12:15 AM
@MatheinBoulomenos hi mathein here ? :D

@KasmirKhaan yeah

@MatheinBoulomenos I want to ask you what is the best book for linear algebra , pure mathematical apporoch
june almost here, and under summer , my plan is to take linear algebra to the next level
i want to start from 0

maybe use the linear algebra section in an abstract algbra book

atm i have lang linear algebra, and roman
I know that !
But I really want to do it from scratch
many things i thought i understood , was not very clear to me
i want to spend a week or two of summer doing just pure linear algebra

But the treatment of most abstract algebra books that do linear algebra is from scratch

12:18 AM
aha
neat :D

and probably in more generality than a lot of linear algbra book, too

okay in 9 days kasmir will start his plan for world domination :D btw i hope u still here under summer , i really want to study with u :D

I heard that some books just work over $\Bbb R$ or $\Bbb C$

yepp that is so true
how is it going with french ?
and the other projects you doing ? :)

I'm sure the textbook my uni recommends for first years works only over $\Bbb R$ and $\Bbb C$

12:19 AM
@ÍgjøgnumMeg hello ! long time no see!

Hey man

I'm too busy with the courses of the semester to worry about French or my projects

Oh I really wish you good luck ! you are a good man :D

3 mins ago, by Kasmir Khaan
@MatheinBoulomenos I want to ask you what is the best book for linear algebra , pure mathematical apporoch

@LeakyNun Hello ! :D

its a good book have its ^^

My linear algebra course used parts of Jacobson "Basic Algebra 1"

am not saying this to be mean or anything but in sweden and germany etc
we do math different
books from USA are good but they do it in slow way, it is good ofc, but we do it faster
not saying its better, but for the genious ones like mathein, its the right way :D

I don't think it's available legally and free

I can check if it is on library
i can have it all summer i think =p

12:23 AM
but it's probably in your school library and it's published by Dover, so it shouldn't be too expensive

@Kasmir where do you study?

Stockholmsuniversity

ahhh nice, how is it there? I considered doing masters at Uppsala or smth
I mean in general.. Swedish education
lol

it is good if you are super smart ><

aha

12:25 AM
otherwise if you are like me , you would have alot of problems
the lectures dont give alot
and they expect you to understand things in two days
courses like monday and then wednesday
for me after couple lectures i come here for help :D

fair

@KasmirKhaan you have read algebra books before, right? Do you have any preference? Most introductory algebra books will have some material on linear algebra

i get lost sometimes too but it depends mor eon the type of course

#DummitAndFoote

>:(

12:26 AM
#DammitMyFoot

@MatheinBoulomenos yes but imeant more like first encounter of the subject

hahaha
#controversial

@ÍgjøgnumMeg thats gunna make him sleepy

i still have DF and Artin

i used to us DF to fall asleep

12:27 AM
I mean understanding what modules are and tensor products ect intuitivly

I've just checked my copy of DF back into the uni library after about 2 years

was not that easy for me

I have to deal with D&F because our professor has mostly been assigning problems from there, much as our in class treatment is different

dami do u fast?

@KasmirKhaan wow youve done more linear algebra than i have im surprised

12:28 AM
Yup. Got about 45 minutes left before I can eat and drink

nice :D

You?

Religious fasting or?

@Faust you can never do enough linear algebra >< all math depends on it almost
yes ! and yes

Yeah it's religious, it's for Ramadan

12:28 AM
haven't use D&F much. I like what I saw from Artin

Ah I see, forgive my ignorance lol

my favorite intro algebra book is Aluffi

Thanks mathein ! :D
ow >< , as I said , dont compare me to you

Lol it's fine :P

from me allufi is very hard ><

12:30 AM
artin is a pretty advanced pace book

I'm just starting intro topology
and idk what I'm doing

And yeah I might read a bit of Aluffi at some point, though I remember talking to my TA and he said that chances are, any dedicated dedicated book on some subtopic of algebra is likely to be better than any general intro to algebra book

if you can understand it i can't imagine having problems with much of any of them

@Daminark I agree with that
you have to start somewhere

that is my point mathein !
i feel am missing alot of the basic stuff
and knowing some other advanced stuff
which makes my understanding of the material very fuzzy

12:31 AM
but once you have a general foundation, you can jump into more specialized literature like for example Atiyah-Macdonald pretty early

BTW mathein ! :D
did you have time to do the 3 problems ?
they are really super important for poor kasmir :D

that will probably be better for learning that subfield of algebra than using just a small chapter in a general book that tries to do it all

