@TruthSerum the Extended Euclidean Algorithm is very useful.

@robjohn hello

wow, i'm popular on the starboard

@BalarkaSen: Yes, I'm going to outline a proof in the smooth case. One issue is that it's very technical, so it's hard to know what details to include and what to skip; another is that it's going to need me to make a bunch of pictures, so it's going to take a while

hello.

@anon

hi

what's up?

morning, @Mike.

morning

yo @Ben

@anon we haven't met in a while

indeed. how goes?

i just googled and saw that it requires Morse theory. i've always wanted to know that stuff. maybe you can give a short intro to the prereqs first and then do the proof, @Mike

@anon Not much. Trying to get back into math after a very long break.

good to hear

@anon you?

finals week in school, not taking any math this semester

No, I'm not going to talk about Morse theory, @Balarka.

@anon oh ok. You in grad school?

nope

ok, if you prefer.

ok. I start in september.

cool

I've wanted to do that but never had the motivation

right. I guess I just wanted to get fitter.

@Balarka: It's used for two technical (but obviously important) parts of the proof. It's just that knowing why these things are true doesn't really help you with understanding what's going on at all.

ah, i see.

@anon when do you apply for grad shool

so you're planning to write up something about the geometric intuition behind it. definitely looking forward to that.

probably a yearish. haven't really thought about it.

@anon

@BenLim are you there now, or heading there in the fall?

:)

@MikeMiller Heading there in the fall. I just visited there and Berkeley in March (plus some other unis in the east coast).

I see

What do you want to work on?

@MikeMiller I applied to UCLA, but withdrew my application when I got other offers.

@MikeMiller I've been thinking of either Ravi Vakil or Brian Conrad as advisors. I'm also open to either Akshay or Zhiwei

ah, algebraic geometry stuff

I'm not too familiar with the various people in the field

@BenLim: I guess UCLA probably wouldn't be a good fit for your tastes; the only person who really does (classical? idk; I mean "not homotopy theory" or "not K-theory") algebraic geometry is Totaro

@MikeMiller Yea. I'm quite surprised that so many of the prospies at Stanford want to do number theory. The latter is not my forte, I have very minimal experience in it.

of course, he's great

number theory is perpetually hip. is stanford's group big?

@MikeMiller Actually, I don't mind algebraic K-theory say. If I had gone to UCLA, probably Rouquier was my first choice. I'm more algebraic minded, and ppl have actually told me that de Jong would be the best fit. Unfortunately, columbia waitlisted me....

ah, I see

@MikeMiller Well there's conrad, venkatesh, soundararajan (analytic stuff) and zhiwei, who's a geometric rep theory guy

ok, didn't know the last two.

soundarajan is a great guy, but he's an analytic number theorists, as far as i knew

@MikeMiller Also I like california. God NY was so hectic....

google says you have daniel bump, and his fantastic beard

@BalarkaSen True. I'm not so much into the analytic stuff.

@MikeMiller That's true. Unfortunately, I think the ppl that usually attend the number theory seminar are akshay, brian and venkatesh

I see

@MikeMiller By any chance do you know brian lawrence who did his undergrad at caltech?

I only know Bump because his automorphic forms book seems pretty canonical; I see people reading it regularly

I don't think I do

is he at UCLA?

No, he's a student of akshay now.

I just assumed caltech ppl know ucla ppl and vice versa :D

oh, I see. no, I did my undergrad in northern california

caltech is far enough that I wouldn't go there except for talks

Right.

and even then I only get to know the graduate students in topology, not anyone in other fields

@MikeMiller i don't know why number theory tends to attract so many ppl.....

@BenLim right. i was planning to meet him a few months earlier. i know him via his 'unofficial' adviser balasubramanian, whom i have met.

people learn some beautiful basic number theory early, and it becomes their first love, and then they continue to like it afterwards

Hi!!!@BalarkaSen how do you meet so many people so easily

@BalarkaSen Is that B. Sury?

no, it's R. Balasubramanian.

@MikeMiller I guess I came to commutative algebra first, then algebraic geometry. I am particularly fond of the functor of points perspective.

@BalarkaSen Oh Ok. I know at least 3 ppl who have been students of Sury. I think Sury's at ISIBang yes?

