Mainly the calculus that isn’t covered in my lectures and homework
#book-recommendations
analog pollen
Riley
analog pollen
It might be lacking
Yeah but in combinations with internet
Does it still come short then?
karmic thorn
Then...you don't even need a book lol
Especially for calculus
There's Paul's Online Math Notes
karmic thorn
analog pollen
Then...you don't even need a book lol
Well I was planning on using this as a reference also
For mechanics
karmic thorn
It might be a fine reference text, I guess
But it's lacking as a book to learn from
analog pollen
Yeah ok thanks
Is there also a book that contains calc 1,2,3 in one book?
Or is it always seperate
willow pecan
Stewart
narrow talon
Is there a canonical order for Evans PDE?
analog pollen
Thanks
narrow talon
Doesn’t seem like a book to read front to back
willow pecan
I guess
narrow talon
Why chapter 5 of all chapters?
willow pecan
Chapter 2 is elliptic pdes
Chapter 5 is where Sobolev stuff begins
3 and 4 are sort of boring
And then the stuff after 5 is good
narrow talon
As in, why is the technical results of Sobolev spaces needed only to then go back to first order nonlinear PDE?
Oh okay gotcha, are 3 and 4 independent of 6 onward?
willow pecan
Chapters 2-4 are pdes with exact solutions
So 3 and 4 sort of are less important
The book is readable in order
But skipping 3 and 4 is pretty common
narrow talon
Okay thanks! Probably the type of thing that’s best to look at while going through the general theory for examples then
willow pecan
By in order I meant skip 3 and 4 completely
analog pollen
Stewart
Does Stewart’s book go advanced?
Or is it like everything that you should know is in the book
willow pecan
What do you mean by advanced
willow pecan
Yes
It's technically one long book with like 16 chapters
But a lot of the time you'll find versions where they cut certain chapters out
broken meadow
or rearrange the order
analog pollen
Yeah so if I would understand every problem in the book for example I would have a great understanding of calc right?
willow pecan
Like you can make it into a single variable calculus book by cutting out everything after chapter 11 or something
analog pollen
Yeah ok
willow pecan
Yes if you can do all the problems that would be very good
broken meadow
especially those in the problems plus section
analog pollen
But the whole version like singel and multi is the good one then
willow pecan
Yes
analog pollen
Ok ok thanks
broken meadow
i would recommend just doing a handful of the regular problems but focusing on knocking out most or many of the problem plus section problems
for each chapter
even the first one
which doesn't even have calculus
analog pollen
Ok thanks guys
broken meadow
ikr hes my favorite actor
No.
absolutely not
tbh i don't know who my favorite tamil actors are
hmm
oh rajinikanth is good yeah i like his older movies
those are good
who is naias
hmm
i am better with the face
oh ok
i might not know then
hmm
sage python
Yeah that's before metal was born
broken meadow
😭
static crest
damn, how does everyone have extensive knowledge of tamil cinema
the only tamil movie I've seen is like that really hyped one that had 2 parts
wooden sparrow
Bahubali?
static crest
oh rip
I did not know about that
I am not familiar with south indian languages
and conflated them
wooden sparrow
It's fine, there are too many languages
gray gazelle
Are tamil and telugu different languages i found out now lol
wooden sparrow
We don't know languages outside our state almost all the time
wooden sparrow
Are tamil and telugu different languages i found out now lol
Yes 
I'm from telangana. I can speak Telugu, but Tamil is like foreign language for me
Or any other language
There are few similar words
gray gazelle
I'm from telangana. I can speak Telugu, but Tamil is like foreign language for m...
Oh thats suprising to hear
wooden sparrow
But most of it, just goes over my head
gray gazelle
Ha ha
wooden sparrow
Let's keep off topic out of this channel
eager phoenix
Can anyone recommend me a course of books that can take me from barely competent at algebra+trig to calc 3 and beyond, possibly to linear algebra and set theory? I like books that explain the why, like why we use pi to calculate circumference, etc. I don't like rote memorization type books. I want to understand the mechanics behind the math and how it makes sense logically. maybe intuitive math is the right description?
static crest
it's definitely really common
but also pretty trash
unless you don't actually plan on doing any math past calc 1/2 in your life
eager phoenix
I've read Axler's would be a good start? what about beyond that?
night knot
Axler's Linear Algebra?
static crest
because stewart's almost sole goal is to impart the knowledge of crunching calculations, the proofs in it aren't particularly great, the exercises don't really motivate much
eager phoenix
precal
static crest
I don't even care if it's a boring read, it's just a bad read
static crest
the goal of calculus is to make people spam computations without much conceptual understanding?
didn't know that
night knot
I mean, half the time even if I have the same edition number, it's still different over the "Early Transcendentals" stuff
eager phoenix
i will transfer to university in about a year for B.S. Comp Sci or possibly Math if I can develop my appreciation of it more.
night knot
and this is why everyone that gets out of Calc-I hates it
static crest
that's one of the worst fucking takes of all time slim
static crest
you get people reinventing trapezoidal sums because they know calculations but have negative conceptual understanding
night knot
more-so there's no thought put into the calculations
static crest
of what they're doing
gray gazelle
Because calculating makes you an engineer
eager phoenix
I don't mind learning to calculate, I just don't want to learn how without the why we calculate the ways we do. I want to develop my math intuition to see at least a beginning of how these formulas and calculations were thought up.
