
field of fractions of Z_p which is a limit works too i think
then you have to topologize
hollow peak
Wait I posed it wrong
I meant to say
Prove the set of closed balls with real radii and p-adic centers is countable
hollow peak
mhm, but the tricky part is that p-adics are uncountable
calm crane
not quite tricky if you explicitly write out the open sets actly
flint forge
oh i've never seen that construction
calm crane
so jus like
flint forge
thats cute
hollow peak
okay well I didn't have that at my disposal
I used a lemma where every ball centered at a point is equal to the ball of equal radius around any interior point of the ball
calm crane
open balls are a+p^nZp
this is direct from definition of the metric tho🤔
hollow peak
then since rationals are dense in the p-adics, the balls end up being countable
hollow peak
I was definitely proud of it 😎
valid moth
Z((x))/(x-p) smh
calm crane
show Z[[x]]/(x-p) is isomorphic to Zp first
kind flower
I mean I do type theory
oo do you maybe have any recs for that??
or just formal systems stuff in general
flint forge
type theory for functional programming is good
as a not-programmer
i can say that its a good intro
sour barn
does anyone know good practice for algebra
I am going through my uni finite mathematics course and although its not necessarily tested on that much, I find my ability to apply all of the various rules including factorization and distribution to be lacking
my brain tends to shut off completely and it is hard to guess what to do next in simplifying an expression or equation
(so far ive done graphs, functions, complex numbers, quadratics, inequalities, exponentials, logarithms, and trigonometry)
karmic thorn
Lang's Basic Mathematics might be worth looking into.
sour barn
ok thankyou
gray gazelle
basic mathematics is great
obsidian valley
@gray gazelle
gray gazelle
cobalt arch
gray gazelle
analytic geometry..... are we talking like, the analysis of PDEs on riemannian manifolds? laplacian stuff?
cobalt arch
No I guess what I want is pretty elementary
obsidian valley
Just give him an RG book
cobalt arch
Wait
gray gazelle
after all analytic geometry is just the study of geodesics and riemannian geometry with a fixed global coordinate system
cobalt arch
I will give you my university's curriculum
dense pewter
let's write one
gray gazelle
ok, but we can do it for nonpositive curvature instead of just curvature zero
unforuntately positive curvature gets fucked
karmic thorn
Coxeter's book is good
It starts with basic Euclidean geo and ends with intro to differential geo
cobalt arch
Vector space ,vectors operations. Linearly dependent and linearly independent vectors. Orientation of the level and space. Coordinate systems in plane and space (general, orthonormal and polar). Coordinate system transformations.
Vector Algebra (internal, external and mixed product, applications in calculating areas and volumes). Line and plane in space (parametric equations, vector equation, equations of line as intersection of planes, Cartesian level equation). A set of parallel levels. Stack of straight intersecting planes. Point distance from the straight line and plane. Distance between lines. Rectangular projections. Second degree surfaces.
The translation is from greek
karmic thorn
Hmmmm, this sounds more like a first course in linear algebra.
obsidian valley
Sounds like linear algebra + high school pre-calculus
cobalt arch
So some terminology might be butchered
obsidian valley
no mention of determinant
use Axler
cobalt arch
Isn't that a la book ?
obsidian valley
That looks like a LA course lol
obsidian valley
rawr
gray gazelle
i mean you just described basic basic LA lol
obsidian valley
uwu
calm crane
ok, but we can do it for nonpositive curvature instead of just curvature zero
there are entire books for hyperbolic geo
gray gazelle
😌
lusty jacinth
would any of you happen to know of any good books for AP Physics C: Mechanics, and AP Physics C: E&M?
or their college equivalent
marble solar
The hardcore good one is griffith's Electrodynamics
I like Halliday and Resnicks physics volumes for freshman physics
@lusty jacinth
lusty jacinth
oooh I thought Griffiths wasn't intro level
lusty jacinth
huh
marble solar
halliday and resnick = heck and wreck
marble solar
I like halliday and resnick
I think it's very good
But the problems can be difficult
If you can do those problems you can probably do the AP problems with ease
They have exercises that are easier than the problems though
Oh there's also this one
The open yale course has really good physics lectures
The course material looks free ish and there's an accompanying text
It's very good
karmic thorn
Yepp, Shankar's books are excellent but I think they don't have exercises. Halliday & Resnick is a brilliant text with loads of exercises.