@BalarkaSen who is that?

yeah, I think he is, @BenLim

Ok. I saw Sound when I was at Stanford, though I didn't speak to him. I think right now he has at least 3 students.

@BalarkaSen Are you a grad student?

that's not so much, compared to some

google tells Sury has collaborated with both Balarsubramanian and his student Adhikari, another person i have met

@BenLim nah

ok.

@BalarkaSen how many people have you met.....

So lucky that you get to met such mathematicians

@BalarkaSen How were you going to meet sound?

well, i was invited to a modular forms school on goa some few months earlier, and i was told that soundarajan was going to come there. but i finally decided against going there.

ok.

@BenLim what sort of questions are you interested in? admittedly i'm pretty ignorant about what today's modern algebraic geometer wants to do

usually representability problems. I find those very beautiful.

You were invited@BalarkaSen wow....but how?

can you give an ex?

Why isn't anyone a answering what I ask.... :( 😢

no, i've gotcha, i just don't know what an example of, like, an open and interesting representability problem is

@MikeMiller Sorry, I misunderstood you.

I wasn't very clear

@MikeMiller Honestly most of the representability problems I know of are already completely understood :(

ah

I think there are interesting problems though considering studying compactifications of these moduli spaces

I'm sure whoever you end up working with will know good problems of that sort, or at least problems of a similar flavor

Yea. Ravi is an expert on this stuff. His papers are often like calculating the chow ring of some moduli space, etc

cool

@MikeMiller These moduli problems often involve like the highfalutin stacks stuff

yeah. stacks are scary

i'm in a homotopy theory seminar and we're talking about the moduli stack of elliptic curves at some point soon

@MikeMiller I'm sure if you asked the derived ppl they will have plenty of representability problems.

@MikeMiller ahhhhh

I think a good starting point would be to understand say, what exactly is meant by "the stack remembers the automorphisms"

i missed yesterday's lecture, since i'm home sick right now :( the speaker defined sites, stacks, and sheaves on them, and tried to get everyone comfortable

You know how ppl often say "obstruction to representability" = "presence of autos"?

yeah

The way I think about this is that the presence of autos stops the moduli functor from being even a sheaf in the first place

So if it's not even a sheaf in some nice topology, no hope of representability

i have to run. the discussion's freaking me out :p

not sure what that means. from being represented by a sheaf? (how do sheaves represent things?)

@BalarkaSen Ok. I'm not saying anything deep at al.

see ya @Balarka

my algebraic geometry is pretty weak, i warn

@MikeMiller No. I'm saying that the moduli functor, say $M_{1,1}$ is not even a sheaf in the (fpqc) topology.

Or if you don't like fpqc, replace it with etale.

ok

@BenLim nah, it's just that i have no idea about this algebraic geometry stuff. i am more into topology.

Representable functors are etale (or even fpqc) sheaves.

gotta go.

@BalarkaSen AG is everything man. :D

@MikeMiller But if you consider the moduli stack

gimme a second to get this business

ok

@MikeMiller math.stackexchange.com/questions/1257517/…

This is a good example.

cute

@MikeMiller

hi

i can elaborate more on my answer

the comments do the job for me

right.

But you see the point is that with the stack.

It remembers "descent data"

The formal definition, you can look it up in vistoli's notes

but basically upon remembering the descent data those two elliptic curves are different.

descent data is supposed to mean what? no formal definition, just a moral one

It's sort of like elliptic curve + data of the cover

Ok that's not a good way to put it.....

See the problem with this stacks stuff is that sooner or later one has to start talking about descent...

haha

don't worry about it then

but yea it's like a formal way to say the two things are different.

I want to say this. There's a functor from $M_{1,1}(Spec Q) \to M_{1,1}(Spec Q \to Spec Q(\sqrt{3}))$, and that $M_{1,1}$ is a stack is precisely that this is an equivalence of categories. The one on the right is the "category of descent data".

i see

The images of the two elliptic curves under this functor are different.

Does someone here have experience with user accounts?

Basically, elliptic curves over Q.

@JayeshBadwaik hello.

and the one on the right is going to be commutative squares of elliptic curves over $\Bbb Q$ and $\Bbb Q(\sqrt{3})$, yes?