Yeah for sure, hopefully after this year I'll be ready to jump in
I've been thinking about what to do for the REU over the summer and homological algebra comes to mind as a possibility

knowing your homological algebra well can help with so many things
alg top, algebraic geometry, algebraic number theory (tbf, it will probably only come up in a second course) and even group theory

@MatheinBoulomenos i n t e n s e

12:34 AM
and working with abelian categories can be helpful for seeing the point of categories

Yeah. I'll probably try to get an idea of what the courses I'm thinking of doing next year will look like, if only to minimize the content overlap. That'll probably inform my final decision, but homological algebra/spectral sequences/group cohomology was the first thing that came to mind along those lines and it seems like a decent candidate

@MatheinBoulomenos mathein ;D answer me :D

homological algebra/spectral sequences/group cohomology sounds really cool

and scary

This is true. I sat in on commutative algebra for a while last quarter

12:36 AM
@KasmirKhaan is it okay if I send it to you this evening (by this evening I mean 21.05.)? I looked at it and I can do that, but I didn't write it down so far

Sadly I got a nasty stomach flu and then got really dehydrated right afterwards (chest pains), so I kinda missed class for a week

@MatheinBoulomenos yes that is fine and thanks ! :D

And that made it very hard to follow from then on, because he did tensor products and directed limits

mathein is my hero _

personally i found it rather dry to learn commutative/homological algebra on its own - but could be just my own personal taste

12:38 AM
But before that he did give an intro to spectral sequences. He didn't really prove that they worked, just showed how you play the game, and used it to prove snake and 5 lemma, but it was a nice explanation and the stuff is definitely intriguing

Though to be fair, I think the only place where you need spectral sequences for group cohomology is the inflation/restriction sequence and you can prove that by other methods

@loch how else would you do it ?

I may be wrong, that's only from my limited experience with group cohomology

Would anyone have a source of a full proof of Cauchy integral theorem, i.e. if $f : \Omega \to \Bbb C$ is holomorphic, where $\Omega$ is a nonempty simply connected open subset of $\Bbb C$, and $\gamma : [0,1]\to\Omega$ is simple and closed, then $\displaystyle \oint_\gamma f(z) \ \mathrm dz = 0$?
Many proofs I encounter are quite handwavy

look at leaky :D doing line integration :D

12:40 AM
@KasmirKhaan :P

haha that was super nice , i enjoyed laurent series alot

@loch that may be a fair point. I haven't seen much of either so far, and I'll probably talk to the professors teaching various classes next year to see if there's a topic which optimizes usefulness, non-redundancy, and interest

and that theory of rouché for finding zeros

@Leaky I think Stein-Shakarchi has it completely?

@LeakyNun have you looked at Freitag's complex analysis book? I don't have it here right now, but I used it for intro complex and all the proofs I saw were careful

12:41 AM
Oh true Freitag is good

@MatheinBoulomenos was uber freitag?

I don't understand the question

But yeah it seems like the best way to prove it is to prove Cauchy integral theorem and then formula on a disk, and use that to get that holomorphic functions are analytic. Then you can use Stokes' theorem and get simply connected domains

Freitag is the surname of a German mathematician

@MatheinBoulomenos I was trying to make a joke on friday

12:42 AM
oh

apparently it didn't get through

sorry for not having humor, but to my excuse, I'm German

eh, does anyone of you have anything that is available online?

freeeetog

Stein is available online if you try hard enough :P
As is Freitag
Actually my copy of Freitag is 100% legal because Springer

12:47 AM
@LeakyNun I can link you the German lecture notes of my prof

nvm, my university is powerful
I got access legally
what happened to path-connected
why does Friday call it arcwise connected
@MatheinBoulomenos

@KasmirKhaan well technically i also learnt commutative algebra first so im not sure what i would do instead :p but in hindsight it wouldve been nice to see a lot of examples in algebraic geometry / number theory on the way - which might not show up if you're just reading e.g. atiyah macdonald (which is a definite read to learn commutative algebra though)
(technically vakil's text on ag introduces the necessary commutative algebra on the way so maybe that's a way- but ive never met a person who learnt commutative algebra that way)

Turns out with Fourier analysis you can prove the Sobolev embedding without integration by parts

hi @loch
I heard vakil

Every day that goes by, my motivation for learning the subject grows

12:53 AM
hi @LeakyNun

hello everyone.

@LeakyNun they're equivivalent for Hausdorff spaces, anyway

I am a beginner at topology.

@MatheinBoulomenos wait, they're different?
I don't see any difference in their definition
calls it an indirect proof

@loch I kinda learnt the commutative algebra I know alongside the number theory I studied for my dissertation, so everything I know is motivated by number theoretic applications lol

12:54 AM
assumes the contrary
proves the original statement directly
proves the original statement directly

I don't understand the reason behind the terminology of discrete topology.

lol

What's so discrete about discrete topology?