marble solar
There is a new edition with exercises @karmic thorn
karmic thorn
Owwww nice!
small haven
Anyone know a good proof based elementary geometry textbook? A modern take on Euclids elements if you will. I’m embarrassed to admit as a college student that my grasp on Euclidean geometry is shaky
hollow jetty
I love Halliday Resnick. My olympiad friend specifically swears by Halliday Resnick pre-Walker. The Cambridge A Levels Physics Coursebook is probably also worth having a look at. A different syllabus but the Cambridge A Levels books were really good back in my day.
valid moth
@small haven perhaps try https://www.amazon.com/dp/0883856190/ref=rdr_ext_sb_ti_sims_1
small haven
Oh, this seems to be exactly what I’m looking for! Thank you
sterile pelican
<@!379423701181136896> perhaps try https://www.amazon.com/dp/0883856190/ref=rdr_...
That is one great book
I have a 2nd hand copy of it and read it right after Gelfand's Geometry & Trigonometry
calm crane
The hardcore good one is griffith's Electrodynamics
thonk this is intro the true hardcore is jacksons
hearty steppe
Has anyone heard of Walter Strauss
hearty steppe
Yea
storm sleet
Any good book recommendations for elementary stochastic processes?
hearty steppe
I’ll definitely second that request
marble solar
Cat man, I'm a huge fan of stein and shakarchi volume 1 for fourier analysis
It touches on PDE as it relates to Fourier stuff
Strauss' PDE seems good, but idk it just never really sat well with me
storm sleet
*stochastic processes and stochastic calculus, for my request
placid sequoia
sorry I was lurking then my cat stepped on my keyboard
obsidian basalt
Is a good practice to sometimes read a math book in a light way (i mean reading the whole theory, looking for examples and maybe doing one or just two exercises per chapter?) I mean not doing it always but do so when you feel in a month where you want to relax a little
calm crane
yes doing every exercise is dumb
calm crane
I am pretty dumb then lol ouch
dw same
hollow current
i actually try to do most of exercises
but like not in one attempth
i can be reading chapter 10 and doing exercises from chap 1
flint forge
i dont think you have to do every exercise but you should do exercises until you feel comfortable with the material
more is better but you're also weighing that against progress
obsidian basalt
Thanks for all advices. Yes i was asking because i am advancing through three books but trying all of them in hardcore (or just doing all exercises) feels a bit overwhelming (or feels like "i am going to end next year with all of them" task)
(Well next year wouldn't be bad considering its december hehe)
valid moth
one heuristic you can try is do as many exercises as needed before you know very quickly how to do the rest
if they're actually 'exercises' in the sense that you always do know how to do them from the start, it's just a matter of practicing the computations, then you can try doing as many as needed before you feel like you've got the hang of the computation
flint forge
This is not quite true
In adv math books exercise can mean ‘actually very hard’
And it’s not always easy to know which is which
There are books where you can understand the material p well and still have to really work at exercises
valid moth
i know, but i was basically going off of stradex's case specifically
marble solar
The metric that I've come up with is give an honest try to 2/3rds the exercises
Actually solve 1/3rd to 1/2 of them on a first passing
static crest
I usually feel like I know when I can get an answer, and don't need to grind certain exercises
especially in like the earlier exercises in a section, which have like parts a) to h), if I feel like I have a good grasp from some of them
I just move to the next question
dapper root
valid moth
chm projected time of hartshorne completion: 2025
hearty steppe
don't feel too bad, I'm probably going to complete Velleman by April at this rate
Of course that could be me over exagerating the amount of effort I am putting into learning proofs with zero formal theoretical mathematics exposure just about
but Chapters 5-7 of Velleman are pretty much analysis level material
induction not so much, but I thought functions were definitely analysis territory
yea but this is not the applied/elementary level understanding of functions you use in precalc and regular calc I believe
where your not playing around with the idea of functions in proof writing
yea
thats basically what I thought of it as
intro to study of functiong in R^n
polar tulip
even "functions on R^n" is too broad for real analysis
you want some sort of regularity, measurable, continuous, differentiable, or something
timber rivet
get the newest edition of the book
flint forge
how does velleman work
i think IBL scripts are the best intro to proof
eh
i don't like toy problems
i think you can start proving actually interesting things
right away
I mean even basic like
pigeon hole stuff
let [n] be the set 1,...,n
show that there is no injection [n]->[m] if m<n
you can do really on day one or two
toxic haven
flint forge
i really dont like NT problems as the be-all-end-all of intro to proofs
that is fun
but again i think you can build theory while you do it
oh thats cool
thats not a toy problem then?