gray gazelle
If you spend too long doing calculus you will slowly lose your identity as a mathematical person
night knot
as an example - computing limits by number chugging
eager phoenix
we have computers to compute, I want to understand metamath so to speak. the logic behind the math.
night knot
you calculate the limit by just throwing numbers
like if I want lim of x² at 0, I'll just shove in 0.001 and -0.001
then keep going on with smaller and nearer values
it's gross
oh
bless you then
this was my Calc-I class
well sure, but then you're taking continuity for granted already
you're not going to hop on to continuity before learning limits
but again, you're taking a good deal of things for granted when doing that
patent ibex
there's a ton of conceptual understanding
if you do a good job getting conceptual understanding in calculus the proofs in intro analysis just fall out
night knot
^
patent ibex
series manipulations
night knot
personally? Riemann sums leading to integrals
patent ibex
yes, riemann sums/fundamental theorem of calculus
sure, but the computational tricks become useful once you stop talking in generalities and work with actual functions
night knot
it really depends on where you take it
patent ibex
i didn't learn this stuff properly in calculus, but looking back a lot of it is accessible
yeah stewart is surprisingly good
night knot
Stewart for me at least was a guilty use book alongside Thomas
a book I'd probably only glance at because I had an assignment out of it
well sure, not fond of it myself
don't make me admit I actually read it for an exam lmao
patent ibex
i opened stewart several years after learning calculus because i forgot how surface integrals worked
night knot
didn't like it much after first checking it out - considering that the only reason I bothered to read it was having to do so for class reasons, I am not really fond of discussing it
plus I had started Apostol a year earlier and was midway through it
patent ibex
random people on the internet have a tendency to like the most abstract dry books
patent ibex
it was sarcastic
south ferry
i feel like i'm being called out
night knot
for me it all just boils down to finding Stewart a jog compared to Apostol
having it as an obligation did not cut any favors
Calc-I as a whole was boring, and half of anything discussed in my class at least was unmotivated
modest dirge
Coincidentally, I opened this channel to ask what the recommended books are for learning calculus
night knot
why do you assume that, exactly
and what does that have to do with people online?
again, I did not like it
I found it boring
nah that's fine lmao
sage python
i opened stewart several years after learning calculus because i forgot how surf...
Heh I had to do a surface integral for the first time the other day
@gray gazelle yeah you were reading wayyyyyy too hard into "guilty use"
He meant it in the sense of when people say "guilty pleasure" to indicate how infrequent it was
Every now and then take a break from the other one and check out Stewart's explanation
The reason I stand by that explanation is that he doubled down on disliking it
Like he's using Stewart as a cheat sheet basically and is eh about it overall is how this comes off to me
modest dirge
What do you guys think of Apostol's calculus book?
night knot
What do you guys think of Apostol's calculus book?
neat, but it did take me a long time to get through it without skipping stuff
sage python
Spivak
If you want proofs
If you don't... Stewart's the standard but it's expensive and I wouldn't imagine it's such high quality that it justifies the price tbh
quick hornet
stewart feels like its only 50% mathematics
and the other 50% is weird pedagogy stuff
sage python
Like books at that level are probably mostly isomorphic. Here's some integrals to do let's draw some thin lines and weird shapes to give cutesy heuristics
But at some point there's not much to say at that levle
modest dirge
Hmm so should I start with Spivak?
sage python
Here's how to compute stuff gg
So if you want computational calc probably just pick any cheap calc book you see
If you want conceptual explanation then yeah Spivak is the correct answer
modest dirge
Ok thank you I'll check it out
static crest
boy do I have a suggestion for you
whole rain
no D&F
static crest
shut
willow pecan
Lang is known for writing lots of books
sage python
Lang Algebra in this case
willow pecan
Lang is not known for writing good books
sage python
Lang I read just a tiny bit of
static crest
d&f on the other hand is known for writing good algebra books
sage python
Some of the "raw" field theory
modest dirge
Hmm I am getting differing opinions
sage python
Mostly to prove the existence of an algebraic closure and whatnot
steel viper
if you want to learn computational calc unironically read pauls notes and the fuckin 3b1b video
night knot
why the D&F hate lmao
steel viper
its long and wordy
sage python
But some of his uses of terminology make me straight up want to punch someone
static crest
why the D&F hate lmao
there's some people in the server who hate d&f too much and reject the one true algebra book sadge
sage python
D&F is like
static crest
their hate blinds them
whole rain
I don't hate it
steel viper
DF is a bit of a snooze fest
static crest
there's at least 3 people in the server who unironically hate it hate it
sage python
Use Jacobson
night knot
their hate blinds them
that's a shame
whole rain
I just "don't like it"
sage python
F[x]: people unironically dislike it
night knot
though, haven't seen Jacobson
sage python
But it's not a book that really creates passion in its hatred you know?
night knot
Artin was nice from glance but I can't find a copy
static crest
nah, when I was about to start algebra, someone in this channel was really railing against d&f, I forget their name
static crest
but they were passionately against d&f
night knot
so the only algebra book I actually have in physical is D&F lmao
static crest
I almost didn't use d&f because of them
steel viper
jacobson is nice but the second half of vol 1 is literally like
the most non standard material
sage python
D&F is something I strongly believe is suboptimal
steel viper
its like lattices lmao
sage python
Because in my mind
static crest
lmfao, lattices?
