00:00 - 08:0008:00 - 15:00

8:19 AM
@0celo7 You didn't mention that...
I don't get it.
Oh I see, it's the "glome". Yes, I have never seen that before.

9:22 AM
hey everybody

10:07 AM
@user1618033 Done. I will never delete my channel again. I just did two versions of Ave Maria. The link is in my profile. =)

10:23 AM
@JasperLoy Awesome!

11:00 AM
@BalarkaSen My course might get cancelled! :(
The prof thought two students was too few

11:13 AM
@Krijn what course

@Huy Class Field Theory

@Huy My prof be mean why?

much course cancel

little students few wow

11:14 AM
you can help me prepare extra work for a student
he's 14 and wants to learn about quaternions for whatever reason

Ah, I have this pretty accessible syllabus for quaternions, written for high school students, but alas, it's in Dutc
h

oh really
one of my students is Dutch
not the same one though unfortunately
I'm kind of worried how I can make him be a bit more down to earth
he's already way too full of himself which leads him to thinking he already understood something when he didn't really
but he's definitely talented

@Huy recognizing when that happens is a sign one's grown up :)
can anyone here help with a question about affine schemes?

@Huy How did he end up over there?

@Krijn: it's a she. I don't know, but she's not officially a student yet. the first semester is basically for getting used to the school and language, and the second semester is a probation time. if all goes well, she'll be an official student of the school in a year.

11:24 AM
just checking. the map $\newcommand{\CC}{\Bbb C}$ $\CC[y]\to\CC[x]$ given by $y\mapsto x^2$ induces a map on the Specs.

@Huy Oh I misunderstood you and mixed up the two students

oki

Give him a very though book on quaternions to bring him down to earth

I think I could also give him a freshmen analysis book that we use at uni and he wouldn't understand a lot very quickly

11:27 AM
"Dear student, I have decided to try to teach you InterUniversal Teichmüller Theory, which should be appropiate at the age of 14."

I'm kinda trying to figure out how he can himself realise that he's not the best yet, without giving him hard problems/books

"Dear student, here are a few tickets to a play at the local Hodge theater. It should be educational."
in particular, that map sends $(y-a)$ to $(x^2-a) = ((x+\sqrt a)(x-\sqrt a)) \subset (x\pm\sqrt a)$.
so the image of, e.g. $\DeclareMathOperator{\Spec}{Spec} (x-\sqrt a)\in\Spec \CC[x]$ under the induced map is $(y-a)$.
can someone verify if this is right? @Huy?

I don't know any algebra

rip huy

rip

11:31 AM
little algebra

I'm writing up some of this stuff for my blog just to get the ideas straight in my head, so I asked because I don't want to put anything false up.
(This was an exercise from Vakil)

wow

ping @Balarka, please scroll up and check if you have time this evening

such bloge

you're jumping on the doge train a few years too late
:P

11:33 AM
I've been on the train since before you were born

@Huy For your student: show that $$4\cdot 3^{2^n}+3\cdot 4^{2^n}$$ is divisible by $13$ if and only if $n$ is even.

all hail Huy the savage shredder, humbler of 14-year-old algebraists and decimator of weird spacetimes
is that Chris's sis?
so what have you been up to, @Huy? haven't talked in a while :)

@SohamChowdhury maybe
;)

@user1618033: do you not consider 0 to be even?
@SohamChowdhury: mostly doing geometric topology from Farb&Margalit, and teaching again since this week.

@Huy I should add that $n$ is a natural number.

11:37 AM
ok

@user1618033 Do you not consider 0 to be a natural number?

I don't either

"we are at an impasse"

Mental.

$\mathbb{N} = \{1, 2, 3, \dots\}$
clearly

11:38 AM
I discussed this topic on chat, I launched the discussion in this chat a along time ago. Zero is clearly an even number.
@Krijn It depends. In our country $0$ is considered a natural number, but things on this matter differ elsewhere.

@Krijn 0 isnt even a numbe, it is just the 'nought'

The question I posted above is a masterpiece and it's not about integrals, series and limits. I wonder how many can solve that.
Maybe no one here?
Hope I'm wrong.
I have to finish some stuff.
BBL
(have fun with my little question)

11:56 AM
@user1618033 reminds of somone we were competing who gives the more accurately big probability ever existed or not he gives a thing like 99,9999999% i gave the 99.... root of the same proportion he can reach ever he didnt understand so i told him that i gave a thing only mathematicians do understand

@Agawa001 :-)
Just for the record, before getting very involved in the area of the calculations of integrals, series and limits, I was doing all kind of Olympiad problems day by day, also including such problems (this particular one is a problem given in some Olympiad).
I think it helped me very much, and I entered pretty strong (in the sense of developing good analytical skills) in the area I'm presently interested in.
I'm highly fascinated by clever solutions in any area of mathematics (of course, I don't count the ones I didn't study).
BBL (I really have to finish some proofs)

12:35 PM
@user1618033 Did you do the notoriously hard ones?
Let $a$ and $b$ be positive integers such that $ab + 1$ divides $a^2 + b^2$. Show that $$\frac{a^2 + b^2}{ab + 1}$$ is the square of an integer.

