Mathematics - 2018-05-30 (page 1 of 3)

Kasmir Khaan

00:00

Let me read it

Faust

lol

i was snarking at you because you said an impossible number

GFauxPas

still Faust you were rude af

Faust

the number is at most 10

ic

GFauxPas

unless the question means that 10 have those 2 and not the third, specifically

Symposium

@GFauxPas maa.org/sites/default/files/pdf/upload_library/22/Ford/… he's doing what they're doing in 2.

Faust

00:01

i am sorry @KasmirKhaan i didnt mean to be rude the question was just pissing me off

Symposium

But that notation is too weird I wouldn't learn it from that page.

Kasmir Khaan

@Faust no problem, and i think you are right, question should not be stated thay way ><

GFauxPas

I just dont see where the numbers are coming from

Faust

the number 17 must be at least 19 for the question to make sense

i think

GFauxPas

it's true for 17 sufficiently large then

Symposium

00:04

lmao @ "thus, inspired..."

Faust

it works for 17 goes to 19

geocalc33

Group theory is the mathematical language of symmetry. For discrete symmetries we use Galois representations and for continuous symmetries we use lie group representations

GFauxPas

00:28

@Symposium I know the guy who put up the proof, he's a character, let me show you a bit more of his personality

proofwiki.org/wiki/Definition:Abuse_of_Notation most of the page is him

I wrote the differentials part

but its lul

Leaky Nun

> Is there an Emma Lehmer Lemma?

GFauxPas

yeah that's his too

Leaky Nun

Emma Lehmer

Emma Markovna Lehmer (née Trotskaia) (November 6, 1906 – May 7, 2007) was a mathematician known for her work on reciprocity laws in algebraic number theory. She preferred to deal with complex number fields and integers, rather than the more abstract aspects of the theory. == Biography == She was born in Samara, Russian Empire, but her father's job as a representative with a Russian sugar company moved the family to Harbin, China in 1910. Emma was tutored at home until the age of 14, when a school was opened locally. She managed to make her way to the USA for her higher education. At UC Berkeley...

Consider $R = k[X]$.

$m : R \to R \otimes_k R : X \mapsto X \otimes 1 + 1 \otimes X$

$i : R \to R : X \mapsto -X$

$e : R \to k : X \mapsto 1$

@loch is back!

$m(X) = X \otimes 1 + 1 \otimes X$

what is associativity supposed to look like

4 hours ago, by Leaky Nun

Now $L := \operatorname{Hom}(T_K, \operatorname{Spec}(K[X,X^{-1}]))$

$\begin{array}{rcl} \operatorname{Hom}_{K-GS}(T_K, \operatorname{Spec}(K[X,X^{-1}])) &\subseteq& \operatorname{Hom}_{K-S}(T_K, \operatorname{Spec}(K[X,X^{-1}])) \\ &=& \operatorname{Hom}_{K-A}(K[X,X^{-1}], \mathcal O(T_K)) \\ &=& \mathcal O(T_K)^\times \end{array}$

I suppose some of my words mean something

what the actual

$k \setminus \{0\}$ embeds into $\Bbb A^2$ by the rule $x \mapsto (x,x^{-1})$

the image is closed as it is the zero locus of $xy-1$

but in the same time, $k \setminus \{0\}$ is an open subset of $\Bbb A^1$

the complement of the zero locus of $x$

@EricSilva ola

Symposium

01:14

@GFauxPas Haha. What the hell @ his reply in the discussion section!

GFauxPas

which discussion section

he is grumpy in general I try not to take it personally

he's the biggest contributor to the wiki by far, he's very talented

Symposium

Click discussion on that page on the abuse of notation.

I don't think it's bad, it's just funny!

Very pompous!

GFauxPas

lol

Symposium

proofwiki.org/wiki/Definition_talk:Abuse_of_Notation

geocalc33

Can somebody help explain to me quotienting on the unit square?

GFauxPas

01:19

context?

what's the quotient? moebius strip etc

did you want an example?

Leaky Nun

@GFauxPas what is your background?

GFauxPas

Ashkenaz

Leaky Nun

I mean mathematical background

GFauxPas

I know i was joking

grad student at NYU

second year

Symposium

I thought for a minute I always wrongly assumed Lehmer (Lehmer's theorem of the degree of cosine or converse of FLT or primality test etc) was male, but it turns out there are multiples of them!