some kids here were talking about a summer class
where you couldn't assume induction
only WOP
which was stupid imo but whatever
@valid moth was in it
werent you
nah
if you want to make students prove induction works
fine
iirc that is how it worked
or at least there was an unreasonable delay
valid moth
yeah, the ross program
idk why they did that lmao
i think i just proved equivalence once and then ignored them and used induction in psets after that
flint forge
learned about a cool problem in dynamics/AT crossover today
mine taught intro topology
my ibl calc class
it was lit
point set is a great intro to proofs topic imo
its like
far less boring than taking point set after you know what youre doing
when it just becomes a meme
i dont like two of those topics
but thats personal
i hate plane geo and solids
idk why
flint forge
you are out of your mind if you think ive read it
valid moth
it's the pointset with cat theo perspective book
valid moth
well i mean, if you've skimmed it lol
flint forge
i like that book
valid moth
ah okay
flint forge
i got math3ma mixed up
Topology and Category Theory right?
its a good book and honestly would be my go-to for point set
flint forge
I think it resembles how modern topologists think
far more than like munkres
and it introduces two dry-ish subjects in a way that makes them less dry together
valid moth
yeah
flint forge
do they
i dont see much of a need for it
outside of rudin ch 1-3
i feel like taking point set to do anything other than AT is weird
you don't need anything more than like, locally metrizable stuff
i think you can cover the topology needed in functional in functional
valid moth
what kind of things do they use topology (seems like both AT and differential smooth manifolds stuff) for in econ anyways
flint forge
if someone is gonna spend a full semester on point set
you can get through the other stuff
like the topology of metric spaces is easy and short
and if you wanna throw in some functional stuff go ahead i guess
but i think the categorical approach is cooler and better motivated
like defining product topologies so that maps are cont.
i disagree
i think thats the second best
everyone needs intro AT
oh sure
yeah
i thought you meant a full course that was minimal point set -> AT
why not
you didnt actually say anything
you can replace any subject with 'self-study it'
Why even take AT at that point
just self study it
who needs to start w riemann integration, just read the defn yourself
oh
i mean all of my classes are 9 weeks
and to me minimal point set is like
2 weeks at most
oh sure
what
why
i don't get your reasoning at all
valid moth
wait, there are people who actually read all of munkres?
flint forge
the correct way to do point set
is to vaguely remember what is true
and google everything else
i think thats fine
1-3 slowly is silly
flint forge
honestly i think if you learn about fundamental groups
you will probably know
if you like AT
lol
ripe granite
why?
flint forge
i think thats true
almost every field requires some level of (co)homology and anything close to manifolds requires more
ripe granite
AT is not the only route to cohomology tho
flint forge
I mean if you wanna get picky and say 'not everyone needs it' the same applies to almost every standard course
but I think knowing cohomology without understanding the topological POV is silly
ripe granite
ok I retract my statement a bit, but what I meant to say is not everyone needs to know singular/cellular cohomology
de rham type stuff is introduced in 'non-AT' settings
flint forge
I think deRham is a pretty limited picture
ripe granite
deRham and sheaf cohomology might be enough for a lot of people
flint forge
again you can say this about any material
ripe granite
true
flint forge
If your argument is no math class should be required unto itself
then i might agree w you, to some extent
lots of people are
AT is required basically nowhere
required
to graduate
its also optional
almost everywhere
afaik
most places make you pick like some subset of quals to take
thats what ive seen so far at least
flint forge
i wish linear algebra were like
a year long class
there so much more of it i wish i knew and people offhandedly mention lemmas ive never seen all the time
hearty steppe
i think IBL scripts are the best intro to proof
IBL scripts?
