Mathematics - 2014-03-20 (page 1 of 4)

Why don't you link these to me before you answer?!

@Mike You're a rep denialist.

You need to spread the wealth.

@Mike yo

@Karl was just getting you in on the conversation

12:28 AM

oops

it died before its prime

@Mike @KarlKronenfeld

so many empty pings.

Unless you're telling mike that you're pinging me...

That makes sense.

I am curious about "projective" and "injective" modules. A projective module $P$ is by definition one for which whenever one have morphisms $\eta:P\to B$ and $\mu:A\to B$ a surjection, we can "factor" $\eta$ with a third $\varphi:P\to A$, as in $\eta=\mu\varphi$, correct?

12:38 AM

yep

@PedroTamaroff How much do you know about exact sequences?

Vewy little. I know what they are for starters. =D

I never fully understood their relevance

There are a couple equivalent definitions using exactness.

@KarlKronenfeld I think Hungerford has them. Let me shee.

@FernandoMartin Nevermind.

equivalent to free => projective

12:44 AM

I think they're useful in homological algebra, but I have no clue about it

@FernandoMartin E.g. projective resolutions

@KarlKronenfeld Yes, that's why I deleted it, it's a bit... not that relevant.

@Fernando They're not so much useful as essential.

Well, that just shows how much algebra I know :)

@Mike =O

12:46 AM

Many (co)homology theories are defined with resolutions, as Karl says, but there are a lot of nice exact sequences associates to homology in various ways the you actually use to compute the homology.

@PedroTamaroff Here are the two equivalent forms I was thinking of: 1, 2 I consider each of those more useful than the definition you gave.

@Fernando Do anything (co)homological and you'll be swimming in exact sequences :)

Oh

I was talking about projective modules, not exact sequences

I wasn't sure whether @FernandoMartin was referring to exact sequences or projective modules so I gave an example of both.

olol. then yeah, that matters more for projective resolutions

12:50 AM

I (think I) understand why exact sequences are essential

btw, I asked this earlier but you guys weren't around

is there a name for a category in which every SEC splits?

do you mean SES?

Haha, yes, SEC is the acronym in Spanish

ah

sucesión exacta corta

yeah he does

@Fernando if there's not one in the nLab article for split exact sequences then probably mof

12:52 AM

also, is there any other known one besides vector spaces?

Yeah I tried nlab but it's down

I would guess the only $R$-modules you can always split for is fields

Shahar

do any of you play piano?

$R$ a field I mean

if only because if doesn't work for the next nicest ring I know, $\Bbb Z$

oh, well no quite only fields. every SES of $0$-modules splits

Silly @Mike, 0 isn't a ring

:)

it is to me!

12:57 AM

@Mike Shun the nonbeliever!

shuuuuuuuuuuuuuuuuuuuuuuuuuun

►

shhhhhhhhhhhhuuuuuuuuuuuuuuuuuuuuuuuu NA

@FernandoMartin (The categories of modules over) Integral domains that aren't fields are ruled out, since every ideal would be a direct summand of $R$.

Hm, not sure what modules over $k\times k$, where $k$, is a field are like. That'd be the most likely nontrivial possibility though.

1:17 AM

@KarlKronenfeld OK, consider modules over $\Bbb Z_2^2$, maybe.

They can't be too complicated since they are up to isomorphism direct sums of the module $k$.

Really?

Take the diagonal $k\subset k\times k$ to give a module $M$ a natural $k$-linear structure.

Ah, true.

I was worried about 0-divisors.

Q: Whenever I try to compile a file using PDFLaTeX, I get "Error, file not found."

0

Whenever I try to compile a file using PDFLaTeX, I get "Error, file not found." I have no idea what is causing this problem. Could it be I am opening a code that uses a package I haven't installed? Usually TeXWorks fixes this automatically by downloading such package. What the issue here?

compiling

I suck at TeX so bad.

@Mike @KarlKronenfeld

@FernandoMartin

1:22 AM

Sounds like you got owned bro

I know nearly nothing about TeX, sorry

WAT

@PedroTamaroff You aren't providing any information in your question.

@KarlKronenfeld What information should I provide?

well, post what you've tried, your thoughts on the problem

1:28 AM

@PedroTamaroff Basically, answer cfr's questions with an edit in your question.

@KarlKronenfeld I think I have found the problem.

