Mathematics - 2018-05-21

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

Faust

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

ÍgjøgnumMeg

#DummitAndFoote

Daminark

>:(

Leaky Nun

12:26 AM

#DammitMyFoot

Kasmir Khaan

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

ÍgjøgnumMeg

hahaha

#controversial

Faust

@ÍgjøgnumMeg thats gunna make him sleepy

Kasmir Khaan

i still have DF and Artin

Faust

i used to us DF to fall asleep

Kasmir Khaan

12:27 AM

I mean understanding what modules are and tensor products ect intuitivly

ÍgjøgnumMeg

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

Kasmir Khaan

was not that easy for me

Daminark

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

Kasmir Khaan

dami do u fast?

Faust

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

Daminark

12:28 AM

Yup. Got about 45 minutes left before I can eat and drink

Kasmir Khaan

nice :D

Daminark

You?

ÍgjøgnumMeg

Religious fasting or?

Kasmir Khaan

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

yes ! and yes

Daminark

Yeah it's religious, it's for Ramadan

MatheinBoulomenos

12:28 AM

haven't use D&F much. I like what I saw from Artin

ÍgjøgnumMeg

Ah I see, forgive my ignorance lol

MatheinBoulomenos

my favorite intro algebra book is Aluffi

Kasmir Khaan

Thanks mathein ! :D

ow >< , as I said , dont compare me to you

Daminark

Lol it's fine :P

Kasmir Khaan

from me allufi is very hard ><

Faust

12:30 AM

artin is a pretty advanced pace book

ÍgjøgnumMeg

I'm just starting intro topology

and idk what I'm doing

Daminark

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

Faust

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

MatheinBoulomenos

@Daminark I agree with that

you have to start somewhere

Kasmir Khaan

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

MatheinBoulomenos

12:31 AM

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

Kasmir Khaan

BTW mathein ! :D

did you have time to do the 3 problems ?

they are really super important for poor kasmir :D

MatheinBoulomenos

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

Daminark

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

MatheinBoulomenos

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

Leaky Nun

@MatheinBoulomenos i n t e n s e

MatheinBoulomenos

12:34 AM

and working with abelian categories can be helpful for seeing the point of categories

Daminark

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

Kasmir Khaan

@MatheinBoulomenos mathein ;D answer me :D

MatheinBoulomenos

homological algebra/spectral sequences/group cohomology sounds really cool

mercio

and scary

Daminark

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

MatheinBoulomenos

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

Daminark

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

Kasmir Khaan

@MatheinBoulomenos yes that is fine and thanks ! :D

Daminark

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

Kasmir Khaan

mathein is my hero _

loch

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

Daminark

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

MatheinBoulomenos

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

Kasmir Khaan

@loch how else would you do it ?

MatheinBoulomenos

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

Leaky Nun

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

Kasmir Khaan

look at leaky :D doing line integration :D

Leaky Nun

12:40 AM

@KasmirKhaan :P

Kasmir Khaan

haha that was super nice , i enjoyed laurent series alot

Daminark

@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

Kasmir Khaan

and that theory of rouché for finding zeros

Daminark

@Leaky I think Stein-Shakarchi has it completely?

MatheinBoulomenos

@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

Daminark

12:41 AM

Oh true Freitag is good

Leaky Nun

@MatheinBoulomenos was uber freitag?

MatheinBoulomenos

I don't understand the question

Daminark

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

MatheinBoulomenos

Freitag is the surname of a German mathematician

Leaky Nun

@MatheinBoulomenos I was trying to make a joke on friday

MatheinBoulomenos

12:42 AM

oh

Leaky Nun

apparently it didn't get through

MatheinBoulomenos

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

Leaky Nun

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

ÍgjøgnumMeg

freeeetog

Daminark

Stein is available online if you try hard enough :P

As is Freitag

Actually my copy of Freitag is 100% legal because Springer

MatheinBoulomenos

12:47 AM

@LeakyNun I can link you the German lecture notes of my prof

Leaky Nun

nvm, my university is powerful

I got access legally

what happened to path-connected

why does Friday call it arcwise connected

@MatheinBoulomenos

loch

@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)

Daminark

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

Leaky Nun

hi @loch

I heard vakil

Daminark

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

loch

12:53 AM

hi @LeakyNun

Mockingbird360

hello everyone.

MatheinBoulomenos

@LeakyNun they're equivivalent for Hausdorff spaces, anyway

Mockingbird360

I am a beginner at topology.

Leaky Nun

@MatheinBoulomenos wait, they're different?