@loch okay I see >< i dont have alot of experience in the field yet but I ll keep that in mind thanks :D

12:56 AM
So, the notion of a discrete set (say a discrete set of points in the plane) being one where you can draw disjoint open balls around each point, is that something you're down for?

@ÍgjøgnumMeg yeah at least for me commutative algebra really shows up in applications in algebraic geometry or number theory (for me more geometry than umber theory) - but i definitely know people who find a lot of algebra intuitive through how they show up in number theory (rather than geometry)

some people use path-connected and arc-connected as synonyms, yes (don't remember what Freitag does, tbh) Others say that a space $X$ is arcwise connected if for each pair of distinct points $a,b \in X$, there exists a topological embedding $f:[0,1] \to X$ with $f(0)=a$ and $f(1)=b$

@MatheinBoulomenos I just made a meme
@MatheinBoulomenos oh, embedding!

all righty its 3 am here

@loch right, I haven't done any algebraic geometry but every time I need to understand something from algebra I just look at $\Bbb Z$ for intuition

12:57 AM
good night yaöö
yall*

lol

good night @Kasmir

I'm one of those guys who find number theory more intuitive than geometry

^
I'm ashamed to say I lack a lot of geometric intuition just because I haven#t done much geometry

1:00 AM
that's usually not a bad idea :p an example of geometric intuition is - if i ask you whether $k[x,y]/(y^2-x^2(x-1))$ is integrally closed - then doing this algebraically might require some thinking (or maybe not - it's not too hard to see this)

but from the geometric point of view this is clear because the curve y^2=x^2(x-1) has a node (singularity) so it's not integrally closed

@ÍgjøgnumMeg same
I haven't seen any algebraic geometry yet, hopefully that will be rectified soon, but differential geometry and I did not get along

i think it's a generational thing too
we have computers to produce plots for us now so it's easy to be lazy if you wanna see something
lol
in low dimensions at least

Tru

trve

what happened with differential geometry?

1:03 AM
2 hard

Hello,guys. can you help me with one thing?

Heavily dependent on the thing

Rip
(initiate strikethrough text)(Also should've learned multivariable calculus but that's for nerds)(end strikethrough text)

1:06 AM
then I'm likely to be in the same position you are!

@loch I found that the problems in the subject for me were either tedious or impenetrable
I could bash some Christ Awful symbols and hate everything

I don't understand the term discrete in discrete topology. What's so discrete about it?

Or I could see a problem with actual geometric content and just have no idea how to set it up
@Mockingbird360 I mentioned something earlier

@ÍgjøgnumMeg
@Daminark where?

Scroll up, basically I asked whether you were happy with the idea of a discrete set of points in the plane defined as being able to draw a disjoint ball around each

1:09 AM
@Daminark Yeah. I think maybe there's a relation between discrete topology and discrete set

@Daminark oh yeah i get that feel sometimes too

i've been finding it weirdly counter-intuitive too; the discrete topology is the finest topology, but I always think of discreteness as being something coarse

@MatheinBoulomenos I see how Friday managed to prove Cauchy Integral Theorem rigorously

There are some nice theorems for sure, Gauss' theorema some Latin word about Gaussian curvature being intrinsic is kinda nifty as long as you don't write down any symbols

because he didn't rely on Green

1:11 AM
And if course there's Gauss-Bonnet

@Daminark egregium

@Daminark ?
You have anything more to say on my question?

@MatheinBoulomenos I also see the fault of my thinking earlier: I thought the $\gamma$ is the problem. It turns out that the domain is the problem, but the functions I usually meet are already meromorphic over $\Bbb C$, so the domain has no problem

Oh right, so if every point in a space is open (that's the discrete topology), than you can separate the points using disjoint open sets, namely the points

@loch not sure if I 100% agree with that example. The proof that for curves smooth and normal are equivalent is nontrivial. Just noting that $\frac{y}{x}=\sqrt{x-1}$ seems simpler to me

1:14 AM
ergh, textbook is typeset in what looks like LaTeX but doesn't have $\bigcup$

@Daminark So you mean the open sets in a topology are those which can be separated from each other with open balls?

i think some ideas in diff geo are really nice e.g. just the whole idea with the definition of the curvature tensor

but i dont really know that much diff geom anyway- once things got hard in the riemannian world with bunch of manipulating tensors etc. i kind of just dozed off

@LeakyNun I don't think any proof that relies on Green can actually work if you don't assume in addition that holomorphic functions are $C^1$ (which you can prove if you don't use Green)