Leaky Nun

01:21

@GFauxPas specialization?

Elcyc

Are locally Lipschitz continuous functions dense in C(R) with respect to $L^\infty$ norm?

GFauxPas

my major is just "mathematics" at the moment

Leaky Nun

but you can't learn everything

GFauxPas

there are subjects that I like more than others

Leaky Nun

like?

GFauxPas

01:23

functional analysis, analysis, calculus

geocalc33

GFauxPas

are my favorite subjects

Leaky Nun

oh we belong to different fields :c

GFauxPas

what's yours?

geocalc what's your question, why it's a tube like that?

Leaky Nun

algebra

GFauxPas

01:24

ah

well let's start without the hole geocalc

Symposium

Didn't like functional analysis. I tried to learn it from Bollobas a while ago, but just got bored.

GFauxPas

we glue the left side and the right side of the square to each other, what do you get

Leaky Nun

to each their own

GFauxPas

I personally have cut out paper squares and played with them in situations like this, you can do the same

if you think it will help

geocalc33

we get a cylinder

GFauxPas

01:25

no

Symposium

@LeakyNun What type of algebra?

GFauxPas

not without the hole

Leaky Nun

@Symposium abstract algebra

geocalc33

we get a tube

we get a cylinder with the caps not there

GFauxPas

I'm thinking of it like an ice cream cone

im not considering the hole

geocalc33

01:26

okay

GFauxPas

do you see why it's an ice cream cone?

go ahead and actually play with a paper square if you need to

geocalc33

so basically you can equate the top bar of the square

so taht shrinks to one point

GFauxPas

when you equate two sides theyre glued together

topologists literally call it gluing

geocalc33

okay

GFauxPas

so if you glue two adjacent sides of a square together, do you see why we get an ice cream cone?

Symposium

01:29

@LeakyNun Who teaches the ANT course in third/fourth year at Imperial now?

geocalc33

Leaky Nun

@Symposium Ambrus Pal

geocalc33

like that??

GFauxPas

oh I was assuming they were gluing the sides together with the dashes

I wasnt reading the heading I was looking at the picture sorry

then its wrong, it's not a cylinder if it has a hole in it...

geocalc33

so by equating certain points to eachother you can make some pretty cool shapes

torus, mobius strip, klein bottle, etc.

GFauxPas

01:36

its how you define things like the torus

it has geometric intuition but you can express it purely algebraically

geocalc33

Leaky Nun

resists urge to talk about torus in algebraic geometry

geocalc33

So what if i had that thing

and it's in the unit square

are the only things you could equate

the corners?

GFauxPas

lol leaky

well its one way to define them, im sure there are many

you mean, what happens if you draw that on a square and pinch all the corners together?

geocalc33

I'm just trying to play around with different curves in the unit square and equate them in different ways to see what shape i get

GFauxPas

01:40

is that what you're asking?

geocalc33

It seems like you can only equate points on the outer edges

GFauxPas

why do you say that?

Elcyc

surely you can equate more than that

GFauxPas

let's say you equate every single point on the border. what do you get?

they're all the same now

Leaky Nun

Let $S \subseteq k[X_1,\cdots,X_n] = A$. Then, for $R$ a $k$-algebra: $$S(R) := \{ v \in R^n \mid \forall f \in S : f(v) = 0 \} = \operatorname{Hom}_{k-Alg}(A/S, R) = \operatorname{Hom}_{k-Sch}(\operatorname{Spec}(R), \operatorname{Spec}(A/S))$$

geocalc33

01:46

a point?

GFauxPas

what happened to the middle part

Leaky Nun

Assuming that $S$ is an ideal, we have the exact sequence $0 \to S \to A \to A/S \to 0$, which gives rise to an exact sequence $0 \to \operatorname{Spec}(A/S) \to \operatorname{Spec}(A) \to \operatorname{Spec}(S) \to 0$

geocalc33

I'm not sure im trying visualise this

Leaky Nun

so $\operatorname{Spec}(A/S)$ is a subset of $\operatorname{Spec}(A) =: \Bbb A^n$

GFauxPas

think of glue

Leaky Nun

01:48

but anyway, $S$ is now a functor from k-Alg to Set

@GFauxPas do you learn algebra?

geocalc33

a sphere?