1:56 AM

@KarlKronenfeld Heya.

Question.

Hi again.

yello

@TedShifrin Hey.

@TedShifrin Almost done.

Yippee @Mike

skullpatrol

1:57 AM

Hello Professor.

Hi, @skull

@KarlKronenfeld Suppose I have a ring $A$ and an abelian group $G$. Then $A[G]$ is an $A$-module in the most obvious way, right? I mean, if we take the additive group of that ring.

@Pedro, was it a graphics file it wasn't finding?

I am not entirely sure, since I know some abelian groups can only be $R$-modules in certain ways.

@PedroTamaroff Yeah

1:59 AM

@TedShifrin I had to compile the thing and install like 10 things.

Like 10 things, eh?

@TedShifrin Yeah, many style packages.

And some other stuff, to compile arrows and whatnot.

Ah, dunno what you're up to.

I just realized I "thinged" too much.

Hehe.

@TedShifrin Our TA is super cool and gives us the TeX file of the practices.

Like for real, like :D

2:01 AM

And we can turn display of solutions off and on. =P

mate.dm.uba.ar/~qvivas/algebra2/#practicas

@KarlKronenfeld When is it interesting to form group rings?

Besides representation theory I dunno.

Oh, what do you do in rep thry?

I am quite ignorant in representation theory. But I think you can "represent" representations using group rings.

@KarlKronenfeld Well, you do group cohomology on $G$-modules.

Lang studies $k[G]$ for $k$ a field in one of his chapters. it seems.

Chapter 18, so pretty tough stuff it must be.

2:33 AM

@KarlKronenfeld

wat

Guys, whats an example of an "infinite" metric space.

@eXtremiity $\Bbb R$.

@eXtremiity presumably a metric space that has infinitely many points

The set, X is infinite in (X,d).

2:36 AM

@KarlKronenfeld The group ring satisfies the following property.

that's going to be roughly every nontrivial metric space you ever encounter

(i) Give an example of infinite metric spaces $(X,d)$
and $(Y,\delta)$
such that every function $f:X\to Y$
is continuous.

I was asking because my question asks this.

For every group (monoid) morphism $\varphi:G\to G'$ there is a unique ring morphism $\bar\varphi:A[G]\to A[G']$ extending $\varphi$ (in the isomorphic copy of $G$ inside $A[G]$), which is the identity on $A$. @KarlKronenfeld

@eXtremiity Do you know about the trivial and discrete topology?

@PedroTamaroff Yep, this is reminiscent of the universal property of polynomial rings.

Wouldn't it be awesome if ever set in $X$ was open? Then $f^{-1}(S)$ would always be open for any $S\subseteq Y$. =)

@KarlKronenfeld I was going to ask if it can be put in a categorical setting, just for the lulz.

2:38 AM

@PedroTamaroff Yeah, you have a functor $A$

That's pretty categorical already

@KarlKronenfeld Ah. From ${\bf Mon}$ to ${\bf Rng}$.

Or, more precisely, to the category of $A$-algebras.

Hold on, so much maths being thrown about. I do know of the discrete metric space.

The trivial metric space, can you please recall that.

@eXtremiity nah

2:40 AM

@eXtremiity There is no trivial metric space, but there is the trivial topology.

@KarlKronenfeld Oh.

Oh, topology. No, we're on metric spaces atm.

Topology is the next topic.

@PedroTamaroff I have to go, but there actually is a good reason to want that precision.

@PedroTamaroff gotta catch all the 'mons

@eXtremiity Well, just use the discrete metric, which really is the discrete topology.

In this course, how is a continuous function defined again ?

2:41 AM

@KarlKronenfeld Damn it, don't leave it hangin'.

@eXtremiity I am no mindreader.

=D

What?

"In this course, how is a continuous function defined again ?"

It's your course, not mine.

OH

Hahahah sorry !!!!

Introductory Real Analysis.

Courses differ from one university to the other

Names are meaningless.

2:42 AM

We've doing closed sets, countable sets. Now we're onto metric spaces.

Still not enough information? That's ok - I'll ask my lecturer.

@eXtremiity Preimage of open sets is open...?

I think we're yet to learn that.

delta-epsilon stuff?

I have read that in a prescribed book. We have not covered it in class though.

Yes, delta-epsilon stuff.

@eXtremiity Dude, you should know how continuity is defined.