sage python
Like alright if you don't know what you're doing use Artin
static crest
they're fun I guess
sage python
If you know what you're doing use Jacobson or Lang
static crest
there's a bunch of cryptography problems based off lattice algebra problems
steel viper
yes
sage python
And I don't think D&F has any real case for it compared to the others
steel viper
stones theorem is neat though
gray gazelle
What if you sort of know what you're doing but sort of not?
static crest
d&f
sage python
It can be read by a child but so can Artin
In what I've read of it yeah I think it's fine
static crest
d&f holds your hand through most of it
anyone can read it
so if you sort of don't know what you're doing, d&f is fine
sage python
For you specifically mirza I think you're probably too strong a student for Artin or D&F
static crest
doubts
sage python
So if you don't like Lang try Jacobson, or possibly this one
static crest
but no algebra tb covering undergrad material has the exercises
modest dirge
http://www.math.stonybrook.edu/~aknapp/download/b2-alg-coverandinside.pdf
this is not basic
sage python
It starts from 0 lol
modest dirge
oh real analysis lol
sage python
Jacobson is more wordsy
Like how to put it
I guess idk Lang that that well but
If you ask someone to define a module
A lot of people might start listing axioms
Jacobson will say "a module is a homomorphism R->End(M)"
median sand
try https://github.com/wenweili/Yanqi-Algebra-3 (you probably have to compile it youself), liwenwei's a good person and scholar alike (x
sage python
So that's what I mean by wordsy
Jacobson would rather use English to describe something than symbols
median sand
A lot of people might start listing axioms
some of them will even use the magical word "operad"
(x
whole rain
@gray gazelle USE ROTMAN 
sage python
Not even about being concise it's more like
sage python
Preference for English over symbols
gray gazelle
Jacobson will say "a module is a homomorphism R->End(M)"
I feel this is just being a smartass/I don't see the point of writing the definition like this.
static crest
like knapp has nice exposition from what I see, its just none of the textbooks have exercises that are as nice as d&f I feel, that's my favorite part of d&f
sage python
8da it's literally a superior way of thinking about it
gray gazelle
I wouldn't mind if it was a remark after the"usual" x definition but
sage python
D&F's exercises I don't remember being anything special
Better than e.g. Aluffi
Tru
whole rain
aren't Aluffi exercises supposed to be hard ?
sage python
But yeah I guess D&F is like, maximally unexceptional in my mind lol
Aluffi's just bad
static crest
Jacobson will say "a module is a homomorphism R->End(M)"
this definition doesn't seem particularly different from defining a module as an abelian group with a ring acting on it in specific ways
whole rain
(never opened it, just what I heard of it)
gray gazelle
Ok sure, maybe it is superior, but it's also weird. Eg there is a way we usually write down the way that elements of R act on elements of M
gray gazelle
Not like phi(r)(m)
sage python
Oh (r+s)(a) = ra + sa etc
sage python
8da notation won't be a problem
static crest
seems pretty decent
whole rain
Oh huh
sage python
You just define a . x = phi(a)(x) gg
The point is that rather than just listing symbolic properties
gray gazelle
Yes, it is basically the same, it just sounds like being terse for the sake of being terse
sage python
You're thinking of modules as actions
sage python
Sure my point is he's using the word action
Rather than listing the axioms lol
That's all I'm saying about Jacobson
static crest
is it possible to not use the word actions
sage python
"It's a map R\times M -> M satisfying
(r+s)(m) = rm + sm
blah blah blah"
static crest
oh true, I guess that is technically right
whole rain
static crest
but I don't really see any point to defining modules without using the word "actions" somewhere
sage python
In fact I'd wager that's how most algebra books would describe it
static crest
like the whole point is that R can act on M
sage python
So Jacobson stands out to me in that regard
night knot
is Jacobson's Basic Algebra?
sage python
Yup
night knot
ah
static crest
actions are the fun part of algebra
imagine not using "actions"
everywhere you can
sage python
So yeah overall with algebra... my flowchart tends to be that
If you don't know anything you should use Artin
sage python
Artin starts early with linear algebra and is good at teaching you why algebra is interesting
night knot
Jacobson is like, $20 each volume
not here lmao
static crest
rip
sage python
Yeah Jacobson doesn't have clean pdfs I feel
night knot
acha
static crest
rip
night knot
Islamabad
night knot
isi lye SBB se pakr leta hoon
sage python
But yeah Jacobson I mostly looked at the very beginning, idk how good it is for field/Galois theory
But in the group theory part he introduces groups and modules simultaneously
night knot
bruh no lmao
sage python
Err
Monoids
Not modules
But yeah he uses the stuff to make the exposition kinda clean
Also one fact that I only know because of Jacobson is
sage python
Monoids
That was a typo
So you might've seen proven in your typical intro to group theory that G/H only really can be given a group structure if H is normal, right?
Well one thing I found out from Jacobson, which is tbh easy once you know it
Is that if you give me any equivalence relation on a group so that multiplication descends to equivalence classes
Then the equivalence class of the identity is a normal subgroup
And the equivalence relation is G/H
Which was kinda cute
And yeah in general I think he kinda divides stuff between groups and monoids to make the presentation slick
Which I appreciated quite a lot
static crest
G/H the set of cosets can be given more structure than just a set
even if H is not normal
sage python
I mean a G-set
static crest
is what dami is saying I think
sage python
No no no F[x]
What I'm saying is
Okay G/H can only be a group in itself when H is normal
But is there some equivalence relation that gives a group that isn't of the form G/H whatsoever?