@MatheinBoulomenos right
there are many ... bad materials on the internet

Yeah that's why I like to prove Cauchy integral theorem only just as general as needed to get analyticity of holomorphic functions, and then finish off the simply connected case with Green
@loch I've only seen curves and surfaces so far, might look into Riemannian a bit, might not

1:17 AM
I'm saying, you don't really need to care about the domain

yeah it's not the easiest proof (compared to a direct proof) , and so hypothetically if i were teaching this i would mention the geometric view but prove it algebraically.

of course this also kind of falls short when your varieties become higher dimensional - because you can have singularities there

it doesn't matter if the curves are ill-behaved
as long as your proof doesn't rely on Green
the domains are what matters
and the domains usually are well-behaved

I'm taking algebraic geometry next year and hopefully that'll be a way to think about geometry that doesn't make me want to die (also it'll be useful for number theory!). Fall is gonna be a bit more classical stuff (varieties), winter is gonna be more commutative algebra and scheme theory

a u t u m n

kvlt

1:20 AM
@Daminark hey?
See my last message to you.
Do you agree with that?

for me differential geometry had a lot of confusing notation (and it didn't help that every author used his own notation) and while there were really cool results, the proofs all seemed to boil down to (imo) ugly calculations or reducing stuff to differential equations and relied on coming up with considering some variations of functionals with I don't have intuition for

maybe another example would be this whole business about associated primes which i found very hard to motivate purely algebraically, but better in geometry

@Mockingbird360 not really, my point is that the notion of a discrete set is one where you can draw a ball around each point and none of them intersect. That's the same thing as saying that in the subspace topology, every point is open

now here's some confusing stuff: in $\Bbb Q_p$, all open balls are closed and vice versa, but the topology is not discrete

1:23 AM
If I have $f:\Bbb R^3 \to \Bbb R$ having continuous partial derivatives and $r \in \Bbb R$, is it true that $\{ \vec v \mid f(\vec v) = r \}$ is locally a plane as long as $r$ is not a critical value of $f$?
@MatheinBoulomenos it's also locally compact but not compact

that's not really counterintuitive
$\Bbb R$ is also locally compact but not compact

$\Bbb R$ is locally compact?

@Daminark i think that's great - i once took an algebraic geometry course which just started off with schemes following ch2 in hartshorne and at the end the lecturer said 'so basically we didn't learn any geometry the whole semester' - so it's good to have examples available (from the classical world)

oh ok

1:26 AM
@Daminark @ÍgjøgnumMeg So what's the motivation developing the idea of open sets in topology?

I'm kind of taking a weird approach to alg geo I think. This semester I'm taking a course on algebraic groups which is done with varities over algebraically closed fields (but we delve into the complications of finite characteristic) and doesn't assume prior knowledge of algebraic geometry. I have read some stuff on schemes before and next semester I'll take a course on schemes
so I'll probably end up knowing stuff about the classical case only for groups

@Mockingbird360 I know about as much topology as you appear to so there's little point in asking me
#nooffence

We did talk about the geometric implications of the Nullstellensatz in our algebra course although the first version we proved was very algebraic (finitely generated algebras over Jacobson rings are Jacobson)

@ÍgjøgnumMeg it's okay

@MatheinBoulomenos that the closed points are the classical affine space?

1:29 AM
that's one implication sure
but also closed subsets correspond bijectively and implication-reversingly to radical ideals (and this restricts to irreducible closed subsets and prime ideals)

aka the galois correspondence?

it's a Galois correspondence yes

@loch I'm not 100% sure how the grad class is gonna go this year. The guy teaching it this year did it last year and at the time told me it was a disaster, so he may change things up, but I'm not sure how. The undergrad class doesn't have commutative as a prereq and is gonna be taught by a geometer who's believes very strongly in concrete things
Last time he taught the class he used a book called "Intro to projective varieties", so yeah

i don't think there's a problem with that ? i dont really know much about alg groups (hopefully i'll learn them next year) but i'd imagine in the classical setting (alg closed/char 0) you don't strictly need schemes (althuogh knowing schemes makes everything better)

i vaguely rmb the scheme structure coming to play in char p though yes

somewhat tangential: one thing i find neat about alg groups which you might know already is to view them as group-valued functors (via functor of points) - apriori when you talk about group schemes its kind of hard (at least to me) to use its defn as a grou