Anyway @gray gazelle you think after chapter 4 I should be good to go with an actual analysis text?
ripe granite
I think spectral theorem(s) + jordan/rational canonical form should be enough to get started
and maybe some stuff on exterior/symmetric/tensor algebras
hearty steppe
you guys really think that the last three chapters of Velleman is not worth reading?
stray veldt
5, 6 and 7?
hearty steppe
5, 6 and 7?
yea
stray veldt
6 is probably one of the most important
you should at least read it until you are sure you understand induction
5 is important for vocab at least
7 is not important
hearty steppe
wdym by vocab
stray veldt
you need to know what surjective, injective etc means
image, preimage etc
and be used to that
also when i say 7 is not important i dont mean ever
but not immediately to do more math
looking at it now, its kinda insane how many exercises this book has
valid moth
why do velleman when you can do https://cdn.discordapp.com/attachments/716264872018706443/769260541947412541/proofs.pdf
stray veldt
i agree
i will add some exercises once this semester is over
and tbf i have no chapter on relations
marble solar
Yeah, I purposefully chose to apply to only schools where I can in some fashion not take an algebra qualifying exam
You'll never make me learn sylow
No I've read like
"Filling Riemannian Manifolds" by gromov
Gromov's books are good to read
He's good at writing, but be ready to fill in details
quartz pawn
Is this your first diff geo class @gray gazelle? jw.
marble solar
If you get sorely confused and nobody knows
You can actually email Gromov
And he will respond
quartz pawn
Do y'all know any good resoureces for understanding diff geo concepts. I want to find some 3D animations to get a feel for some of the concepts that my book is explaining but I can't really find any.
I understand some of them intuitively but I would still like to see some animations.
marble solar
Schaum's outline to differential geometry
Is a good text with lots of examples + pictures
quartz pawn
Vector Calculus by Susan Colley
marble solar
That's for the classical stuff Don
marble solar
If you're doing modern stuff, it's harder
quartz pawn
Thx
marble solar
some classes touch on classical DG to reinforce the calc 3 material @gray gazelle
quartz pawn
There are some diff geo concepts that get introduced but for the most part it's a multivariable calc text. It introduces the concepts of torsion curvature moving frames etc.
marble solar
e.g. at my CC the honors section to Calc 3 was intro to classical DG
quartz pawn
And has some stuff in the back related to manifolds, differential forms etc.
marble solar
We made it up to the fundamental forms
Yes! If you want to set up to go do spivak's calculus on manifolds
You need calc 3 + classical DG, linear algebra, differential equations, and some introduction to topology
quartz pawn
The concept of moving frames makes sense but the book is talking about viewing it from a geometric sense as a limiting plane and I want to see these planes converge that it's talking about
Well, it talks aobut the osculating frame at a point P on a path as being the limiting position of two other points P_1 and P_2 on the same path near P as they converge towards it
I mean that's what I interpreted it as too
And I'm pretty sure that's what it is also, but it's also some sort of "limit plane"
same with the tangent plane
I get what they are but I want to sort of see this limiting behavior that they're talking about and watch these planes converge.
It makes sense intuitively because in the singe variable case it's the tangent line is the "limiting line" of the secant lines.
So it make sense that sort of geometry extends to multivariable case but I just want a picture basically.
cause it's kind of hard to visualize.
Noice
thx lol
I mean literally lol
molten wave
why does that look like a weird song album cover
quartz pawn
lol i was thinking the same thing
looked like an album cover
for like ambient music or smooth jazz or something
broken meadow
the wave...
hearty steppe
@stray veldt you think the psets in Velleman chapter 5 are not as important as understanding the vocab
flint forge
doing exercises is how you become fluent in vocab
hearty steppe
thats what i thought
ill just skip chapter 7 since a lot of people tell me not to focus on that with Velleman
gray gazelle
btw, the other day i was talking about how springer mycopy is unavailable. i went on the springerlink website today, and it is still not available, but the UI has changed (they added options to buy ebook/softcover), so i'm guessing they're changing the website right now and mycopy is temporarily unavailable
gray gazelle
FROM WHERE CAN I STUDY ABSOLUTE VALUE AND INEQUALITY HIGHER ALGEBRA
hey, inside voice
that said
khan academy?
obsidian valley
HIGHER ALGEBRA
I'm not sure this is a great jumping off point to learn absolute value, but hey
gray gazelle
i havent done the test in a while
okay so
what the FUCK do i do with these ~215 pages of differential geometry lecture notes
like
is it time for me to become the next spivak 
quick hornet
gray gazelle
i'd post the full thing here if it wouldn't be an extreme dox
hollow jetty
Why do you attach your home address to your notes? 🤔