Turns out no
Yeah probably
Though keep in mind there's a volume 2 of Jacobson
south ferry
basic algebra 2 has a nice copy 👀
gray gazelle
@static crest you should put "(poros)" in your nickname so people know who you are
south ferry
since chapter 1 is categories, it is clearly the superior tome
static crest
I see, so that's what you mean dami
sage python
Like I think he does rep theory, commalg, Lie algebras, etc
south ferry
i read the ToC
sage python
But yeah Jacobson or Knapp for you Mirza if you don't like Lang
south ferry
in reverse
and it looked the same
so i was confused
but he just lists the contents of both books for some reason
sage python
Lol she's not a 4 year old
static crest
it is
sage python
Or maybe she is but she's smarter than the generic 4 year old
static crest
lol
sage python
Mirza I'm basically interacting with you like Hegel
Calling you a child
And then throwing books at you to read and being like "Get good"
This is the Amin training arc
You'll become a fields medalist by the end of it
no u
south ferry
it has colours though 🥺
static crest
so a crank
got it
not mutually exclusive with crankery
same
crayons are also a nice snack
on the side
static crest
is this fucking gallian
south ferry
is this fucking gallian
yes
steel viper
Mirza I'm basically interacting with you like Hegel
south ferry
where is this from?
south ferry
for posting cringe
🥺 i only sent it as an example
i wish (x)f didn't look wrong to me
then i could have (x)f;g no problm
btw what's wrong with gallian?
don't wanna read it to figure out
dapper root
Isn't it just like too easy?
broken meadow
from someone who used it in a class for a first course in abstract algebra
it is just
kinda light
south ferry
nvm i think i can see
broken meadow
yep
like i mean this is ok for like intro look at algebra but oh well
i lowkey feel like i learned nothing this semester from my algebra course but thats ok
dapper root
Group actions r based
When I realized it gives you free maps into symmetric groups I was like
south ferry
Group actions r based
ik
south ferry
i think we motivated them with the sylow theorems? so not much motivation
dapper root
I disagree
Sylow is so based that's great motivation
that's only halfway sarcastic
it definitely doesn't capture the bigger picture at all tho hahaha
static crest
Isn't it just like too easy?
it treats you like you're in preschool, takes way too long to introduce some concepts, introduces certain concepts in really strange ways, has shit notation
dapper root
in other fields groups exist mainly to act on shit
static crest
Sylow is pretty useful in cases
it's just really annoying
to grind
Sylow is nice to use to guarantee the existence of certain groups
and shit
dapper root
I like it, but it is totally useless in most times you're using groups like
as someone who likes groups for gropus
I like it
but for most ppl for which groups exist to act on spaces
south ferry
i think the problem is that most exercises are just, "find all groups of order 345, up to isomorphism"
or sth
dapper root
and be automorphism groups and stuff
Sylow is just trash
The thing is I like thos problems 🤩
static crest
Sylow is just trash
based
static crest
unbased
south ferry
F
static crest
lmao
I was about to sya
if you wanna do algebra and geometry
just do the based stuff
dapper root
learn a ton about commutative rings
south ferry
commie rings are nice 
static crest
read hartshorne with chmonkey
dapper root
One does not "read" Hartshorne
static crest
lowmath hartshorne reading session
dapper root
one "suffers" through Hartshorne
south ferry
in highschool ofc
dapper root
Suggesting Hartshorne for ur middle aged white mom reading group
static crest
I wonder if I ever (d)evolve to a point where I end up opening hartshorne
gray gazelle
i might read a few chapters of vakils book at some point
gray gazelle
oh, why not?
dapper root
it's so verbose, and it has the stuff I want
static crest
I will apply algebraic geometry to electrical engineering, somehow
static crest
lol
static crest
maybe it's just an AG textbook thing
dapper root
and like I've found it in Vakil twice
gray gazelle
get vakil to send an email to every graduate student with comprehensive solutions to his ag book somehow
static crest
it's wild how few options there are for algebraic geometry textbooks
whole rain
Is hartshorne this wild ? Every single time I see people speaking about it, they're like that's a super duper hard to read book
static crest
lol
dapper root
Shika
...
hahaha
I can talk at length on this
I've probably spent like... 500 hours on it since May of last year
static crest
don't make chmonkey talk about hartshorne
whole rain
I've never opened it myself, I'm just curious
static crest
you'll regret it
dapper root
and I've fucking hurt
I don't think 500 is even enough
I spent like 12 -14 hours a day over winter break
static crest
sadge
dapper root
and spent multiple days on a single problem
There's just a lot about it man...