2:44 AM

I agree with @Mike

OK, you want every map to be continuous. Think about what that would mean in terms of delta-epsilon. Maybe fiddle with it on paper, play with some metrics you know.

Ahhh, delta-epsilon definition. Yes, ok. I know it

Well, fiddle with it a bit.

I'll let Mike do it.

Alternatively, if that is continuous, can you show $\{x\}$ is open?

@PedroTamaroff I think he can get it on his own.

2:47 AM

Yes, @mike is right. I need to fiddle with this. It's all basic stuff as you've seen.

I just don't want to be using the wrong definitions. Thanks guys.

Thanks for your help. Going to look into it.

2:59 AM

@Pedro Done with that essay. Now I need to write two pages of another. Then prepare for my topology final.

@Mike What was the essay about?

Comparing two religious leaders. Needed to use a bunch of journal arguments to back up my argument.

took wY too long

@Mike What leaders? =D

It was a more focused topic than that but I don't wanna get into it.

his holiness the Dalai Lama and ayatollah Khomeini

@Mike @FernandoMartin had his topology final not so much ago.

3:07 AM

Well, my topology course was way more elementary than his I think

ORLY.

ya rly

@Mike

@FernandoMartin You should ask Ximena to let you make a mini-class on the Steinberg relations and Whitehead's lemma.

Hahaha

Students would hate me

Our last JTP made a class about some Google stuff with matrices.

3:12 AM

Yeah, that's more or less standard stuff

So you can totally do it.

Group presentations and quotienting stuff are definitely way out of the course's league IMO

@FernandoMartin Nah man, I just mean to explain how elementary matrices behave and how you can decompose stuff into those.

It's pretty elementary yet super important stuff.

Everybody knows Gauss' algorithm, say.

The cool part is how we kill off the remaining diagonal.

What do you mean?

@FernandoMartin That Whitehead's lemma tells you how to kill the diagonal Gauss' algorithm gives with elementary matrices.

3:17 AM

If I understood correctly, Whitehead's lemma is useful for rings which are not fields

Where "elementary" means row operations.

@FernandoMartin I am not sure. If $a_1\cdots a_n=1$ then every $a_i$ is invertible.

Regardless of the ring being a field or not.

Then I must have not understood the lemma correctly

Let me look for it

I can tell you what the lemma says.

@Mike You know, Lang is not that bad after all.

@TedShifrin Herro!

Yessah?

How are classes going?

3:24 AM

Almost at Stokes's Theorem :)

Cool!

Stokes's is a crowning moment =D

And doing cool stuff tomorrow am in diff geo :)

Because of the video, I'm going to try to do partitions of unity for a few minutes ...

@TedShifrin Like in Mike Spivak's book?

I found that part really interesting.

No, like in my book :) no chains. I disapprove of what Spivak did for a first course.

@TedShifrin I didn't get to that part in your book.

But Spivak does that in chapter 3, where he integrates in space, not over chains.

3:28 AM

Nope. You missed out on the hairy ball theorem and such.

Well, I looked it up

Oh, you meant partitions of unity.

I guess one wants to do it just with row-addition elementary matrices

Hi @Fernando

Hi @Ted

3:30 AM

@FernandoMartin Yes, that's what I told you! =D

12 mins ago, by Pedro Tamaroff

Where "elementary" means row operations.

That's the interesting thing, I guess.

And $$\begin{pmatrix}u&0\\0&u^{-1}\end{pmatrix}=E^{21}(u^{-1})E^{12}(1-u)E^{21}(-1) E^{12}(1-u^{-1})$$

is badass on its own.

I still fail to see how one could motivate that in a LA course

@FernandoMartin Well, you can take some time to think about it. I don't know at the moment.

Yikes ... WTH.

@TedShifrin Yes, Whitehead.

oh, trying to see what generates $SL(2)$?

3:33 AM

@TedShifrin Kinda, =)

${\rm SL}(n,k)$ is generated by elementary matrices.

Now that's why you want to use row operations

Instead of the obvious scaling elementary matrices

Right.

Well, everything is gen by elem matrices ... But you want only two specific ones here ...

@Pedro: Sorry if you already mentioned this stuff the other day but I suck at understanding things without writing them down and thinking about them for a while

@FernandoMartin Dude, don't be Canadian.

3:35 AM

sorry mate

@Fernando, so do the rest of us.