GFauxPas

yup, very good

you ever make a mobius strip out of paper?

geocalc33

yeah

GFauxPas

try to describe how you did it in words to me, and we'll try to translate that into algebra

if you want

geocalc33

we take the unit square, and fold it along the diagonal crease of y=x. Then cut it straight across to the bottom right corner. then glue the pieces together but we have to twist it

something like that

GFauxPas

02:00

you're thinking too hard lol

let me try

you twist opposite sides and glue them together

sound about right? ;)

geocalc33

yeah :)

GFauxPas

okay

so we have to think about what it means to twist one side and not the other

have any ideas?

think linear algebra

how do we flip the orientation of a line segment

geocalc33

we have a matrix that performs a rotation

GFauxPas

you dont need it if you're just flipping the orientation, it's overkill

we want to flip which way an arrow goes

I mean, you can

will give you the right answer

geocalc33

a negative sign

GFauxPas

02:04

right

so

a mobius strip is described by

$I \times I / x \sim -x$

actually thats not a good way to write it

$I^2 / (x,0) \sim (-x,1)$

that's better

on the "zero" side, we're the same as the negative of the "one" side

$I^2 = [0..1]^2$

make sense?

geocalc33

yeah

GFauxPas

okay

now lets do one that's common in topology but you can't actually create in the 3rd dimension

it's a 3d object but you have to make it in 4d

first let's make a torus, which we can do without complication

any ideas of how to make a torus? It's harder than the examples we've done so far

geocalc33

yeah glue the left side and right side and then stretch the cylinder thing so that the circles are glued together

GFauxPas

nice!

though I personally glue the top and bottom one and do it the other way, but that doesn't matter

okay so opposite sides are glued together

and you want to glue the two circles at the end together

but instead of gluing them together in a straightforward way

we'll negate it the same way we did for a mobius strip

so one circle will be oriented clockwise and the other will be oriented counterclockwise

and then we glue them together

this twist-and-glue needs 4 dimensions to work

geocalc33

okay

GFauxPas

02:13

this is called the Klein bottle

like a mobius strip, it has the property that if you walk all the way around, you'll come back to the original position upside-down

:)

that's what "non-orientable" means in this comic

smbc-comics.com/comic/2012-10-13

geocalc33

so how would you quotient this shape? like what would you equate to what?

posting image momentarily

pretend it's rotated so that it's like a square

GFauxPas

idk what that is

geocalc33

it's some lines on a square

GFauxPas

I dunno what you can do with it

geocalc33

just like equating successive diagonal lines going up the square yields a helical curve that winds around the cylinder thing

I'm wondering what sets of points to equate for this shape

GFauxPas

02:25

I mean in topology you generally are interested in loops and the question to ask is "I did this, is the loop still a loop"

geocalc33

I guess you could do lots of things like make it into a torus

and see how the curves are stretched around the torus

GFauxPas

just keep in mind that anything you're analyzing is going to be "up to stretching without tearing" so don't get hung up on angles or lengths

geocalc33

yeah :)

so what could you analyze after stretching the curves onto the torus

GFauxPas

02:42

I'm not good at topology i just happen to be here :P

i can talk about loops, but not sure how to deal with lattices like the thing you posted

loops are very important because you can give them a group structure

geocalc33

like a loop meaning the initial point is glued to the final point?

GFauxPas

ya

given a parameterization $T$ on unit time, it means $T(0) = T(1)$

geocalc33

I remember hearing that closed curves were more interesting than open ones from a prof

GFauxPas

well as I said, you can give loops a group structure

Al t.

Hello, Could someone tell me why the terms $c_1c_2 \Delta w_1 \cdot \Delta w_2 + c_2 c_1 \Delta w_2 \cdot \Delta w_1 + ... $ were added here

$(\sum c_k \Delta w_k, \sum c_j \Delta w_j) = (c_1 \Delta w_1 + ...+ c_n \Delta w_n, c_1 \Delta w_1 + ...+ c_n \Delta w_n) \\ = \iiint c_1^2 \Delta w_1 \cdot \Delta w_1 + ... + c_n^2 \Delta w_n \cdot \Delta w_n + c_1c_2 \Delta w_1 \cdot \Delta w_2 + c_2 c_1 \Delta w_2 \cdot \Delta w_1 + ... d \mathbf{x} = \\ \sum (\Delta w_j, \Delta w_j) c_j^2 + 2 \sum \sum_{j\neq k} (\Delta w_k, \Delta w_j) c_kc_j$?