No
it's just frustrating
static crest
I also spent that much time a day over winter break (on d&f), but my textbook was far easier thankfully
dapper root
WHY IS CHOW'S LEMMA AN EXERCISE
AND IN II.4!!!!!1
Bruh every proof of that spends like 2 pages before it
static crest
!1!11!one!!
south ferry
static crest
"schematically dense subschemes"
I think this is nonsense to everyone in the channel
south ferry
idk
dapper root
I think II.2.4 is the one that says that Spec is adjoint to global sections
let me check
ayyyyy
I'm right
whole rain
"schematically dense subschemes"
I think this is nonsense to everyone in the cha...
accurate tho
dapper root
Yes, I know a lot of these exercises by their number
because you spend like 10 hours on them
static crest
dapper root
and it gets burned into your brain
south ferry
this i think
if X is a scheme and Spec A an affine scheme, then there is a bijection between scheme isomorphisms X → Spec A and ring isomorphisms A → O_X(X)
whole rain
and you meet someone learning algebraic geometry
dapper root
Oh that's in Vakil I think
static crest
are there any non-masochists in algebraic geometry
dapper root
Or well...
dapper root
Muf, that is like a corollary of this
dapper root
more generally the result is that
Hom(X,Spec A) = Hom(A, O_X(X))
it's very similar
but restricts to the case of isomorphisms
whole rain
I've heard good thing about shafarevich's basic algebraic geometry, what do you think about it chmonkey ?
dapper root
I've heard about it
Probably decent
it's not as hard as Hartshorne
I think the best intro scheme book I've found
is Algebraic Geometry and Commutative Algebra by Bosch
it's like 1/2 CA in the front
and then Schems for the latter half
but unlike pretty much every source
he will show you how to do the ugly tedious stuff
that's the biggest problem getting into it
No one wants to write down the really ugly crap like verifying cocycle conditions or how to glue schemes
since it's just a lot of data and tedious
but if you're new to it, and they say "one can verify that blah blah"
you don't know how tf to do that
south ferry
more generally the result is that
Hom(X,Spec A) = Hom(A, O_X(X))
i think he proves the general one
dapper root
Ah yeah
yup
That result is treu even if X is just a locally ringed space lol
but you can't prove it the way you want to with a scheme
since you can't just cover X with affine schemes haha
south ferry
yea idk any of that lol
whole rain
south ferry
it's mostly a joke anyways
whole rain
yeah math's a joke
south ferry
lol
sage python
Oh we talking AG now?
static crest
no pls
steel viper
broken meadow
Yes
sudden kindle
Tfw you take ag but dont do any work and learn nothing
ripe granite
Tfw you take ag but dont do any work and learn nothing
start writing up hartshorne solutions
youre a grad student now
write em all up before you go to grad school
and flex on everyone else
sudden kindle
Nah I'll learn AG by osmosis
ripe granite
is it possible 
but dont you want to be the person
who looks at a problem
and says "oh, this is just hartshorne 2.7.6"
sudden kindle
No 
ripe granite
sudden kindle
I already have plans for what to learn in the summer
ripe granite
?
sudden kindle
I want to learn class field theory 
ripe granite
ripe granite
I think im implicitly doing some class field theory right now?
sage python
You know what I'm gonna find a way to sell out but to a job which has low time commitment and just spend the rest of my time learning math instead of doing research
That way there's no pressure to be original and I can just learn cool things forever
Finessed the system
ripe granite
sage python
@ripe granite what kinds of stuff?
ripe granite
milnor K theory
sudden kindle
@ripe granite what is K2 group
ripe granite
idk it just seems like a generators and relations presentation for galois cohomology
K1 can be motivated pretty well I think
K2 just seems like
oh, we have these relations in the brauer group
and then you just set the higher K groups to be something similar
sudden kindle
I watched a talk where they said the number of ways you can write n as a sum of 5 squars is just some known constant times the order of K2(R) where R is the ring of integers in the real quadratic field Q(sqrt(n)).
ripe granite
yeah idk
the sum of 5 squares expression might correspond to a norm on some algebra
which would correspond to something in the brauer group
which would be something in K2
also idk anything about K groups of number fields/arithmetic stuff yet
we've only been dealing with function fields/curves
you might want to ask namington
🤷
hearty steppe
@sudden kindle what is your motivation for learning class field theory?
sudden kindle
I want to
hearty steppe
Nice, it’s cool to see people learn math just cuz XD
ripe granite
chad
sage python
Tbf PTY is into number theory so it's not too much of a surprise
hearty steppe
Gona just learn this random math topic don’t mind me
sudden kindle
It's not random, I've been wanting to for a while
hearty steppe
Im curious what insights you can gain from it
sudden kindle
Let me give a better answer. Last year I learned basic algebraic number theory and class field theory is the natural next step
sage python
Do you know Galois theory?
hearty steppe
Nope
sage python
Yeah then it'll be impossible to explain this to you
I guess hmm
So if you give me one field inside another
You can asks for automorphisms of the larger field which restrict to the identity on the smaller field
These form a group called (in nice cases) the Galois group
Class field theory is concerned with understanding certain types of field extensions where this group is abelian
hearty steppe
So a proper subset of a field that relates to the larger field it belongs to is this automorphism you speak of that gives us a Galois group?