I don't see any difference in their definition

calls it an indirect proof

ÍgjøgnumMeg

@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

Leaky Nun

12:54 AM

assumes the contrary

proves the original statement directly

proves the original statement directly

Mockingbird360

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

mercio

lol

Mockingbird360

What's so discrete about discrete topology?

Kasmir Khaan

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

Leaky Nun

Daminark

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?

loch

@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)

MatheinBoulomenos

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 $f(1)=b$

Leaky Nun

@MatheinBoulomenos I just made a meme

@MatheinBoulomenos oh, embedding!

Kasmir Khaan

all righty its 3 am here

ÍgjøgnumMeg

@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

Kasmir Khaan

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

ÍgjøgnumMeg

^

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

loch

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

Daminark

@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

ÍgjøgnumMeg

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

Daminark

Tru

ÍgjøgnumMeg

trve

loch

what happened with differential geometry?

ÍgjøgnumMeg

1:03 AM

2 hard

Balarka Sen

Hi

Mockingbird360

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

ÍgjøgnumMeg

Heavily dependent on the thing

Daminark

Rip

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

Mockingbird360

@ÍgjøgnumMeg It's about topology.

ÍgjøgnumMeg

1:06 AM

then I'm likely to be in the same position you are!

Daminark

@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

Mockingbird360

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

Daminark

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

Mockingbird360

@ÍgjøgnumMeg

@Daminark where?

Daminark

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

Mockingbird360

1:09 AM

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

loch

@Daminark oh yeah i get that feel sometimes too

ÍgjøgnumMeg

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

Leaky Nun

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

Daminark

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

Leaky Nun

because he didn't rely on Green

Daminark

1:11 AM

And if course there's Gauss-Bonnet

Leaky Nun

@Daminark egregium

Mockingbird360

@Daminark ?

You have anything more to say on my question?

Leaky Nun

@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

Daminark

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

MatheinBoulomenos

@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

ÍgjøgnumMeg

1:14 AM

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

Mockingbird360

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

loch

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

MatheinBoulomenos

@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)

Leaky Nun

@MatheinBoulomenos right

there are many ... bad materials on the internet

Daminark

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

Leaky Nun

1:17 AM

I'm saying, you don't really need to care about the domain

loch

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

Leaky Nun

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

Daminark

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

Leaky Nun

a u t u m n

ÍgjøgnumMeg

kvlt

Mockingbird360

1:20 AM

@Daminark hey?

See my last message to you.

Do you agree with that?

MatheinBoulomenos

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

ÍgjøgnumMeg

@Mockingbird360 math.stackexchange.com/questions/2614268/…

loch

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

Daminark

@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

MatheinBoulomenos

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

Leaky Nun

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

MatheinBoulomenos

that's not really counterintuitive

$\Bbb R$ is also locally compact but not compact

Leaky Nun

$\Bbb R$ is locally compact?

MatheinBoulomenos

yeah

loch

@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)

Leaky Nun

oh ok

Mockingbird360

1:26 AM

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

MatheinBoulomenos

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

ÍgjøgnumMeg

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

#nooffence

MatheinBoulomenos

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)

Mockingbird360

@ÍgjøgnumMeg it's okay

Leaky Nun

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

MatheinBoulomenos

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)

Leaky Nun

aka the galois correspondence?

MatheinBoulomenos

it's a Galois correspondence yes

Daminark

@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

loch

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…

Daminark

@MatheinBoulomenos that's real slick

loch

1:37 AM

@Daminark ive never heard of the book :p

MatheinBoulomenos

@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"

Leaky Nun

continuous space...

Daminark

Whut?

Leaky Nun

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

loch

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

Leaky Nun

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

Daminark

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?

Leaky Nun

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

where $ad-bc \ne 0$

loch

lol

Daminark

2:08 AM

I see

loch

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

Leaky Nun

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

MatheinBoulomenos

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

Leaky Nun

well you would get some theorems for free

MatheinBoulomenos

sure, it gives you a better understanding of Möbius transformations

Leaky Nun

2:10 AM

@MatheinBoulomenos isn't maths about understanding?

MatheinBoulomenos

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

Leaky Nun

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

MatheinBoulomenos

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

Leaky Nun

and then, somehow you need to make this arbitrary geometric object called "circle or line"

because you don't want to play with "infinity"

Daminark

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

MatheinBoulomenos

2:13 AM

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

Leaky Nun

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

MatheinBoulomenos

$k$-algebra homomorphisms

Leaky Nun

hmm

MatheinBoulomenos

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 $\operatorname{Aut}(\overline{\Bbb C}) \cong \operatorname{Aut}_{\Bbb C}(\Bbb C(x)) \cong \operatorname{PGL}_{2}(\Bbb C)$

Daminark

2:43 AM

Damn