Isn't just a dot product?

geocalc33

02:48

And you can't give non-loops a group structure?

GFauxPas

well I've never tried, let's google

no you cant

you need to fix a point and parameterize the loops to end and start at the point, then you can multiply them

but otherwise they wont obey group laws

how would you multiply them if they werent loops, theyd have to start where the other ended and that's very restrictive

the requirement that you fix a point for the loops to have in common isnt that big a deal because in general the choice of the point doesn't affect anything

geocalc33

you could glue the square with the lattice onto a torus and then all four points would be at the same spot on the torus and then you could assign a group structure

GFauxPas

I don't understand the lattice so no comment

geocalc33

i mean like the four corner points would end up all in the same spot

four corner points of the square

GFauxPas

I dunno, I guess check the group axioms on what you're trying to do

loch

04:05

Assuming obvious compatibility relations, here you have a morphism of functors $Hom(-,G) \rightarrow Hom(-,\mathbb{G}_m)$ (on the category of affine schemes over $k$). Now let $R = G$, and consider the image of the identity. This recovers your map $G\rightarrow \mathbb{G}_m$.

(This is basically Yoneda)

@LeakyNun This is how you show how given an affine variety, the locus $\{f\ne 0\}$ is also an affine variety (These opens are sometimes called principal opens/distinguished opens and they are a basis for your Zariski top - they are important!)

@LeakyNun you have to careful by what you mean by exact. Your first exact sequence is exact as a sequence of $A$-modules. I don't think your second exact sequence makes sense (even if, say, you identify this with the opposite category of fin gen. $k-$algebras - this is not an abelian category!)

Eulb

04:24

@BalarkaSen sup

Daminark

Hey there nerds

skull

yo wazzup?

Daminark

04:41

Not much, how about you?

loch

Hey

Eulb

did anyone listen to any cool albums this week?

link

Daminark

Here's a link

Eulb

04:59

@Daminark think*

Alex

@Daminark What number theory are you learning atm?

And from where?

skull

we apparently have a mod from Warwick now...

perhaps, that's what keeping Alec away

};-)

Zee

Yo skull

Sup my man

skull

yo

Zee

How are you ?

Eulb

05:09

isn't skull a girl?

Zee

Raiders fans are all guys

Eulb

(assuming that statement is correct) I was lied to then. (but it's not apparently) so nvm.

unsure how true that is^

Zee

Lol it’s not

Eulb

reverse the order of those messages btw.

lol

actually

skull

raiderettes exist

Daminark

05:11

@Alex I'm working with a prof of mine at the moment, he wants me to learn p-adics next since he wants to prove Kronecker-Weber locally

Zee

Are you one ? :):):):)

Eulb

fixed

Zee

I just kicked out of school :)

Daminark

Though I feel I should review some things from last session a bit better, in particular one of his proofs of quadratic reciprocity didn't quite stick. I think it'll help also if I go over the proof that cyclotomic polynomials are irreducible, since the two were similar in spirit

skull

for what?

Zee

05:13

Bad grades

Silent

@anon, is it true: Let $\{a_n\}$ be real sequence. If $|a_{n+1}-a_n|<\frac1{3^n}$ for all $n$, then $a_n$ convergent.

Alex

@Daminark Reading Neukirch then?

Zee

It’s a bit freeing , now I could do research

Eulb

wait what

Zee

^

Eulb

05:16

grad school?

Daminark

I'm looking through Neukirch to supplement our sessions, along with the book my prof recommended, Ireland & Rosen

Eulb

you can't get kicked from undergrad because you're just a number to them that supplies monet$$$.

Zee

Ya grad

But am not even a PhD student

Eulb

what are you

Zee

It don’t really bother me , I never was a good student , I’ll show these bastards that I don’t need them to do research

Eulb

05:18

that's an unhealthy state of mind.

Zee

Why ?

Eulb

because it's right at the edge of a downward spiral for your academic career.

if you don't like courses and feel you're misrepresented you can ask to be accommodated.

Zee

That career is over , I never cared for it much anyway , I just wanna Get some theorems

Eulb

that's not how it works.