With regard to class field theory the focus is with commutative operations?
sudden kindle
Class field theory is a about understanding the abelian extensions of a number field (data outside your base field) using the arithmetic data inside your base number field
A field extension is just a larger field containing your base field
sudden kindle
Like how R contained in C is an extension of R
But if your base field is Q, there are a lot more extension between Q and C
hearty steppe
Makes sense to me now
sudden kindle
Example Q(sqrt2)
Algebraic number theory in part is about studying those field extensions of Q (which are called number fields)
Class field theory straight up tells you everything you wanted to know about abelian extension
hearty steppe
I definitely see the motivation now, so your into number theory
sudden kindle
Yes
hearty steppe
I might consider learning some number theory sooner or later
sudden kindle
Follow your heart uwu
hearty steppe
Seems to be the most popular topic on this server
I mean I’m relatively open but I’m gona prioritize what is gona motivate me to use maths to dissect biology in highly abstract ways
And I think topology is a good area, especially for my project of trying to analyze hydrocarbon chain orientations
Or if I wanted to do more neuroscience, orientations of synaptic networks and emergence of larger subsystems of them
There’s also cool stuff about motivating what’s going on with microtubules in cells
So I think anything involving geometries, dimensions, stuff of that sort will be helpful
Also combinatorics
No it’s not this is before we even get to proteins
But protein folding is an interest too
Certain orientations cause amine groups to bind to specific orientations of hydrocarbon chains and that’s the tricky part here
Like this is before we even get to amino acids tbh
You need amino acids to get proteins
You need amine groups to bind to the hydrocarbons before you get to amino acids but then all the amino acids are different because there is all kinds of convulsions going on with orientations of the chains bound to amine groups where you get other complex molecules and well yea… very convoluted situation here
But it’s interesting. Apparently this is an unsolved problem in biology
We don’t know how this shit is happening with missing intermediary processes we are unaware of basically
icy frost
Hello! I want to learn some topology and I really want a introductory PDF/book on it. Not too harsh but not too hand-wavy. Any suggestions?
gray gazelle
As in, point set topology? I think basically everyone reads munkres
icy frost
Okay! I will look it up!
gray gazelle
I guess munkres is kind of dry though. I'm not sure there are many other popular point set topology books though
icy frost
But what other topologies are there apart from point set topology?
frozen arch
Hey
Does anyone use the book named "Cambridge IGCSE and O Level Additional Mathematics"
If someone does let me know
;-;
gray gazelle
I feel like I've heard people refer to things like algebraic topology as just "topology"
icy frost
Oh okay, so point set topology can be considered as the “standard” topology?
glossy grove
I guess munkres is kind of dry though. I'm not sure there are many other popular...
Where do you get a copy of Munkres?
gray gazelle
Where do you get a copy of Munkres?
Pdf or physical copy?
icy frost
Well I just found a pdf but I don’t know if it’s legal
gray gazelle
Oh okay, so point set topology can be considered as the “standard” topology?
I guess something like that, when people say "topology" I do generally think of point set topology
icy frost
Okay thank you!
glossy grove
Pdf or physical copy?
Anything, honestly. On Amazon, a physical, used copy costs 300$ o.O
gray gazelle
Munkres is pretty good imo
gray gazelle
Anything, honestly. On Amazon, a physical, used copy costs 300$ o.O
get the indian version then its like 10$ softcover
icy frost
Just watch Indian YouTube guys
frozen arch
Few responses
gray gazelle
add maths?
frozen arch
add maths?
Yea Additional Mathematics
glossy grove
What is that?
glossy grove
,rotate
hasty eagleBOT
gray gazelle
So up to high school math?
frozen arch
So up to high school math?
Yes
icy frost
Wow, you wrote that?
gray gazelle
Because there's nothing to discuss 
hasty eagleBOT
glossy grove
Wow, you wrote that?
Who wrote what?
karmic thorn
Hello! I want to learn some topology and I really want a introductory PDF/book o...
Hatcher's notes on point set topology are often recommended
gray gazelle
Anything, honestly. On Amazon, a physical, used copy costs 300$ o.O
how do they justify that price for a book??
frozen arch
So by the way you guys asked, you don't have add maths right?
glossy grove
how do they justify that price for a book??
I assume they don't print it anymore, for some reason.
icy frost
Hatcher's notes on point set topology are often recommended
Okay! I will definitely check it out!
frozen arch
So by the way you guys asked, you don't have add maths right?
@gray gazelle ?
karmic thorn
frozen arch
Also its funny that my chemistry teachers name is "Mannan"
And we have a manan here
As well
☑️
glossy grove
they don't? I thought munkres was a popular book
It is, but I have never seen a new copy for sale.
icy frost
Might be off topic but is point set topology a prerequisite for algebraic topology?
karmic thorn
I think some understanding is necessary? Not entirely sure how much.
glossy grove
Might be off topic but is point set topology a prerequisite for algebraic topolo...
You should know some group theory though, I think.
icy frost
Well I have self studied some algebra but I don’t know how much I should know
gray gazelle
One thing I will say, I tried to read hatcher for algebraic topology, and it was weirdn
icy frost
Imma snatch that real quick
gray gazelle
Hey guys, any recommendations for a book to start algebra from scratch?
scratch as in you don't know what a variable is ?
how you manipulate equations or ?
Like, literally from basics
I'm working on my basics, and I'll steadily move onto the higher level stuff
karmic thorn
Khan Academy
exotic cradle
Just watch Indian YouTube guys
i won't recommend munkres even though it is a classic. If you are aiming at self studying topology munkres will let you confused sometimes as some proofs are dry. the book is dry and lacks intuition. you can try to find modern ressources covering the topic in an intuitive way
icy frost
i won't recommend munkres even though it is a classic. If you are aiming at se...