@MikeMiller That's how I remember the statement, stack remembers the automorphisms.

@BenLim Hello!

No, not really.

oh.

@MikeMiller Let me see how to explain it, should be simple here because there's only one element in the cover

It's really

an elliptic curve E over $Spec Q(\sqrt{3})$

together with an isomorphism $\phi : Spec Q(\sqrt{3}) \times_{Spec Q} Spec Q(\sqrt{3})$

satisfying cocycle

ohh

and basically to say that two objects is the same

ok, now i understand your automorphism/descent data comment.

is basically saying that there is an automorphism over Q that takes one to the other.

So because there's none between the two I constructed, the two are different objects in the category of descent data.

yeah.

@MikeMiller You can trace through the definitions in vistoli, but I think the intuition here is more important.

homepage.sns.it/vistoli/descent.pdf pg 71

@JayeshBadwaik how are you

@MikeMiller So yea

thanks!

@MikeMiller I hope that made the stacks shit less scary.

a bit, yes

Whenever someone says to me "stack" I just think of like M_{1,1}

@BenLim I'm doing okay. Qualifiers in two months time or so. What about you?

Or if they say QCoh is a stack in the fpqc topology, I just think it just means quasi-coherent sheaves can be glued in that topology

It's like everytime I attend a rep theory talk and they say let g be a semisimple lie algebra, h a Cartan subalgebra, etc

I just translate everything into sl_2.

@JayeshBadwaik Waiting to go to grad school I guess.

sure

@BenLim aha, cool.

@MikeMiller It seems this stacks stuff seems so prevalent nowadays.

@JayeshBadwaik Where are you attending?

I mean, where do you go for grad school @JayeshBadwaik

@MikeMiller burttotaro.wordpress.com/2013/11/28/…

saw that a while ago. funny

@Vrouvrou hey. Did you have a question?

@robjohn

hello

I have one.

Q35 here

isn't "2" a more minimum?

@ADG but $n=3$ fails, so $n_0=2$ does not satisfy the condition that for all $n\ge n_0$, ...

@robjohn very thanks :D.

@BenLim I attend grad school at Tata Institute of Fundamental Research.

@robjohn It would be great if you could change the mathjax links to "https" instead of "http" since firefox has stopped loading http content on https pages by default.

@JayeshBadwaik TIFR?

@ADG Yes

hello

Hi.

Hi @ADG

Anyway, I'm off now, see you guys later.

okggttyl

@robjohn something like this

if you have a matrix, and you swap the ith row with the jth row and the ith column with the jth column, does the resulting matrix have the same characteristic polynomial?

@Lepidopterist hint: the char poly is the det of A-xI

but permutating A is not the same as permutating A-xI

how to tag multiple lines with (1),(2),(3)... in latex?

@Lepidopterist think harder about my hint

@ADG on the mainsite? are they in separate $$ things?

i understand that determinants only change sign by permutation

yes

but again, permutating A-xI twice and having an unchanged determinant doesn't imply that permutating A will be the same. That is if p is two permutations applied to a matrix, that det(p(A)-xI) is the same as det(p(A-xI))

I'm not sure I understand your second sentence.

we know that permutating a matrix twice under the determinant leaves the determinant unchanged

yes

so if p is one such even permutation, that det(p(A-xI)) is unchanged

if det(p(A)-xI) is the same as det(p(A-xI)) is the same as det(A-xI), then you're done no?

and you just agreed that det(p(A)-xI) equals det(p(A-xI))

i didn't agree with that

is that obvious? i must be stupid at the moment

> det(p(A)-xI) is the same as det(p(A-xI))

direct quote from you

i meant that det(p(A-xI) is the same as det(A-xI)

typo

in fact p(A)-xI is the same as p(A-xI)

(*)

which is what I wanted you to see

@ADG ?

just unpinging myself

(*)

oh

thanks @anon, it is obvious indeed

:)

the head of my math department once submitted a few-page paper which basically amounted to the fact that taking transposes commutes with squaring a matrix

was he embarassed?

no

#yolo

the head of the department hasn't always been a he btw...