Alex

@Daminark Are local fields even defined in that text?

Zee

05:21

Why not ?

Eulb

how will you sustain yourself financially if you're not paid to do mathematics?

Zee

That the issue , but I’ll find the time , I’ll kill my social life

skull

can't you appeal their decision?

Daminark

Looking through contents, yeah it's in chapter 2

Zee

Maybe sell drugs make money or foot prison in which case I can still do math !

Eulb

05:22

@Zee where's the $$$$$ coming from, my dude?

yeah you're already spiralling.

good luck.

Alex

@Daminark I mean Ireland

Eulb

@everyone someone talk some sense into this guy

Zee

I can appeal, I might but am sick of them

Eulb

you're either being super edgy or have fully lost it.

Zee

Come on now , I don’t think what am saying is irrational

Alex

05:24

@Eulb Zee is a known troll

skull

let your wounds heal, my brother

Zee

It may be weird but it makes sense

Eulb

@Alex oh

Zee

@Alex now am sick of you too

skull

1 min ago, by skull

let your wounds heal, my brother

Eulb

05:25

@Zee proof?

Zee

What’s the point ?

Alex

@Daminark Oh I see, I think I misread. You are supplementing your sessions with the pair of books, not supplementing Ireland and your meetings with Neukirch :P

Zee

It don’t matter , I appreciate you sympathy @Eulb but it really don’t matter

Eulb

prove that $\mathrm{what you're saying} \in \mathbb Q$.

Zee

Math is much more important than a bunch of buerocrats

skull

05:26

true

Daminark

Yeah tru

Eulb

truest

Zee

You only need time to do mathematics nothing else , time is not easy to get but you can get it

I’ll get married and have a social life when I publish my first paper

skull

be true to thyself

Eulb

@Zee that's not how it works.

Zee

05:28

^

Why not ?

Why do things have to be complicated ?

Eulb

you don't simply "get married."

Zee

Well it’s not an equality but an inequality

I can only get married on or after I publish

Maybe I’ll find me a rich girl wholl support me :)

Eulb

mathematics is a collaborative effort.

skull

sugar mamas are tough to find

Zee

That don’t mean you have to physically be around people , reading research papers is a collaborative effort

skull

05:31

too many girls are looking for sugar daddy's

Zee

True skull , am banking on my sense of humor

(Don’t bank too much pal )

skull

:-D

Eulb

@Zee I'm not going to discuss this with you anymore because you don't seem to want to come to your senses. (or because you're trolling)

Zee

What do you want me to say ?? Am faced with reality and that is how I deal with it , do you have a better way ?

Silent

@Daminark, is it true that : There exists a pair of disjoint subsets of $\Bbb Q$ such that both of them are dense in $\Bbb R$?

Zee

05:36

Just take the even index in any sequence it still has the same limit silent

skull

Yes, actually, there are better ways than venting on the internet

:-)

Zee

Ok

Am not venting am just talking ...

skull

I know, no offence intended

Zee

Non taken , o just wanted to share this thing , am not sure why though

Silent

@Zee You mean Dyadic? Let Dyadic rationals form set $\Bbb D$, how do we know $\Bbb Q-\Bbb D$ dense in $\Bbb R$?

Zee

05:39

Start listing the fractions and skip every other number

Then any limit , you can still get

As a sequence of those fractions

Silent

@Zee Is it true that $\{\frac{a}{m^n}: a\in \Bbb Z, n\in \Bbb N\}$ is dense in $\Bbb R$, where $m$ is fixed natural number?

Zee

Sorry , can you write it without Latex

Silent

well, it will be very hard to read!

trying

Zee

I think you can get countable many such disjoint sets that are dense in the reals actually

Silent

Is it true that the set of a/m^n, where a is any integer and n is natural number, is dense in reals, where m is fixed natural number?

@Zee so, {a/p^n} for any prime p will be dense subset of R right?

Thanks for your help!

Zee

05:46

Wait am not sure

Silent

ok

Zee

Am sure about what I said but idk about what you just said

Silent

@Zee ok so there are countably many disjoint subsets of Q which are all dense in R, is that what you are sure about?

Zee

ya

Silent

thank u

Zee

05:49

You are most welcome silent

Transcript for

$Mathematics$ Mathematics