Okay! I will try to find some intuitive resources online then!
quick hornet
hm interesting, thats the opposite of my experience with munkres
maybe i just took it when i was used to its style of presentation though
gray gazelle
Khan Academy
I cannot focus on videos, I need something that engages my hands, so a book came to mind
karmic thorn
I see. OpenStax has freely available books for Algebra 1, 2. You could check them out.
icy frost
hm interesting, thats the opposite of my experience with munkres
Well I might try to read it then and if it becomes too hardcore level pro math for me then I might try to switch to some more intuitive book
I think that I could borrow my math teachers topology book. However, I don’t really know what book that will be until like next week
quick hornet
it strikes me as fairly standard difficulty as far as mathematics texts go
not particularly easy but its no hartshorne
exotic cradle
for books
quick hornet
hey, sorry, for TOS reasons we cant allow the posting of links to pirated materials
discord admins will yell at us
queen rampart
I just started picking up topology
I tried both Lee's topological manifolds and munkres
munkres was so dry I got bored after a chapter
I've heard lee is narrow and does topology from a manifold perspective, so maybe after lee if your aim is algebraic topology or w/e you'd want hatcher or djungdi or kelley or willard
but I am enjoying lee
gray gazelle
Why are your takes so bad all the time
queen rampart
^^^
it just goes "here's a topology, these are called open sets. now moving on...."
I get that defining open sets independent of a metric space is going to be a little abstract but it's just like bam, done
gray gazelle
you might have been reading it upside down
queen rampart
I'm reading Lee's Topological Manifolds
that one was much nicer in its exposition so far
heard good things about janich, brown's topology and groupoids, topology without tears, and people seem to say willard/kelley/djungdji are good but not for a first time
but equally I have that awful habit of flitting between books whenever I get the slightest bit stuck and never actually do maths 
quick hornet
it just goes "here's a topology, these are called open sets. now moving on...."
yes this is the correct approach
what do you want it to do
queen rampart
karmic thorn
Topology Without Tears is okay as far as I read it
queen rampart
I feel like just saying open sets are ones that are in T, and T has these properties isn't very clear if you've not seen topology before
karmic thorn
Minus the horrible font
queen rampart
my current choice of books:
analysis - knapp's basic analysis
topology - lee's topological manifolds
algebra - aluffi's algebra, rotman's advanced modern algebra
quick hornet
alright i dont mind introducing neighbourhoods to motivate open sets
i think munkres only uses the term "neighbourhood" like
prisma snow
I've never seen a topology book that doesn't at least motivate the definition by saying it's a generalization of open sets in R which has these properties. I'm sure even Munkres must do that.
quick hornet
5 chapters into the topology section
prisma snow
Yay for Rotman!
karmic thorn
Aluffi 🤮
queen rampart
is aluffi really that bad lmao
karmic thorn
I didn't like it but then I'm a math newb
Category theory felt intimidating
Even the bare basics
gray gazelle
Topology Without Tears is okay as far as I read it
whoever name that 👌
queen rampart
this is all munkres says about open sets before moving onto X = {a, b, c} and then defining finer/coarser topologies
karmic thorn
That doesn't feel so bad if you've probably seen some metric space topology in analysis, right?
gray gazelle
(1) is redundant
karmic thorn
I think you need at least one of them?
gray gazelle
empty union is empty set; empty intersection is X
karmic thorn
Oh
prisma snow
I used Topology Without Tears. I think it's an easy book for people who aren't necessarily to mathematically mature. But it was a little heavy in explaining proof techniques and stuff too.
I would agree with empty union. Not everyone would define the empty intersection that way. Then intersection is dependent on the space you are working in.
queen rampart
yeah it seems every book either devotes 100 pages to "here's how to do the most basic of proofs" or is just a carbon copy of rudin or lang
sacred tulip
karmic thorn
I would agree with empty union. Not everyone would define the empty intersection...
Oh yeah, Munkres said something about empty intersection in the prelims chapter
gray gazelle
they do say intersection of a collection of elements of T.
prisma snow
Yeah, but that doesn't mean the definition of intersection has to depend on T
That's a convention you can adopt, but it's definitely not standard to define the empty intersection that way.
quick hornet
in practice you almost always need to check (1) by a separate case
so might as well mention it separately
same reason group theory books present "closure" as an axiom
karmic thorn
Closure is redundant since group operations are binary operations, and hence closed by definition?
quick hornet
yes, theyre functions G × G → G
karmic thorn
dark solstice
narrow talon
I don’t think a modern book on point set topology at the grad level really exists right now
At least not that I’ve found. Lee is probably closest, and even Folland contains everything I’ve ever needed thus far
queen rampart
I don’t think a modern book on point set topology at the grad level really exist...
djungdji?
I can't spell his name
the only things I've heard negatively about it is that it's hard.
narrow talon
Is that really modern? It’s not even in print anymore
narrow talon
As far as I know
queen rampart
I mean...please correct me if I'm wrong (as you can tell, I know very little topology) but isn't point set topology basically the same as it's always been?
any modern changes would be far beyond point sent topology
narrow talon
Yeah, by modern I mean not uncomfortable to read
willow pecan
narrow talon
Retypsetting Kelly would make it pretty lit, but as it stands everything is sort of eh
Also, does point set topology really take more than 100 pages 
I’m not so sure
queen rampart
yeah it seems like it has the same problem as analysis; do you really need to devote all that space to saying open sets and coverings (respectively: series converging)
narrow talon
I suppose most people learn this stuff for the first time in undergrad
But I wouldn’t be upset if there was an Atiyah Macdonald esque problem book on topology
For analysis, I suppose there’s Rudin
patent ibex
You can replicate that by treating most of the theorems in Munkres as exercises (other than the Urysohn-type ones)
narrow talon
Yeah I suppose you’re right
smoky surge
so i love elements of statistical learning but sometimes i feel like HTF has a fetish with not clearly defining variables
willow pecan
No
gray gazelle
lmao that's so based
narrow talon
so i love elements of statistical learning but sometimes i feel like HTF has a ...
