@0celo7 You didn't mention that...

@0celo7 What's surprising about that?

I don't get it.

Oh I see, it's the "glome". Yes, I have never seen that before.

hey everybody

@JasperLoy Awesome!

@BalarkaSen My course might get cancelled! :(

The prof thought two students was too few

@Krijn what course

@Huy Class Field Theory

your prof be mean

@Huy My prof be mean why?

much course cancel

little students few wow

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.

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

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

@Huy I can help you with some integrals, series, limits ...

"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

much dead

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

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.

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

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)

@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)

@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)

@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.

Consider $\DeclareMathOperator{\Z}{\Bbb Z}\Spec \Z[i]\to\Spec\Z$ given by the inclusion of $\Z$ into $\Z[i]$.

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?

@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

@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.

@BalarkaSen I asked my prof to talk about mollifiers, smoothings, etc.

@SohamChowdhury I think so.

then what the hell is going on there?

(I just asked on main)

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.

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)

$3$ 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?

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?

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.

go think

see ya

Nah, probably won't.

math.stanford.edu/~vakil/725/class10.pdf

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.

Hi guys

I have a Banach Spance $C_{b}(\mathbb{R})$ of bounded continuous real-valued functions on $\mathbb{R}$ with the supremum norm.

It's possible to find a closed bounded equi-continuous subset of $C_{b}(\mathbb{R})$ that is not compact?

I cannot see any example of that

It follows inmediately that if a subest $U$ is bounded and closed then $U$ is compact, so I don't see any sense to that question.. or am I wrong?

why does that follow

Hi @Huy

hi @Balarka

because if $X$ is compact then $X = \bigcup_{i=1}^{m} U_{i}$, then is clearly bounded and if we take the complement of X, that is, $\varline{X}$ we got $\emptyset$ which is strictily open in a bounded space, then $X$ is closed. Indeed, $X$ si compact. Right?

what

so you say if X is compact, indeed X is compact ???

@Huy I meant by definition

I don't follow

@Huy it's Heine-Borel Theorem

???

That's only for a subset of $\Bbb R^n$. You're working with subsets of $C_b(\Bbb R)$.

^

@Huy working on analysis...

??

I proved the sequential definition of continuity is equivalent to epsilon-delta, I proved closed subsets of compact spaces are compact, and that O(n) is comact

I like the annoying question multiple marks. Very useful.

I do them all the time when correcting students' homework

there are some not-so-interesting problems here

is very helpful for them indeed

also there's a typo

I don't know what it should be

which one

@0celo7 Easy-peasy.

@BalarkaSen what does $\lim_{x\to\infty}f(x)=\infty$ mean for higher dimensions

$\|f(x)\|$, he meant, of course.

and $x\to\infty$?

|x|

$\|x\|$, similarly.

@Huy do you have any ideas about this?

Bonus: show that such maps are equivalent to maps $f : \Bbb R^m \to \Bbb R^n$ which extend to one-point compactifications.

Not too hard, but that's how I think of properness.

I don't know what a one-point compactification is

hm, maybe I have a closer look later

cleaning kitchen atm

also I'm assuming it should be $f:R^m\to R^n$

yes

as in ii)

@0celo7 Just that $f$ extends to a map $S^m \to S^n$, where you identify $\Bbb R^k$ with $S^k$ minus the north pole.

Extends continuously, I mean.

Hey guys

Is it possible to simplify k(k+1) >2n more for finding approx value of k ?

@TheLittleNaruto: It's an inequality, so you are asking to approximate the smallest such k?

@TedShifrin Yes

Is n pretty big?

Yes

Consider It a very large number

So then k(k+1) is essentially k^2, and the answer would be sqrt(2n).

How would I expect that answer ? Can you explain a little ?

Good evening everybody.

hi @Owatch

evening Owatch

Something's odd, you don't show up in the chatroom users bar Ted.

But yeah, hey.

(Also TheLittleNar)