@MatheinBoulomenos that's real slick

1:37 AM
@Daminark ive never heard of the book :p

@loch no there's nothing wrong with it. I just think it's weird that I will have some knowledege about algebraic groups but not much about varieties which aren't necessarily algebraic groups
We also talked a lot about the functor point of view and it's really cool
it also works for algebraic groups you just have representability in a smaller subcategory
also for a lot of examples I've seen, the definition as a functor is really elegant and coordinate-free, whereas even showing that it is a variety may be painful and inelegant (that inelegance usually translates into the part where you show that the functor is representable)
Like, say you have a finite-dimesional $k$ algebra $A$, then the group of units $A^\times$ is an algebraic group as the functor $R \mapsto (A \otimes_k R)^\times$
showing that $A$ is an algebraic group involves choosing a basis and embedding $A$ into $\operatorname{End}_k(A)$
oh btw, sadly I have no photo available, but that is some prime r/badmathematics material: in a physics problemset at my university, there was the great definition "a manifold is a continuous space with a smooth boundary"

continuous space...

Whut?

I hate how people teach Mobius transformations without pointing out how they relate to linear transformation up one dimension
this phenomenon is really common
I mean, the connection is so beautiful and fundamental
why would people not make it
worse still, people who introduce the wrong domain and image for Mobius transformations

yeah! one reason why i really like this point of view because e.g. the scheme $\mathrm{Spec}k[x_1,\ldots,x_m]/(f_1,\ldots,f_n)$ is really the same thing as the functor from $k$-algebras to sets sending each $k$-algebra $A$ to the set of $m-$tuples of points $a$ in $A$ which satisfies $f_i(a) = 0$ for all $i$ - so a scheme is really the zero set of a bunch of polynomials (but where you allow yourself to vary your where your points live in) - which looks more innocent than the locally ringed space definition

1:51 AM
that make them "undefined" at a point on $\Bbb C$
I should probably rant on math.SE
I just feel like mobius transformations are being taught the wrong way everywhere

Wait so, the way I've seen it done is that Mobius maps are often defined on the sphere/CP^1, how have you seen it?

@Daminark A mobius transformation is a map $f(z) = \dfrac{az+b}{cz+d}$, defined at $\Bbb C \setminus \{-\frac dc\}$
where $ad-bc \ne 0$

lol

2:08 AM
I see

i guess maybe one potential reason why people might choose to do it this way is because he or she might think that introducing this whole point at infinity thing is going to confuse people who just learnt that you can't do that in real analysis

but not motivating it by linear transformations in $\Bbb C^2$ at all?

I don't really see what that gives you for complex analysis

sure, it gives you a better understanding of Möbius transformations

2:10 AM

yeah, but you can only do finitely many stuff in a course

e.g. inverse of mobius is mobius; mobius is generated by three bois
@MatheinBoulomenos entirely unmotivated
here, let's look as this special class of maps

the first thing is something that follows from formal properties of group actions

and then, somehow you need to make this arbitrary geometric object called "circle or line"
because you don't want to play with "infinity"

@LeakyNun we didn't talk much about linear maps on C^2, though it happened in algebra for finite fields to talk about PGL

2:13 AM
for any field $k$, one has that $\operatorname{Aut}_k(k(x)) \cong \operatorname{PGL}_2(k)$

@MatheinBoulomenos which category are you in, on the left hand side?

$k$-algebra homomorphisms

hmm

since the functor sending each Riemann surface to its field of meromorphic functions is a category equivalence to the oppposite category of the category of finitely generated extensions of $\Bbb C$ of transcendence degree $1$, we get an isomorphism $\operatorname{Aut}(\overline{\Bbb C}) \cong \operatorname{Aut}_{\Bbb C}(\Bbb C(x)) \cong \operatorname{PGL}_{2}(\Bbb C)$

2:43 AM
Damn

2:58 AM
hi Demonark. Damn you, too.

I never asked for you to be damned!
Rude
How's everything going?

Doing just fine. Just baked a rhubarb cake.

Ah, nice!

3:14 AM
It's "obvious" that an open set is a neighbourhood of each of its points right?

Yes.

Okay good
lol

And it's even true.

In the sense that a neighbourhood of a point $p \in U$ (with $U$ open) is a set $V$ containing $U$, so you can just take $V = U$?
or.. a subset $V$ of your space $X$

Sure. Sometimes you want a smaller open set, but for what you asked that's fine.

3:16 AM
Okay thanks :)

1 hour later…
4:35 AM
is any1 here

4:51 AM
hi

2 hours later…
6:59 AM
hi

7:56 AM

8:24 AM
I got an interesting strategy to evaluate an integral I've been stuck on
Anybody want to hear me talk about it...

8:37 AM
:-(

8:50 AM
How to find number of positive integers x,y satisfying $7(x+y)= xy-1$