Lol, HTF I feel like is just the book to read once you already know what to do and know what the variables are
Still pretty great tho, not dissing it
sudden kindle
But I wouldn’t be upset if there was an Atiyah Macdonald esque problem book on t...
Theres a problem textbook called Elementary Topology by Oleg Viro et al
Literally contains no proofs except for some outlines in hints
queen rampart
maybe I'm just looking for validation or maybe I just want to rant, but is aluffi's algebra just hugely overrated?
quick hornet
ive went on a rant here before about how much i dislike the text
but some people like it.
queen rampart
yeah I've seen nothing but positive things except "well it's not a very good category theory text"
so I feel like it's a me problem that I'm just not grokking it
quick hornet
its exercises are lacking and its prose is overly flowery
and i have a lot of tolerance for flowery prose
i learned algebra out of d&f lmao
queen rampart
hah, that's my path
over the years I've tried them all, but I finally clicked with D&F up until composition series
since they did not explain normal subgroups well in the slightest
somehow it's just not explaining the algebra at all and the category theory bits are really confusing?
I can't tell when their diagrams are diagrams and when they're objects in a comma category
I guess if I just use a more standard algebra text and then, some point down the line, a standard cat theory text...I'd be able to join them up again then
quick hornet
i kinda picked up the necessary category theory for algebra "naturally" as i went
the theorem statements just started gradually using more and more diagrams and categorical terms
and i got the hang of it
i did eventually read riehl's category theory in context
but not super seriously
queen rampart
yeah everything I've seen so far is just...I feel like if I learnt this in algebra, I learnt the cogs in category theory, and was told "btw X is just Y" then I'd be ok sure
quick hornet
i skipped the exercises
queen rampart
whereas rn I'm not getting the algebra or the category theory
but yeah ty
this was mostly me seeking validation but
I'll go back to knapp and rotman
storm sleet
Aluffi seemed weird
queen rampart
I have tried very hard and I just cannot vibe with it
the algebra seems off and the category theory seems off too
not wrong, just...oddly explained
smoky surge
Lol, HTF I feel like is just the book to read once you already know what to do a...
Hahahah honestly I’m comfortable with the intuitive concepts so I wanted a more mathmatecial understanding but holy dick sometimes it takes me like a thirty minutes to get through two pages
narrow talon
Hahahah honestly I’m comfortable with the intuitive concepts so I wanted a more ...
Yeah book is dense at
narrow talon
isn't there a slightly more friendly version of ESL?
Indeed! It’s absolutely excellent tbh
storm sleet
HTF?
smoky surge
It’s good for my math maturity I think struggling is good
narrow talon
Hastie Trevor and Friedman Elements of Statistical Learning
queen rampart
idk ESL/ISL were always something I read about online all the time, but were never ever mentioned in my research group (I'm in a research group with MLers)
people (my supervisor included) swore by bishop
queen rampart
I took bishop on a plane
queen rampart
I got through 1 chapter in 8 hours
narrow talon
Bishop is also great, but totally different vibe
queen rampart
which I did like
narrow talon
wym it's not ML?
smoky surge
I think the beginning is stats
queen rampart
as in, it was clearly written about probability and stats and not jumping on the ML fad
smoky surge
It goes into ML later right
narrow talon
Err, and what's the delineation?
queen rampart
it didn't jump straight into perceptrons and regression
one of the exercises of the first chapter was, in about 3 or 4 parts, to derive the formula for a gaussian distribution
narrow talon
Bishop is of course the Bayesian version of ESL (that and maybe Murphey) and though I didn't find it super rigorous, it's probably the least hand-wavey in terms of basically deriving everything from MLE
queen rampart
which was really quite eye opening
narrow talon
lol, reminds me on that book which tries to be the Godel Escher Bach of probability theory
smoky surge
See I just don’t have a lot of exposure to anything Bayesian
So idk how I would do with bishop
queen rampart
I think it more cemented that I just really hate probability
narrow talon
I find Bishop far less dry than ESL
queen rampart
and stats
smoky surge
ESL can be very dry
queen rampart
my entire phd is in fucking stats and I hate it
narrow talon
Nailed it
queen rampart
computational biology
smoky surge
Ahhhh
narrow talon
Probability != stats though
queen rampart
parameter sensitivities of genetic network ODE models under sexual reproduction
smoky surge
True
smoky surge
Bishop is of course the Bayesian version of ESL (that and maybe Murphey) and tho...
Yea there are so times in ESL where I’m trying to figure out where everything came from feels like steps get skipped
queen rampart
I don't have any real grounding in bayesian anything - hell, I still can't understand bayes theorem, but bishop didn't feel like it skipped things
it was just classical maths and it was hard
queen rampart
if anything I found the endless discussions about how bayesian bishop was to be more confusing
I've still got no clue why frequentist vs bayesian vs ??? matters so much
narrow talon
Very different perspectives
smoky surge
It’s how stuff gets derived right
narrow talon
In Bayesian setting, you’re trying to find a random variable, in frequent it’s, you’re trying to find a constant