@TedShifrin Elementary matrices here are $E^{ij}(\alpha):=\alpha E^{ij}+I$.

No others. =D

Right @Pedro

@TedShifrin You know about the Steinberg relations?

Nah

3:36 AM

I feel like a preacher repeating this over and over.

No preaching ...

Guillermo told me the K-theory course was going through stuff real quick

$$E^{ij}(\alpha)E^{ij}(\beta)=E^{ij}(\alpha+\beta)\\
[E^{ir}(\alpha),E^{rj}(\beta)]=E^{ij}(\alpha\beta)\\
[E^{il}(\alpha),E^{kj}(\beta)]=1$$

@TedShifrin

@FernandoMartin Didn't like that?

Tochi was tochi'ed.

He likes that

Hahah

I just mentioned it since we're talking about K-theoretic-ish stuff

@FernandoMartin Right.

Combinatorics tomorrow was cancelled.

Darn it.

I have to return a book nevertheless.

3:38 AM

What was the first class about?

Well, bedtime for this bonzo. Night, all :)

Do you have anything to do?

Good night @Ted

@FernandoMartin I missed it, but I will do the practice.

mate.dm.uba.ar/~jbarmak/combinatoriap1.pdf

I will probably don't go to Algebra II next Tuesday and to Combinatorics.

I think Jonathan is a cool guy. =)

I don't know him personally but he seems to be brilliant

3:41 AM

@FernandoMartin I guess so.

@TedShifrin Am I too late?

I'm too late.

@Mike Do you know how to count?

As long as it's not too hard, yes.

3:56 AM

@Mike Anagrams of AAABBBCCDE such that the D is always to the left of E.

I am counting them, but my methods sucks.

My first option was the following:

Consider bins ____ D ____ E ____

Then start putting stuff in there.

=D

But I need to take care of order.

So I have to "pile up" the balls.

No, that's correct.

So, count the number of ways of arranging AAABBBCC.

Then count the number of places to put D and E in that mess.

Multiply.

Ah, I was doing it backwards.

So I have 9!/3!3!2!.

Great. Got it.

Perfect.

Wait, that's only the first part.

I know.

Also, no. 8! up top.

4:00 AM

Yeah, sorry. I wanted $$\binom{8}{3,3,2}$$

"I wrote a few papers on Koszul algebras, but I really don't understand the definition of Koszul algebras."

Takayuki Hibi (Joint Summer Research Conference, Snowbird, UT, June 15, 2006)

@FernandoMartin =D

Mariano's last student graduated with a thesis on Koszul algebras.

Badass stuff.

Did he understand the definition?

I suppose so!

But it was some craaaaaazy stuff.

If Takayauki didn't...

Oh, I guess his famly name is Hibi.

4:03 AM

@Eugene Look who's visiting.

Eugene

hi

@Mike Here's an old "friend".

lo

lo

lo

4:24 AM

holis

@Bill olakeace

estoy contando

mate.dm.uba.ar/~jbarmak/combinatoriap1.pdf

ya tuviste clase?

no, tenia mañana pero jonathan cancelo

asi que falto a la teorica de algebra y voy el martes.

ah que lindo eh

ola ke ase bill

4:26 AM

is it ok if I ask for a paper here?

i don't have access

@Bill I think it's OK.

usá el proxy de la facu

@Bill no te pongas celoso

no esta disponible

uh

4:26 AM

es un memoirs de ams

books.google.com.ar/…

es para la tesis?

Anyone?

si y no

@Mike

@robjohn

wat

books.google.com.ar/…

need access to that paper

do you have access by your uni's library?

4:29 AM

what paper?

I'm at home now so no

But you'll only rarely get access to things through google boks

Use MathSciNet if your uni has access or jstor

It's not in jstor

i'll come back tomorrow morning

"Let $a_n$ be a sequence of complex numbers such that it's generating function is rational. Then $a_n$ satisfies a linear recurrence."

Slightly trivial, but cool nevertheless.

@FernandoMartin Did you get a chance to use the Lang line?

What Lang line?

@FernandoMartin In a scale of 1 to Lang...?

@PedroTamaroff what do you need that paper for?

4:35 AM

Ahhh, haven't used it yet

@Mike Not me, @Bill

robjohn

@PedroTamaroff yes?

qwr

anyone feeling kansas