@Krijn it's a problem I know. I could have told you that it is a problem I never saw it before, but this would have brought me no satisfaction. :-)
Cheating in mathematics is the worst possible thing.
@Krijn All kind of problems I had in my books.
BBL (back to my work)

12:57 PM
@Krijn some say that it's one of the toughest problems given in the history of IMO, maybe even the toughest.
I'm out to buy some food for my dogs. Later.

@Krijn You can look at Neukirch's Algebraic Number Theory.
@Huy I still like Jacobson's Basic Algebra best.

1:15 PM
Consider $\DeclareMathOperator{\Z}{\Bbb Z}\Spec \Z[i]\to\Spec\Z$ given by the inclusion of $\Z$ into $\Z[i]$.
now the fiber over every prime which is 1 mod 4 has two points, e.g. $f^{-1}((5)) = \{(1+2i),(1-2i)\}$
over primes congruent to 3 mod 4 (or over 2), there is one point.
in this pdf (not his main notes), Vakil asks the reader to compare the degree of the "residue" field extension with the number of points in the fiber. I'm guessing these will be the same except at $2$, but:
the weird thing is, isn't $\Z[i]/(3)$ isomorphic to $\Bbb F_{3^2}$?, since $x^2+1$ is irred mod $3$
@Krijn, if you're studying class field theory in the near future, I'd love to join you :)
can anyone here help with that question?

1:43 PM
@BalarkaSen Oh, well. It's pretty nontrivial.
It involves some bump function trickery

@0celo7 Yes, it's not immediately obvious. Uniform convergence does not mean uniform convergence of derivatives.

It's an exercise in Spivak and I can't find a reference online

So it's believable that you need to play with the specific sort of functions aka bump functions.
@0celo7 What is?

Maybe I'll have more of an idea after my analysis course this semester

test

1:46 PM
@BalarkaSen Prove that the Riem. distance can be defined with smooth curves and nothing changes.

and now my gravatar is back to normal. weird.

@0celo7 Ah, ok.

@SohamChowdhury I think so.

then what the hell is going on there?

1:48 PM
Is it not branched over $2$ though?
That is your question: it's branched over $2$ so you'd expect the degree to be $1$, but it's actually $2$?

no, the degree over $3$ should not be $2$ because the fiber over $3$ has one point
and vice versa with $5$ happens, also weird

algebra sounds like gibberish to me

@SohamChowdhury Yikes, I meant branched over $3$. Not $2$.

what does branched mean?

Not as many points in a particular fiber as a generic fiber has.

1:51 PM
hm. but it should be a generic fiber, right? it doesn't split or ramify

(generic fiber of $\Bbb Z \hookrightarrow \Bbb Z[i]$ is of cardinality 2)

idea: add the dimensions of residue fields at points in the fiber. this fixes 5, since $\Z/(1+2i)$ and $\Z/(1-2i)$ are degree-1 extensions of $\Z/5$.

@SohamChowdhury Oh. Wait a second.
It's unramified at $3$?

but this is sort of cheating
um ... yes?

(branched is synonymous to ramified, BTW)

1:52 PM
$3$ doesn't split in $\Z[i]$
you mean split or ramified?

I meant ramified. $3$ is a Gaussian prime, that is true.

only 2 ramifies, right?

The geometric picture here is unclear. In topology, I'd define ramified points to be the ones which have preimage cardinality less than a generic preimage cardinality (in this case, 2). So I would call 3 ramified. What's the definition here, again?
I forgot. (I think I have run into this issue before)

there is a picture.

A "fat" point, IIRC?

1:57 PM
for ramified points, yes
but I mean, a picture of this map
it's like a branched cover

I know, I have seen it.
But I can't remember why $3$ does that.
(Hence, tell me what "ramifies" mean in algebra?)

so two lines, which join up into one at Gaussian primes
ramifies = prime factorization is not squarefree
split = nontrivial prime factorization, but squarefree
inert is the remaining case

$3$ is inert, thus, yes?

OK. This is a weird class of points in the codomain I don't have a geometric intuition for. What do these correspond to in topology? Hmm.

2:01 PM
go think
see ya

Nah, probably won't.

I gotta get more work done in what I am actively studying.
Which includes a bunch of exercises from Mike I am delaying for a couple days. :S
@Soham There's no direct analogy, because this simply cannot happen over algebraically closed fields. But I can't find any indirect analogies either.
Maybe someone smart (@Krijn?) can come up with something.

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