i was correct, that is the important thing

i haven't seen a female head before, forgive my ignorance

there seems to be a crusade to get more women interested in math now

yeah

personally i'm more concerned about the disparity in the mining industry

women seem less interested in tonka trucks and digging than men, and i have a feeling it's connected to discrimination

@JayeshBadwaik I thought that MathJax had some problem working on https pages. Perhaps they've fixed that.

maybe i should have said Caterpillar trucks

when you mention tonka trucks it brings to mind the differences in the boy's and girl's aisles at the toy stores

I believe lego used to be less "pinkified" for girls, for instance

Hello. I have a question about terminology. I have a definition about a certain "ordering" of sets (well it's not just any ordering, it also imposes some other restrictions but it doesn't really matter). I proved that each of the concerned sets has an "ordering" satisfying the definiton. Can I say "[this proves] the definition is well-founded"? Or is there a better term than well-founded?

the little girls i knew as a kid loved pink legos, i'm sure they would be disappointed to know that the boys have taken over their lego colors too

i think you mean well-defined

@Andrew123321

@Andrew123321 was there any reason to suspect the definition to be ill or ambiguous in the first place?

I suppose if the definition implicitly claims to be applicable to all of the "concerned sets," then you do need to justify such an ordering is possible on all of the concerned sets

an abstract algebra teacher I helped TA with emphasized "well-defined" too much in my opinion. the only time it need be discussed is in the context of defining maps to or out of collections of equivalence classes. (actually I remember thinking this true of one other context too but can't remember what it was.)

hi

hi

@Lepidopterist I thought so :-) Ok, thank you, I'll write that the "ordering" is well-defined

if you are stating the area of a 2d shape in centimeters in surd form - would the unit still be written as cm^2 so for example sqrt(4) cm^2

or is it simply cm

@anon Yes, it's not that obvious that the definition is always applicable

How can I show [0,1] meets the Heine-Borel property (other than by knowing it is closed and bounded)

I need to show every open cover of $[0,1]$ has a finite subcover?

ah I think I just found a math.se answer

@JayeshBadwaik I have updated my local bookmark... is there an https site I can test this on?

Can anyone help me on that question ? Thanks.

@JayeshBadwaik it just requires adding one "s", if I am not mistaken

@hippa how to start a new irc chanel ? The one we used yesterday expired, too...

@Ramanewbie no -_-

Hello @Ramanewbie @Hippalectryon :)

@evinda Hello :-)

What's up? @Hippalectryon

@evinda The sky :P

And that V

@ADG math.stackexchange.com/review/close/407872, you should try to type it e_e

Hi @evinda

is somebody who can give me an concrete answer at math.stackexchange.com/questions/1271828/…

?

I don't want another method, just explanations

@Lucas What's wrong with the current answer ?

@Hippalectryon it don't ask at my question

look at yellow

...

is something wrong?

@Lucas It's not very clear... anyway, you have a new answer.

@Hippalectryon These days I felt the need to attend some far far more advanced mathematics than the one I attend now. After publishing my book I'll definitely try to do that.

Presently all gets reduced to having the proper ideas (in the ends it's all about ideas). Just writing the proofs is not exciting anymore.

@Chris'ssis Define 'more advanced'

@Hippalectryon more advanced than what I use at the moment, some very deep stuff (and hope to also be very hard to understand).

@Hippalectryon :D

@Ramanewbie What's going on?

@Chris'ssis WHat do you mean by 'deep' though ? How are sheaves any deeper than integrals of arcsin of fractional parts ?

@Hippalectryon I was suggested to try some fractional calculus (by some mathematicians).

@Chris'ssis Not a bad idea, but I wouldn't call that any 'deeper'.

@Hippalectryon in the sense that sheaves are about using some completely different ideas to formalize complex analysis.

[not that i know about them]

@Hippalectryon I was also expecting you to recommend me a profound study in the area of elliptic functions. Actually this is what I plan to do in the near future in terms of math. :-)

@Chris'ssis I'd just recommend you to get a better view of classical things you might have not worked on so far (matrices, whatever)

@BalarkaSen Well they open broader possibilities and a new mindset, but I don't know if that makes them 'deeper'

'deeper' is vague